SOME RESULTS ON THE ADMISSIBLE REPRESENTATIONS OF NON-CONNECTED REDUCTIVE p-ADIC GROUPS

DAVID GOLDBERG AND REBECCA A. HERB

Abstract. We examine the theory of induced representations for non-connected reductive p-adic groups for which G/G^{0} is abelian. We first examine the structure of those representations of the form Ind_{P^{0}}^{G}(σ), where P^{0} is a parabolic subgroup of G^{0} and σ is a discrete series representation of the Levi component of P^{0}. Here we develop a theory of R–groups, extending the theory in the connected case. We then prove some general results in the theory of representations of non-connected p-adic groups whose component group is abelian. We define the notion of cuspidal parabolic for G in order to give a context for this discussion. Intertwining operators for the nonconnected case are examined and the notions of supercuspidal and discrete series are defined. Finally, we examine parabolic induction from a cuspidal parabolic subgroup of G. Here we also develop a theory of R–groups, and show that these groups parameterize the induced representations in a manner that is consistent with the connected case and with the first set of results as well.

1. Introduction

The theory of induced representations plays a fundamental role within representation theory in general. Within the theory of admissible representations of connected reductive algebraic groups over local fields, parabolic induction is used to complete classification theories, once certain families of representations are understood [3,8,9, 10]. The theory of admissible representations on non-connected reductive groups over nonarchimedean local fields has been addressed in part in [1,4,6,11], among other places. We will study certain aspects of parabolic induction for disconnected groups whose component group is abelian.

Let F be a locally compact, non-discrete, nonarchimedean field of characteristic zero. Let G be a (not necessarily connected) reductive F -group. Thus G is the set of F -rational points of a reductive algebraic group defined over F. Let G^{0} be the connected component of the identity in G. We assume that G/G^{0} is finite and abelian.

Our goal is to address three major points. The first is an extension of the results of [6] to the case at hand. This entails a study of induction from a parabolic subgroup of G^{∘} to G. In particular, suppose that P^{∘} = M^{∘}N is a parabolic subgroup of G^{∘},

Goldberg was partially supported by National Science Foundation Fellowship DMS-9206246 and National Science Foundation Career Grant DMS-9501868. Research at MSRI supported in part by National Science Foundation Grant DMS-9022140. Herb was partially supported by National Science Foundation Grant DMS-9400797.

and let σ_{0} be an irreducible discrete series representation of M^{∘}. We are interested in the structure of π_{0} = Ind_{P^{∘}}^{G}(σ_{0}). In [1] Arthur suggests a construction, in terms of the conjectural local Langlands parameterization, of a finite group whose representation theory should describe the structure of π_{0}, when G/G^{∘} is cyclic. In [6] the case where G/G^{∘} is of prime order is studied, and there is a construction, on the group side, of a finite group R_{G}(σ_{0}) which (along with an appropriate 2–cocycle) parameterizes the components of π_{0}. It is also shown there that R_{G}(σ_{0}) must be isomorphic to Arthur’s group R_{ψ,σ_{0}}, if the latter exists. One cannot confirm the existence of R_{ψ,σ_{0}} without proofs of both the local Langlands conjecture and Shelstad’s conjecture [12] that R_{ψ_{0},σ_{0}} is isomorphic to R_{G^{∘}}(σ_{0}). (See [1] for the precise definitions of R_{ψ,σ_{0}} and R_{ψ_{0},σ_{0}}.) Here, by extending the definition of the standard intertwining operators (cf Section 4) we show we can construct a group R_{G}(σ_{0}) in a manner analogous to [6], and show that it has the correct parameterization properties (cf Theorems 4.16 and 4.17). An argument, similar to the one given in [6] shows that if G/G^{∘} is cyclic, then R_{G}(σ_{0}) must be isomorphic to R_{ψ,σ_{0}}, assuming the latter exists (cf Remark 4.18).

The second collection of results is an extension of some standard results in admissible representation theory to the disconnected group G. In order to develop a theory consistent with the theory for connected groups, one needs to determine an appropriate definition of parabolic subgroup. There are several definitions in the literature already, yet they do not always agree. We use a definition of parabolic subgroup which is well suited to our purposes. Among the parabolic subgroups of G we single out a collection of parabolic subgroups which we call cuspidal. They have the property that they support discrete series and supercuspidal representations and can be described as follows. Let P^{0} be a parabolic subgroup of G^{0} with Levi decomposition M^{0}N and let A be the split component of M^{0}. Let M = C_{G}(A). Then P = MN is a cuspidal parabolic subgroup of G lying over P^{0}. We also say in this case that M is a cuspidal Levi subgroup of G. Thus cuspidal parabolic subgroups of G are in one to one correspondence with parabolic subgroups of G^{0}.

Using our definitions we can prove the following. Let M be a Levi subgroup of G and let M^{0} = M ∩ G^{0}.

Lemma 1.1. (i) If M is not cuspidal, then M has no supercuspidal representations, i.e., admissible representations with matrix coefficients which are compactly supported modulo the center of M and have zero constant term along the nil radical of any proper parabolic subgroup of M. (ii) If M is cuspidal and π is an irreducible admissible representation of M, then π is supercuspidal if and only if the restriction of π to M^{0} is supercuspidal. (iii) If M is not cuspidal, then M has no discrete series representations, ie. unitary representations with matrix coefficients which are square-integrable modulo the center of M.

ADMISSIBLE REPRESENTATIONS OF NON-CONNECTED REDUCTIVE p-ADIC GROUPS 3

(iv) If M is cuspidal and π is an irreducible unitary representation of M, then π is discrete series if and only if the restriction of π to M^{0} is discrete series.

Using Lemma 1.1 it is easy to extend the following theorem from the connected case to our class of disconnected groups.

Theorem 1.2. Let π be an irreducible admissible (respectively tempered) representation of G. Then there are a cuspidal parabolic subgroup P = MN of G and an irreducible supercuspidal (respectively discrete series) representation σ of M such that π is a subrepresentation of Ind_{P}^{G}(σ).

Let P_{1} = M_{1}N_{1} and P_{2} = M_{2}N_{2} be cuspidal parabolic subgroups and let σ_{i} be irreducible representations of M_{i}, i = 1, 2, which are either both supercuspidal or both discrete series. By studying the orbits for the action of P_{1} ×P_{2} on G, we are able to extend the proof for the connected case to our situation and obtain the following theorem.

Theorem 1.3. Let P_{1} = M_{1}N_{1}, P_{2} = M_{2}N_{2}, σ_{1}, σ_{2} be as above. Then if π_{1} = Ind_{P_{1}}^{G}(σ_{1}) and π_{2} = Ind_{P_{2}}^{G}(σ_{2}) have a nontrivial intertwining, then there is y ∈ G so that

M_{2} = yM_{1}y^{−1} and σ_{2} ≃ yσ_{2}y^{−1}.

The third question of study is the structure of π = Ind_{P}^{G}(σ) when P = MN is a cuspidal parabolic subgroup of G and σ is a discrete series representation of M. We show that, as in the connected case, the components of π are naturally parameterized using a finite group R. As in the connected case we first describe a collection of standard intertwining operators R(w, σ) which are naturally indexed by w ∈ W_{G}(σ) = N_{G}(σ)/M, where

N_{G}(σ) = {x ∈ N_{G}(M) : σ^{x} ≃ σ}.

We prove that there is a cocycle η so that

R(w_{1}w_{2}, σ) = η(w_{1}, w_{2})R(w_{1}, σ)R(w_{2}, σ), w_{1}, w_{2} ∈ W_{G}(σ).

Let σ_{0} be an irreducible component of the restriction of σ to M^{0}, and P^{0} = M^{0}N = P ∩ G^{0}. Then σ ⊂ Ind_{M^{0}}^{M}(σ_{0}) so that

π = Ind_{P}^{G}(σ) ⊂ Ind_{P}^{G}(Ind_{M^{0}}^{M}(σ_{0})) ≃ Ind_{P^{0}}^{G}(σ_{0}).

Using the intertwining operators and R-group theory developed earlier for the representation Ind_{P^{0}}^{G}(σ_{0}), we can prove the following results. First, the collection

{R(w, σ), w ∈ W_{G}(σ)}

spans the commuting algebra of π. Second, let Φ_{1}^{+} be the set of positive restricted roots for which the rank one Plancherel measures of σ_{0} are zero and let W (Φ_{1}) be the group generated by the reflections corresponding to the roots in Φ_{1}. Then W (Φ_{1}) is naturally embedded as a normal subgroup of W_{G}(σ) and W_{G}(σ) is the semidirect product of W (Φ_{1}) and the group

R_{σ} = {w ∈ W_{G}(σ) : wα > 0 for all α ∈ Φ_{1}^{+}}.

Finally, R(w, σ) is scalar if w ∈ W (Φ_{1}). This proves that the operators R(w, σ), w ∈ R_{σ}, span the intertwining algebra. But we can compute the dimension of the space of intertwining operators for Ind_{P}^{G}(σ), again by comparison with that of Ind_{P^{0}}^{G}(σ_{0}), and we find that it is equal to [R_{σ}]. Thus we have the following theorem.

Theorem 1.4. The R(w, σ), w ∈ R_{σ}, form a basis for the algebra of intertwining operators of Ind_{P}^{G}(σ).

Just as in the connected case, we show that there are a finite central extension

1 → Z_{σ} → R̃_{σ} → R_{σ} → 1

over which η splits and a character χ_{σ} of Z_{σ} so that the irreducible constituents of Ind_{P}^{G}(σ) are naturally parameterized by the irreducible representations of R̃_{σ} with Z_{σ}-central character χ_{σ}.

Finally, we give a few examples which point out some of the subtleties involved in working with disconnected groups. For instance, we show that if we do not restrict ourselves to cuspidal parabolic subgroups, then the standard disjointness theorem for induced representations fails. Examples such as these show why one must restrict to induction from cuspidal parabolic subgroups in order to develop a theory which is consistent with that for connected groups.

Many interesting problems involving disconnected groups remain. For example, the question of a Langlands classification is still unresolved, and some of the results here on intertwining operators may help in this direction. One also hopes to remove the condition that G/G^{∘} is abelian, and extend all the results herein to that case. Problems such as these we leave to further consideration.

The organization of the paper is as follows. In §2 we give the definition of parabolic subgroup and prove Lemma 1.1 and Theorem 1.2. The proof of Theorem 1.3 is in §3. The results on induction from a parabolic subgroup of G^{0} to G are in §4, and the results on induction from a parabolic subgroup of G to G, including Theorem 1.4, are in §5. Finally, §6 contains examples that show what can go wrong when we induce from parabolic subgroups of G which are not cuspidal.

The first named author would like to thank the Mathematical Sciences Research Institute in Berkeley, California, for the pleasant and rich atmosphere in which some of the work herein was completed.

2. Basic Definitions

Let F be a locally compact, non-discrete, nonarchimedean field of characteristic zero. Let G be a (not necessarily connected) reductive F -group. Thus G is the set of F -rational points of a reductive algebraic group over F. Let G^{0} be the connected component of the identity in G. We assume that G/G^{0} is finite and abelian.

The split component of G is defined to be the maximal F -split torus lying in the center of G. Let A be any F -split torus in G and let M = C_{G}(A). Then M is a reductive F -group. Now A is called a special torus of G if A is the split component of M. (Of course A is an F -split torus lying in the center of M. The only question is whether or not A is maximal with respect to this property.)

Lemma 2.1. Let A be a special torus of G^{0}. Then A is a special torus of G.

Proof: Let M = C_{G}(A) and M^{0} = C_{G^{0}}(A) = M ∩ G^{0}. Write Z(M) and Z(M^{0}) for the centers of M and M^{0} respectively. Now A is the maximal F -split torus lying in Z(M^{0}) and A ⊂ Z(M). Suppose A′ is the maximal F -split torus lying in Z(M). Then A ⊂ A′. But A′ is a torus, so it is connected. Hence A′ ⊂ Z(M)∩M^{0} ⊂ Z(M^{0}). Thus A′ ⊂ A and so A′ = A is the split component of M.

Remark 2.2. The converse of Lemma 2.1 is not true. For example, let G = O(2) = SO(2) ∪ wSO(2) where SO(2) ≃ F^{×} is the group of 2 × 2 matrices

d(a) = ( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) , a ∈ F^{×}, and w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

satisfies wd(a)w^{−1} = d(a^{−1}), a ∈ F^{×}. Let A = {d(1)}. Then M = C_{G}(A) = G and Z(M) = {±d(1)}. Thus A is a special torus of G. However M^{0} = C_{G^{0}}(A) = G^{0} and Z(M^{0}) = G^{0} is an F -split torus. Hence A is not the maximal F -split torus in Z(M^{0}) and so is not special in G^{0}.

If G is connected, then A is a special torus of G by the above definition if and only if A is the split component of a Levi component M of a parabolic subgroup of G. We will define parabolic subgroups in the non-connected case so that we have this property in the non-connected case also.

Let A be a special torus of G and let M = C_{G}(A). Then M is called a Levi subgroup of G. The Lie algebra L(G) can be decomposed into root spaces with respect to the roots Φ of L(A):

L(G) = L(G)_{0} ⊕ ∑_{α∈Φ} L(G)_{α}

where L(G)_{0} is the Lie algebra of M. Let Φ^{+} be a choice of positive roots, and let N be the connected subgroup of G corresponding to ∑_{α∈Φ^{+}} L(G)_{α}. Since elements of M centralize A and L(A), they preserve the root spaces with respect to L(A). Thus M normalizes N. Now P = MN is called a parabolic subgroup of G and (P, A) is called a p-pair of G. The following lemma is an immediate consequence of this definition and Lemma 2.1.

Lemma 2.3. Let P^{0} = M^{0}N be a parabolic subgroup of G^{0} and let A be the split component of M^{0}. Let M = C_{G}(A). Then P = MN is a parabolic subgroup of G and P ∩ G^{0} = P^{0}.

Lemma 2.4. Let P be a parabolic subgroup of G. Then P^{0} = P ∩ G^{0} is a parabolic subgroup of G^{0}.

Proof: Let P = MN be a parabolic subgroup of G and let A be the split component of M. Let M^{0} = C_{G^{0}}(A) = M ∩ G^{0}. Let A_{1} be the split component of M^{0}. Then A ⊂ A_{1} so that C_{G^{0}}(A_{1}) ⊂ C_{G^{0}}(A) = M^{0}. But A_{1} is in the center of M^{0}, so that M^{0} ⊂ C_{G^{0}}(A_{1}). Thus C_{G^{0}}(A_{1}) = M^{0} so that A_{1} is a special torus in G^{0} and M^{0} is a Levi subgroup of G^{0}. Let Φ and Φ_{1} denote the sets of roots of L(A) and L(A_{1}) respectively. For each α_{1} ∈ Φ_{1}, the restriction rα_{1} of α_{1} to L(A) is non-zero since C_{G^{0}}(A_{1}) = C_{G^{0}}(A) = M^{0}. Let Φ^{+} be the set of positive roots used to define N. Then Φ_{1}^{+} = {α_{1} ∈ Φ_{1} : rα_{1} ∈ Φ^{+}} is a set of positive roots for Φ_{1} and

∑_{α∈Φ^{+}} L(G)_{α} = ∑_{α_{1}∈Φ_{1}^{+}} L(G)_{α_{1}}.

Thus P^{0} = M^{0}N is a parabolic subgroup of G^{0}.

We say the parabolic subgroup P of G lies over the parabolic subgroup P^{0} of G^{0} if P^{0} = P ∩ G^{0}. We will also say the Levi subgroup M of G lies over the Levi subgroup M^{0} of G^{0} if M^{0} = M ∩ G^{0}. Lemma 2.4 and its proof show that every parabolic (resp. Levi) subgroup of G lies over a parabolic (resp. Levi) subgroup of G^{0}.

Remark 2.5. There can be more than one parabolic subgroup P of G lying over a parabolic subgroup P^{0} of G^{0}. For example, define G = O(2) and G^{0} = SO(2) as in Remark 2.2. Then A = {d(1)} and A_{0} = SO(2) are special vector subgroups of G corresponding to parabolic subgroups P = O(2) and P_{0} = SO(2) respectively. Both lie over the unique parabolic subgroup SO(2) of G^{0}.

Lemma 2.6. Let P^{0} = M^{0}N be a parabolic subgroup of G^{0} and let A be the split component of M^{0}. Let M = C_{G}(A) and let P = MN. Then if P_{1} is any parabolic subgroup of G lying over P^{0} we have P ⊂ P_{1}. Further, M is the unique Levi subgroup lying over M^{0} such that the split component of M is equal to A.

Proof: Write P_{1} = M_{1}N where M_{1} lies over M^{0}. Let A_{1} be the split component of M_{1}. Then A_{1} ⊂ A so that M = C_{G}(A) ⊂ C_{G}(A_{1}) = M_{1}. Clearly M_{1} = M if and only if A_{1} = A.

Remark 2.7. Lemma 2.6 shows that there is a unique smallest parabolic subgroup P of G lying over P^{0}. Although it is defined using a Levi decomposition P^{0} = M^{0}N of P^{0}, it is independent of the Levi decomposition. Recall that if M_{1}^{0} and M_{2}^{0} are two Levi components of P^{0} with split components A_{1} and A_{2} respectively, then there is n ∈ N such that A_{2} = nA_{1}n^{−1} and M_{2}^{0} = nM_{1}^{0}n^{−1}. Now if M_{i} = C_{G}(A_{i}), i = 1, 2, we have M_{2} = C_{G}(A_{2}) = nC_{G}(A_{1})n^{−1} = nM_{1}n^{−1}, and M_{2}N = nM_{1}n^{−1}N = M_{1}N since M_{1} normalizes N.

Let Z be the split component of G. We let C_{c}^{∞}(G, Z) denote the space of all smooth complex-valued functions on G which are compactly supported modulo Z. We say f ∈ C_{c}^{∞}(G, Z) is a cusp form if for every proper parabolic subgroup P = MN of G,

∫_{N} f(xn)dn = 0 ∀x ∈ G.

Let ^{0}A(G) denote the set of cusp forms on G. We say G is cuspidal if ^{0}A(G) ≠ {0}. We know that every connected G is cuspidal.

Lemma 2.8. G is cuspidal if and only if the split component of G is equal to the split component of G^{0}. Moreover, if G is cuspidal, then a subgroup N of G is the nilradical of a proper parabolic subgroup of G if and only if N is the nilradical of a proper parabolic subgroup of G^{0}. If G is not cuspidal, then G has a proper parabolic subgroup G_{1} with nilradical N_{1} = {1}.

Proof: First suppose that G and G^{0} have the same split component Z. Let f ≠ 0 ∈ 0A(G^{0}). Define F : G → C by F (x) = f(x), x ∈ G^{0}, and F (x) = 0, x ∉ G^{0}. Then F ∈ C_{c}^{∞}(G, Z) and is non-zero. Let P = MN be any proper parabolic subgroup of G. Then N ⊂ G^{0}, so for all n ∈ N, x ∈ G, xn ∈ G^{0} if and only if x ∈ G^{0}. Thus for x ∉ G^{0},

∫_{N} F (xn)dn = 0

while for x ∈ G^{0},

∫_{N} F (xn)dn = ∫_{N} f(xn)dn.

Now P^{0} = P ∩ G^{0} = M^{0}N is a parabolic subgroup of G^{0}. Suppose that P^{0} = G^{0}. Then P lies over G^{0} so that by Lemma 2.6, G ⊂ P. This contradicts the fact that P is a proper parabolic subgroup of G. Thus P^{0} = M^{0}N is a proper parabolic subgroup of G^{0}. Since f is a cusp form for G^{0} we have

∫_{N} f(xn)dn = 0 ∀x ∈ G^{0}.

Thus F is a non-zero cusp form for G.

The above argument also showed that if P = MN is a proper parabolic subgroup of G, then P^{0} = M^{0}N is a proper parabolic subgroup of G^{0}. Conversely, if P^{0} = M^{0}N is a proper parabolic subgroup of G^{0} and P = MN is any parabolic subgroup of G lying over P^{0}, then clearly P ≠ G.

Conversely, suppose that G and G^{0} do not have the same split component. Let Z be the split component of G^{0} and define G_{1} = C_{G}(Z). By Lemma 2.6, G_{1} is a proper parabolic subgroup of G. Further since G_{1} lies over G^{0} its nilradical is N_{1} = {1}. Now if F is any cusp form on G and x ∈ G, we have

F (x) = ∫_{N_{1}} F (xn)dn = 0.

Thus G has no non-zero cusp forms and so is not cuspidal.

Example 2.9. Let G = O(2) as in Remarks 2.2 and 2.5. Then SO(2) is a cuspidal parabolic subgroup of G and O(2) is not cuspidal.

We can sum up the proceeding lemmas in the following proposition.

Proposition 2.10. Let P^{0} = M^{0}N be a parabolic subgroup of G^{0}. Then there is a unique cuspidal parabolic subgroup P = MN of G lying over P^{0}. It is contained in every parabolic subgroup of G lying over P^{0}, and is defined by M = C_{G}(A) where A is the split component of M^{0}.

Now that we have parabolic subgroups of G, we want to study parabolic induction of representations. Many of the most basic notions of representation theory are defined in [13, chapter 1] for any totally disconnected group. In particular, admissible representations of G are defined and the following is an easy consequence of the definition.

Lemma 2.11. Let Π be a representation of G. Then Π is admissible if and only if Π|_{G^{0}}, the restriction of Π to G^{0}, is admissible.

Further, the results of Gelbart and Knapp regarding induction and restriction between a totally disconnected group G and an open normal subgroup H with G/H finite abelian can be applied to G and G^{0}. If π is any admissible representation of G^{0} on V, we will let Ind_{G^{0}}^{G}(π) denote the representation of G by left translations on H = {f : G → V : f(gg_{0}) = π(g_{0})^{−1}f(g), ∀g ∈ G, g_{0} ∈ G^{0}}.

Lemma 2.12. (Gelbart-Knapp [5]) Let Π be an irreducible admissible representation of G. Then Π|_{G^{0}} is a finite direct sum of irreducible admissible representations of G^{0}. Let π be an irreducible constituent of Π|_{G^{0}} which occurs with multiplicity r. Then

Π|_{G^{0}} ≃ r ∑_{g∈G/G_{π}} π^{g}

where G_{π} = {g ∈ G : π^{g} ≃ π}.

Lemma 2.13. (Gelbart Knapp [5]) Let π be an irreducible admissible representation of G^{0}. Then there is an irreducible admissible representation Π of G such that π occurs in the restriction of Π to G^{0} with multiplicity r > 0. Let X denote the group of unitary characters of G/G^{0} and let X(Π) = {χ ∈ X : Π ⊗ χ ≃ Π}. Then

Ind_{G^{0}}^{G}(π) ≃ r ∑_{χ∈X/X(Π)} Π ⊗ χ

is the decomposition of Ind_{G^{0}}^{G}(π) into irreducibles and r^{2}[X/X(Π)] = [G_{π}/G^{0}].

The following result was proved by Gelbart and Knapp in the case where the restriction is multiplicity one [5]. Tadic [14] refined their result in the connected case. We now prove the more general result.

Lemma 2.14. Suppose that G is a totally disconnected group, and H is a closed normal subgroup, with G/H a finite abelian group. If Π_{1} and Π_{2} are irreducible admissible reprsentations of G, which have a common constituent upon restriction to H, then Π_{2} ≃ Π_{1} ⊗ χ, for some character χ with χ|_{H} ≡ 1.

Proof: If the multiplicity of the restrictions is one, then this result holds by Gelbart-Knapp [5]. In particular, if |G/H| is prime, the statement is true. We proceed by induction. We know the Lemma holds when |G/H| = 2. Suppose the statement is true whenever |G_{1}/H| < n. Suppose |G/H| = n. We may assume n is composite, so write n = km, with 1 < k < n. Let H ⊂ G_{1} ⊂ G with |G/G_{1}| = k. If Π_{1}|_{G_{1}} and Π_{2}|_{G_{2}} have a common constituent, then, by our inductieve hypothesis, there is a χ with χ|_{G_{1}} ≡ 1 with Π_{2} ≃ Π_{1} ⊗ χ. Since (G/G_{1}ˆ) ⊂ (G/Hˆ), we are done, in this case.

Now suppose that τ is an irreducible subrepresentarion of both Π_{1}|_{H} and Π_{2}|_{H}. Then, there are constituents Ω_{i} ⊂ Π_{i}|_{G_{1}} so that τ ⊂ Ω_{i}|_{H}. By the inductive hypothesis Ω_{2} = Ω_{1} ⊗ χ, for some χ of G_{1} whose restriction to H is trivial. But, since G_{1}/H ⊂ G/H is abelian we can extend χ to a character η of G/H. Note that (Π_{1} ⊗ η)|G_{1} has Ω_{1} ⊗ χ ≃ Ω_{2} as a constituent, so, as we have seen above, Π_{1} ⊗ η ⊗ ω ≃ Π_{2}, for some character ω of G whose restriction to G_{1} is trivial. Thus, the statement holds by induction.

Let (π, V) be an admissible representation of G and let A(π) denote its space of matrix coefficients. We say π is supercuspidal if A(π) ⊂ ^{0}A(G). Of course if G is not cuspidal, then ^{0}A(G) = {0} so that G has no supercuspidal representations. If P = MN is any parabolic subgroup of G, define V (P) = V (N) to be the subspace of V spanned by vectors of the form π(n)v − v, v ∈ V, n ∈ N. Then we say π is J-supercuspidal if V (P) = V for every proper parabolic subgroup P of G. If G is not cuspidal, then by Lemma 2.8 there is a proper parabolic subgroup G_{1} of G with nilradical N_{1} = {1}. For any admissible representation (π, V) of G, V (G_{1}) = V (N_{1}) = {0} ≠ V so that π is not J-supercuspidal. Thus G has no J-supercuspidal representation.

Suppose now that G is cuspidal and let (π, V) be an irreducible admissible representation of G. Let (π_{0}, V) denote the restriction of π to G^{0}.

Lemma 2.15. Assume that G is cuspidal. Then π is J-supercuspidal if and only if π_{0} is J-supercuspidal if and only if any irreducible constituent of π_{0} is J-supercuspidal.

Proof: Since G is cuspidal, by Lemma 2.8 the set of nilradicals of proper parabolic subgroups is the same for both G and G^{0}. Thus π is J-supercuspidal if and only if π_{0} is J-supercuspidal. Moreover, since by Lemma 2.12 the irreducible constituents of π_{0} are all conjugate via elements of G, it is clear that π_{0} is J-supercuspidal if and only if every irreducible constituent of π_{0} is J-supercuspidal if and only if any irreducible constituent of π_{0} is J-supercuspidal.

Lemma 2.16. Assume that G is cuspidal. Then π is supercuspidal if and only if π_{0} is supercuspidal if and only if any irreducible constituent of π_{0} is supercuspidal.

Proof: Let Z denote the split component of G. By Lemma 2.8 it is also the split component of G^{0}.

Assume that π is supercuspidal. Thus A(π) ⊂ ^{0}A(G). Let f_{0} be a matrix coefficient of π_{0}. Then there is a matrix coefficient f of π so that f_{0} is the restriction of f to G^{0}. Now f ∈ ^{0}A(G). Since f smooth and compactly supported modulo Z, so is f_{0}. Further, by Lemma 2.8 the nilradicals of proper parabolic subgroups are the same for both G and G^{0}. Thus f_{0} will satisfy the integral condition necessary to be a cusp form on G^{0}. Hence A(π_{0}) ⊂ ^{0}A(G^{0}) so that π_{0} is supercuspidal.

Conversely, suppose that π_{0} is supercuspidal. Let π_{1} be an irreducible constituent of π_{0}. Then π ⊂ Ind_{G^{0}}^{G}(π_{1}) so that every matrix coefficient of π is a matrix coefficient of the induced representation. But since G^{0} is a normal subgroup of finite index in G, the restriction of Ind_{G^{0}}^{G}(π_{1}) to G^{0} is equivalent to ∑_{x∈G/G^{0}} π_{1}^{x}. Thus matrix coefficients of the induced representation can be described as follows. Let f be a matrix coefficient of Ind_{G^{0}}^{G}(π_{1}) and fix g ∈ G. Then there are matrix coefficients f_{x} of π_{1}^{x}, x ∈ G/G^{0}, (depending on both f and g) so that for all g_{0} ∈ G^{0},

f(gg_{0}) = ∑_{x∈G/G^{0}} f_{x}(g_{0}).

Since π_{1} is supercuspidal, so is π_{1}^{x} for any x ∈ G/G^{0}, and so each f_{x} ∈ ^{0}A(G^{0}). Thus the restriction of f to each connected component of G is smooth and compactly supported modulo Z. Also, if N is the nilradical of any proper parabolic subgroup of G, then

∫_{N} f(gn)dn = ∑_{x∈G/G^{0}} ∫_{N} f_{x}(n)dn = 0

since by Lemma 2.8, N is also the nilradical of a proper parabolic subgroup of G^{0}.

Proposition 2.17. Assume that G is cuspidal and let π be an irreducible admissible representation of G. Then π is supercuspidal if and only if π is J-supercuspidal.

Proof: This is an immediate consequence of Lemmas 2.15 and 2.16 and the corresponding result in the connected case.

We now drop the assumption that G is cuspidal. Let P = MN be a parabolic subgroup of G and let σ be an admissible representation of M. Then we let Ind_{P}^{G}(σ) denote the representation of G by left translations on

H = \{f ∈ C^{∞}(G, V) : f(gmn) = δ_{P}^{− \frac{1}{2}} (m)σ(m)^{−1}f(g), ∀g ∈ G, m ∈ M, n ∈ N\}.

Here δ_{P} denotes the modular function of P.

Theorem 2.18. Let π be an irreducible admissible representation of G. Then there are a cuspidal parabolic subgroup P = MN of G and an irreducible supercuspidal representation σ of M such that π is a subrepresentation of Ind_{P}^{G}(σ).

Remark 2.19. We will see in Corollary 3.2 that the group M and supercuspidal representation σ in Theorem 2.18 are unique up to conjugacy.

Proof: Let ρ be an irreducible constituent of the restriction of π to G^{0}. Then π ⊂ Ind_{G^{0}}^{G}(ρ). Since ρ is admissible, there are a parabolic subgroup P^{0} = M^{0}N of G^{0} and an irreducible supercuspidal representation τ of M^{0} such that ρ ⊂ Ind_{P^{0}}^{G^{0}}(τ). Thus

π ⊂ Ind_{G^{0}}^{G}(ρ) ⊂ Ind_{G^{0}}^{G}(Ind_{P^{0}}^{G^{0}}(τ)) ≃ Ind_{P^{0}}^{G}(τ).

Let P = MN be the unique cuspidal parabolic subgroup of G lying over P^{0}. Let σ be an irreducible admissible representation of M such that τ is contained in the restriction of σ to M^{0}. By Lemma 2.16, σ is supercuspidal. By Lemma 2.13 applied to M and M^{0} we have

Ind_{M^{0}}^{M}(τ) ≃ s ∑_{η∈Y/Y (σ)} σ ⊗ η

where Y is the group of unitary characters of M/M^{0}. Since τ is contained in the restriction of σ ⊗ η to M^{0} for any η, all the representations σ ⊗ η are supercuspidal. Now

π ⊂ Ind_{P^{0}}^{G}(τ) ≃ s ∑_{η∈Y/Y (σ)} Ind_{P}^{G}(σ ⊗ η).

Thus π ⊂ Ind_{P}^{G}(σ ⊗ η) for some η.

Let A(G) = ∪_{π}A(π) where π runs over the set of all admissible representations of G. Similarly we have A(G^{0}) and because of Lemma 2.11 it is clear that if f ∈ A(G), then f|_{G^{0}} ∈ A(G^{0}). Define the subspace A_{T} (G^{0}) ⊂ A(G^{0}) as in [13,§4.5]. It is the set of functions in A(G^{0}) which satisfy the weak inequality. Define

A_{T} (G) = {f ∈ A(G) : l(x)f|_{G^{0}} ∈ A_{T} (G^{0}) for all x ∈ G}

where l(x)f denotes the left translate of f by x. In other words, A_{T} (G) is the set of functions in A(G) which satisfy the weak inequality on every connected component of G. If π is an admissible representation of G, we say π is tempered if A(π) ⊂ A_{T} (G). The following lemma is easy to prove using the properties of matrix coefficients of π and π_{0} from the proof of Lemma 2.16.

Lemma 2.20. Let π be an irreducible admissible representation of G. Then π is tempered if and only if π_{0} is tempered if and only if any irreducible constituent of π_{0} is tempered.

Let π be an irreducible unitary representation of G and let Z be the split component of G. We say that π is discrete series if A(π) ⊂ L^{2}(G/Z). Every unitary supercuspidal representation is discrete series since its matrix coefficients are compactly supported modulo Z.

Lemma 2.21. If G is not cuspidal, then G has no discrete series representations. If G is cuspidal, then π is discrete series if and only if π_{0} is discrete series if and only if any irreducible constituent of π_{0} is discrete series.

Proof: Suppose that G is not cuspidal. Then the split component Z of G is a proper subgroup of the split component Z_{0} of G^{0}. Let π be any irreducible unitary representation of G. Then there is an irreducible unitary representation π_{1} of G^{0} so that π is contained in Ind_{G^{0}}^{G}(π_{1}). Thus as in the proof of Lemma 2.16, for any matrix coefficient f of π and any g ∈ G, we have matrix coefficients f_{x} of π_{1}^{x}, x ∈ G/G^{0}, so that for all g_{0} ∈ G^{0},

f(gg_{0}) = ∑_{x∈G/G^{0}} f_{x}(g_{0}).

Let ω be the Z_{0}-character of π_{1}. Then for any z ∈ Z_{0}, g_{0} ∈ G^{0}, we have

f(gg_{0}z) = ∑_{x∈G/G^{0}} f_{x}(g_{0}z) = ∑_{x∈G/G^{0}} ω^{x}(z)f_{x}(g_{0}).

Thus z ↦ f(gg_{0}z), z ∈ Z_{0}, is a finite linear combination of unitary characters of Z_{0}, and cannot be square-integrable on Z_{0}/Z unless it is zero. Now if f is square-integrable on G/Z, then g_{0} ↦ f(gg_{0}) is square-integrable on G_{0}/Z for all coset representatives g ∈ G/G^{0}, and z ↦ f(gg_{0}z) must be square-integrable on Z_{0}/Z for almost all g_{0}, so that f(gg_{0}z) must be zero for almost all g_{0}, z, and f = 0.

Suppose that G is cuspidal. Let π be a discrete series representation of G. Let f_{0} be a matrix coefficient of π_{0}. Then there is a matrix coefficient f of π so that f_{0} is the restriction of f to G^{0}. Since f is square-integrable on G/Z, certainly f_{0} is square-integrable on G^{0}/Z.

Conversely, suppose that π_{0} is discrete series. Let π_{1} be an irreducible constituent of π_{0}. For any matrix coefficient f of π we have

∫_{G/Z} |f(g)|^{2}d(gZ) = ∑_{g∈G/G^{0}} ∫_{G^{0}/Z} |f(gg_{0})|^{2}d(g_{0}Z).

As above, for fixed g ∈ G, we have matrix coefficients f_{x} of π_{1}^{x}, x ∈ G/G^{0}, so that for all g_{0} ∈ G^{0},

f(gg_{0}) = ∑_{x∈G/G^{0}} f_{x}(g_{0}).

Thus

(∫_{G^{0}/Z} |f(gg_{0})|^{2}d(g_{0}Z))^{\frac{1}{2}} ≤ ∑_{x∈G/G^{0}}(∫_{G^{0}/Z} |f_{x}(g_{0})|^{2}d(g_{0}Z))^{\frac{1}{2}} < ∞

since every f_{x} is square-integrable on G^{0}/Z. Thus f is square-integrable on G/Z.

The following theorem can be proven in the same way as Theorem 2.18 using Lemmas 2.20 and 2.21.

Theorem 2.22. Let π be an irreducible tempered representation of G. Then there are a cuspidal parabolic subgroup P = MN of G and an irreducible discrete series representation σ of M such that π is a subrepresentation of Ind_{P}^{G}(σ).

Remark 2.23. We will see in Corollary 3.2 that the group M and discrete series representation σ in Theorem 2.22 are unique up to conjugacy.

3. Intertwining Operators

Let G be a reductive F -group with G/G^{0} finite and abelian as in §2. If π_{1} and π_{2} are admissible representations of G, we let J(π_{1}, π_{2}) denote the dimension of the space of all intertwining operators from π_{2} to π_{1}. We want to prove the following theorem.

Let (P_{i}, A_{i}), i = 1, 2, be cuspidal parabolic pairs of G with P_{i} = M_{i}N_{i} the corresponding Levi decompositions. Let σ_{i} be an irreducible admissible representation of M_{i} on a vector space V_{i}, and let π_{i} = Ind_{P_{i}}^{G}(σ_{i}), where we use normalized induction as in §2. Let W = W (A_{1}|A_{2}) denote the set of mappings s : A_{2} → A_{1} such that there exists y_{s} ∈ G such that s(a_{2}) = y_{s}a_{2}y_{s}^{−1} for all a_{2} ∈ A_{2}.

Theorem 3.1. Assume that σ_{1} and σ_{2} are either both supercuspidal representations, or both discrete series representations. If A_{1} and A_{2} are not conjugate, then J(π_{1}, π_{2}) = 0. Assume A_{1} and A_{2} are conjugate. Then

J(π_{1}, π_{2}) ≤ ∑_{s∈W} J(σ_{1}, σ_{2}^{y_{s}}).

Corollary 3.2. Let σ_{1} and σ_{2} be as above. Then J(π_{1}, π_{2}) = 0 unless there is y ∈ G such that M_{1} = M_{2}^{y}, σ_{1} ≃ σ_{2}^{y}.

Corollary 3.3. Let σ_{1} and σ_{2} be irreducible discrete series representations, and suppose that π_{1} and π_{2} have an irreducible constituent in common. Then π_{1} ≃ π_{2}.

Proof: In this case J(π_{1}, π_{2}) ≠ 0 so by Corollary 3.2 there is y ∈ G such that M_{1} = M_{2}^{y}, σ_{1} ≃ σ_{2}^{y}. Thus

π_{1} = Ind_{M_{1}N_{1}}^{G}(σ_{1}) = Ind_{M_{2}^{y}N_{1}}^{G}(σ_{2}^{y}) ≃ Ind_{M_{2}N′_{2}}^{G}(σ_{2})

where N′_{2} = N_{1}^{y^{−1}}. Now P_{2} = M_{2}N_{2} and P′_{2} = M_{2}N′_{2} are two cuspidal parabolic subgroups of G with the same Levi component M_{2}. We will see in Corollary 5.9 that there is an equivalence R(P′_{2} : P_{2} : σ_{2}) between π_{2} = Ind_{P_{2}}^{G}(σ_{2}) and π′_{2} = Ind_{P′_{2}}^{G}(σ_{2}). Thus

π_{1} ≃ π′_{2} ≃ π_{2}.

Let W (σ_{1}) = {s ∈ W (A_{1}, A_{1}) : σ_{1}^{s} ≃ σ_{1}}. We say σ_{1} is unramified if W (σ_{1}) = {1}.

Corollary 3.4. Assume that σ_{1} is discrete series. Then J(π_{1}, π_{1}) ≤ [W (σ_{1})]. In particular, if σ_{1} is unramified, then π_{1} is irreducible.

In order to prove Theorem 3.1 for discrete series representations we will need the following results about dual exponents. Let (π, V) be an irreducible representation of G and let V′ denote the algebraic dual of V. For x ∈ G, let π(x)^{t} : V′ → V′ denote the transpose of π(x). Let (P, A) be a cuspidal parabolic pair in G. Then a quasi-character χ of A is called a dual exponent of π with respect to (P, A) if there is a nonzero ϕ ∈ V′ such that for all \overline{n} ∈ \overline{N}, a ∈ A,

π(\overline{n})^{t}ϕ = ϕ and π(a)^{t}ϕ = δ\overline{P}^{\frac{1}{2}} (a)χ(a)ϕ.  (∗)

We will write Y_{π}(P, A) for the set of all dual exponents of π with respect to (P, A).

Let V = ∑_{i=1}^{k} V_{i} be the decomposition of V into G^{0} invariant subspaces and let π_{i} be the irreducible representation of G^{0} on V_{i}, 1 ≤ i ≤ k.

Lemma 3.5. Let (P, A) be a cuspidal parabolic pair in G. Then

Y_{π}(P, A) = ∪_{i=1}^{k}Y_{π_{i}}(P^{0}, A).

Proof: Let χ ∈ Y_{π}(P, A) and let ϕ ∈ V′ be a nonzero functional satisfying (*). Then there is 1 ≤ i ≤ k such that ϕ_{i}, the restriction of ϕ to V_{i}, is nonzero. Now for any v_{i} ∈ V_{i} and g_{0} ∈ G^{0} we have

< π_{i}(g_{0})^{t}ϕ_{i}, v_{i} >=< π(g_{0})^{t}ϕ, v_{i} > .

Now since \overline{N} and A are both contained in G^{0} it is easy to check that

< π_{i}(\overline{n})^{t}ϕ_{i}, v_{i} >=< ϕ_{i}, v_{i} > and < π_{i}(a)^{t}ϕ_{i}, v_{i} >= δ\overline{P}^{0}^{\frac{1}{2}}(a)χ(a) < ϕ_{i}, v_{i} > (∗∗)

for all v_{i} ∈ V_{i}, \overline{n} ∈ \overline{N}, a ∈ A. Thus χ ∈ Y_{π_{i}}(P^{0}, A).

Now assume that χ ∈ Y_{π_{i}}(P^{0}, A) for some 1 ≤ i ≤ k. and let ϕ_{i} ∈ V′_{i} be a nonzero functional satisfying (**). Now π is contained in the induced representation Ind_{G^{0}}^{G}(π_{i}) so we can realize π on a subspace V of

H = {f : G → V_{i} : f(gg_{0}) = π_{i}(g_{0})^{−1}f(g), g ∈ G, g_{0} ∈ G^{0}}

with the action of π given by left translation on the functions. Now define ϕ ∈ V′ by

< ϕ, f >=< ϕ_{i}, f(1) >, f ∈ V.

Then ϕ ≠ 0 and it is easy to check that for all g_{0} ∈ G^{0}, f ∈ V, we have

< π(g_{0})^{t}ϕ, f >=< π_{i}(g_{0})^{t}ϕ_{i}, f(1) > .

From this it easily follows that since ϕ_{i} satisfies (**), ϕ satisfies (*). Thus χ ∈ Y_{π}(P, A).

Lemma 3.6. Assume that G is cuspidal and let π be a discrete series representation of G. Then Y_{π}(P, A) ∩ Â = ∅ for every cuspidal parabolic pair (P, A) ≠ (G, Z).

Proof: This follows easily from Lemma 2.21, Lemma 3.5, and the corresponding result for connected groups.

Let (P_{0}^{0}, A_{0}) be a minimal p-pair in G^{0} and let P_{0} be the cuspidal parabolic subgroup lying over P_{0}^{0}. We will call (P_{0}, A_{0}) a minimal parabolic pair in G.

Lemma 3.7. Let (P, A) be any parabolic pair in G. Then there is x ∈ G^{0} such that P_{0} ⊂ xP x^{−1}, xAx^{−1} ⊂ A_{0}.

Proof: Let M = C_{G}(A) and let P^{0} = P ∩ G^{0}, M^{0} = M ∩ G^{0}. Let A′ be the split component of M^{0}. Thus A ⊂ A′. Now (P^{0}, A′) is a p-pair in G^{0} so there is x ∈ G^{0} such that P_{0}^{0} ⊂ xP^{0}x^{−1} and xAx^{−1} ⊂ xA′x^{−1} ⊂ A_{0}. Now M_{0} = C_{G}(A_{0}) ⊂ C_{G}(xAx^{−1}) = xMx^{−1} so that P_{0} = M_{0}P_{0}^{0} ⊂ (xMx^{−1})(xP^{0}x^{−1}) = xP x^{−1}.

We will say a parabolic pair (P, A) is standard with respect to the minimal parabolic pair (P_{0}, A_{0}) if P_{0} ⊂ P and A ⊂ A_{0}. We say (P, A) is semi-standard with respect to (P_{0}, A_{0}) if A ⊂ A_{0}.

Fix a minimal parabolic pair (P_{0}, A_{0}) of G. Let N_{G}(P_{0}, A_{0}) denote the set of all elements of G which normalize both P_{0} and A_{0}. Write W_{G}(P_{0}, A_{0}) = N_{G}(P_{0}, A_{0})/M_{0}^{0}. If (P, A) is any parabolic pair of G which is standard with respect to (P_{0}, A_{0}) and M = C_{G}(A), write N_{M}(P_{0}, A_{0}) = M∩N_{G}(P_{0}, A_{0}) and W_{M}(P_{0}, A_{0}) = N_{M}(P_{0}, A_{0})/M_{0}^{0}.

Lemma 3.8. We can write P as a disjoint union

P = ∪_{w∈W_{M} (P_{0},A_{0})}wP^{0}.

Proof: We first prove the result when P = G. Let y ∈ G. Then (yP_{0}^{0}y^{−1}, yA_{0}y^{−1}) is a minimal parabolic pair in G^{0} and hence is conjugate to (P_{0}^{0}, A_{0}) via an element of G^{0}. Thus there is g ∈ G^{0} such that gy normalizes both A_{0} and P_{0}^{0}. But then gy also normalizes M_{0} = C_{G}(A_{0}) and P_{0} = M_{0}P_{0}^{0}. Thus gy ∈ N_{G}(P_{0}, A_{0}).

Thus the coset yG_{0} has a representative n ∈ N_{G}(P_{0}, A_{0}) which depends only on the coset w of n in W_{G}(P_{0}, A_{0}). Since N_{G}(P_{0}, A_{0}) ∩ G^{0} = M_{0}^{0}, n_{1}, n_{2} ∈ N_{G}(P_{0}, A_{0}) determine the same coset of G^{0} in G just in case they are in the same coset in W_{G}(P_{0}, A_{0}). Thus we have the disjoint union

G = ∪_{w∈W_{G}(P_{0},A_{0})}wG^{0}.

Now let (P, A) be arbitrary. There is a subset θ of the simple roots Δ of (P_{0}^{0}, A_{0}) corresponding to P^{0} so that P^{0} = P_{θ}^{0}. Note that W_{G}(P_{0}, A_{0}) acts on Δ and that w^{−1}P_{θ}^{0}w = P_{wθ}^{0}. Let w ∈ W_{G}(P_{0}, A_{0}) and suppose that P ∩ wG^{0} ≠ ∅. Let wg_{0} ∈ P ∩ wG^{0}, g_{0} ∈ G^{0}. Then wg_{0} normalizes P^{0} so that w^{−1}P^{0}w = g_{0}P^{0}g_{0}^{−1} is conjugate to P^{0} in G^{0}. But w^{−1}P^{0}w = P_{wθ}^{0} is a standard parabolic subgroup of G^{0} and so P_{wθ}^{0} = g_{0}P^{0}g_{0}^{−1} = P^{0}. Hence g_{0} ∈ N_{G^{0}}(P^{0}) = P^{0}, wθ = θ, and w has a representative in P. Hence w normalizes A = {a ∈ A_{0} : α(a) = 1 for all α ∈ θ}. But N_{G}(A)∩P = M. Thus w ∈ W_{M}(P_{0}, A_{0}).

In order to prove Theorem 3.1 we extend the proof of Silberger in [13, §2.5] to the disconnected case. Fortunately many of the technical results on intertwining forms needed are proven in [13, §1] for any totally disconnected group and so can be directly used in our case. We follow Silberger’s notation. Let (P_{1}, A_{1}) and (P_{2}, A_{2}) be cuspidal parabolic pairs in G. We may as well assume that they are standard with respect to a fixed minimal parabolic pair (P_{0}, A_{0}).

We need to study the orbits for the action of P_{1} × P_{2} on G given by y · (p_{1}, p_{2}) = p_{1}^{−1}yp_{2}. Recall that G^{0} = ∪_{v}P_{0}^{0}vP_{0}^{0} where v runs over W (G^{0}, A_{0}) = N_{G^{0}}(A_{0})/M_{0}^{0}. Thus, G = ∪_{w}wG^{0} = ∪_{w,v}wP_{0}^{0}vP_{0}^{0} = ∪_{w,v}P_{0}^{0}wvP_{0}^{0} where w ∈ W_{G}(P_{0}, A_{0}), v ∈ W (G^{0}, A_{0}). Now since P_{0}^{0} ⊂ P_{i}, i = 1, 2, each double coset P_{1}yP_{2} can be represented by y = wv, w ∈ W_{G}(P_{0}, A_{0}), v ∈ W (G^{0}, A_{0}). Write W_{i} = W_{M_{i}}(P_{0}, A_{0}), i = 1, 2.

Lemma 3.9. Let O = P_{1}w_{0}vP_{2} be an orbit of P_{1} × P_{2} in G where w_{0} ∈ W_{G}(P_{0}, A_{0}) and v ∈ W (G^{0}, A_{0}). Then for w ∈ W_{G}(P_{0}, A_{0}), O∩wG^{0} is empty unless w ∈ W_{1}w_{0}W_{2}. If w ∈ W_{1}w_{0}W_{2}, then O ∩ wG^{0} = wO_{w} where O_{w} is a finite union of orbits of (w^{−1}P_{1}^{0}w) × P_{2}^{0} in G^{0}, all of which have fixed dimension d_{O} equal to the dimension of the orbit O_{0} = (w_{0}^{−1}P_{1}^{0}w_{0})vP_{2}^{0}.

Proof: Using Lemma 3.8 we can write

P_{1}w_{0}vP_{2} = ∪_{w_{1},w_{2}}w_{1}P_{1}^{0}w_{0}vw_{2}P_{2}^{0}

where w_{i} ∈ W_{i} = W_{M_{i}}(P_{0}, A_{0}), i = 1, 2. But

w_{1}P_{1}^{0}w_{0}vw_{2}P_{2}^{0} = w(w^{−1}P_{1}^{0}w)w_{2}^{−1}vw_{2}P_{2}^{0}

where w = w_{1}w_{0}w_{2} ∈ W_{G}(P_{0}, A_{0}). Thus w_{1}P_{1}^{0}w_{0}vw_{2}P_{2}^{0} ⊂ w_{1}w_{0}w_{2}G^{0} and for fixed w ∈ W_{1}w_{0}W_{2},

P_{1}w_{0}vP_{2} ∩ wG^{0} = w ∪_{w_{2}} (w^{−1}P_{1}^{0}w)w_{2}^{−1}vw_{2}P_{2}^{0}

where w_{2} runs over elements of W_{2} such that ww_{2}^{−1}w_{0}^{−1} ∈ W_{1}. Finally,

(w^{−1}P_{1}^{0}w)w_{2}^{−1}vw_{2}P_{2}^{0} = w_{2}^{−1}(w_{0}^{−1}P_{1}^{0}w_{0})vP_{2}^{0}w_{2}

so that

dim(w^{−1}P_{1}^{0}w)w_{2}^{−1}vw_{2}P_{2}^{0} = dim(w_{0}^{−1}P_{1}^{0}w_{0})vP_{2}^{0} = d_{O}.

Because of Lemma 3.9 each orbit O has a well-defined dimension d_{O}. For each integer ν ≥ 0, let O(ν) denote the union of all orbits of dimension less than or equal to ν. Set O(−1) = ∅.

Lemma 3.10. For each ν ≥ 0, O(ν) is a closed set in the p-adic topology. Further, if O is an orbit of dimension d, then O ∪ O(d − 1) is also closed in the p-adic topology.

Proof: Using Lemma 3.9 we see that O(ν) ∩ wG^{0} = wO_{w}(ν) where O_{w}(ν) is the union of all orbits of (w^{−1}P_{1}^{0}w) × P_{2}^{0} in G^{0} having dimension less than or equal to ν. This set is closed in the p-adic topology by [13, pg. 93]. Thus O(ν) is a finite union of closed sets, hence is closed. Similarly, if O has dimension d, then [O ∪ O(d − 1)] ∩ wG^{0} = w(O_{w} ∪ O_{w}(d − 1)) where O_{w} ∪ O_{w}(d − 1) is a finite union of sets of the form O′ ∪ O_{w}(d − 1) where O′ is an orbit of (w^{−1}P_{1}^{0}w) × P_{2}^{0} in G^{0} of dimension d. These are also closed by [13, page 94].

Let E = V_{1} ⊗V_{2} where V_{i} is the space on which σ_{i} acts, i = 1, 2. Let T be the space of all E-distributions T^{0} on G such that

T^{0}(λ(p_{1})ρ(p_{2})α) = T^{0}(δ_{1}(p_{1})^{\frac{1}{2}} δ_{2}(p_{2})^{\frac{1}{2}} σ_{1}(p_{1})^{−1} ⊗ σ_{2}(p_{2})^{−1}α)

for all (p_{1}, p_{2}) ∈ P_{1} × P_{2}, α ∈ C_{c}^{∞}(G : E). Here λ and ρ denote left and right translations respectively and δ_{i} is the modular function of P_{i}, i = 1, 2 The first step in the proof of Theorem 3.1 is the inequality [13, 1.9.4]

I(π_{1}, π_{2}) ≤ dim T .  (3.1)

Here I(π_{1}, π_{2}) is the dimension of the space of “intertwining forms” defined in [13, §1.6]. It is related to the dimension of the space of intertwining operators by I(π_{1}, π_{2}) = J(π̃_{1}, π_{2}) where π̃_{1} is the contragredient of π_{1}. [13, 1.6.2]

If O is an orbit of dimension d, write T (O) for the vector space of T^{0} ∈ T with support in O ∪ O(d − 1) and T_{ν} for the space of those with support in O(ν). We have

T = ∑_{O} T (O)

and

dim T ≤ ∑_{O} dim(T (O)/T_{d(O)−1}).  (3.2)

Lemma 3.11. Suppose that (P_{1}, A_{1}) and (P_{2}, A_{2}) are semi-standard cuspidal parabolic pairs in G. Then M_{1} ∩ P_{2} = (M_{1} ∩ M_{2})(M_{1} ∩ N_{2}) = ^{∗}P_{1} is a cuspidal parabolic subgroup of M_{1} with split component ^{∗}A_{1} = A_{1}A_{2}.

Proof: We know from the connected case [13, p 94] that ^{∗}P_{1}^{0} = M_{1}^{0} ∩ P_{2}^{0} is a parabolic subgroup of M_{1}^{0} with split component ^{∗}A_{1} = A_{1}A_{2} and Levi decomposition ∗P_{1}^{0} = (M_{1}^{0} ∩ M_{2}^{0})(M_{1}^{0} ∩ N_{2}). Thus there is a cuspidal parabolic subgroup ^{∗}P_{1} of M_{1} with split component ^{∗}A_{1} and Levi decomposition ^{∗}P_{1} = ^{∗}M_{1} ^{∗}N_{1}. Here ^{∗}M_{1} = C_{M_{1}}( ^{∗}A_{1}) = C_{M_{1}}(A_{1}A_{2}) = M_{1} ∩ C_{G}(A_{2}) = M_{1} ∩ M_{2} and ^{∗}N_{1} = M_{1}^{0} ∩ N_{2} = M_{1} ∩ N_{2}.

Clearly ^{∗}P_{1} = (M_{1} ∩ M_{2})(M_{1} ∩ N_{2}) ⊂ M_{1} ∩ P_{2}. Thus we need only show that M_{1} ∩ P_{2} ⊂ ^{∗}P_{1}.

Let x ∈ M_{1} ∩ P_{2}. Using the Levi decomposition of P_{2} we can write x = m_{2}n_{2} where m_{2} ∈ M_{2}, n_{2} ∈ N_{2}. Since m_{2}n_{2} ∈ M_{1} = C_{G}(A_{1}) we have m_{2}n_{2}a_{1}n_{2}^{−1}m_{2}^{−1} = a_{1} for any a_{1} ∈ A_{1}. This implies that a_{1}^{−1}n_{2}a_{1}n_{2}^{−1} = a_{1}^{−1}m_{2}^{−1}a_{1}m_{2}. But since (P_{1}, A_{1}) and (P_{2}, A_{2}) are semi-standard, we have A_{1} ⊂ A_{0} ⊂ C_{G}(A_{2}) = M_{2}. Thus a_{1}^{−1}n_{2}a_{1}n_{2}^{−1} ∈ N_{2} and a_{1}^{−1}m_{2}^{−1}a_{1}m_{2} ∈ M_{2}. Hence a_{1}^{−1}n_{2}a_{1}n_{2}^{−1} = a_{1}^{−1}m_{2}^{−1}a_{1}m_{2} ∈ N_{2} ∩ M_{2} = {1}. Thus n_{2} and m_{2} both commute with a_{1} so that n_{2} ∈ M_{1} ∩ N_{2} and m_{2} ∈ M_{1} ∩ M_{2}.

Return to the assumption that (P_{1}, A_{1}) and (P_{2}, A_{2}) are standard with respect to (P_{0}, A_{0}) and that y ∈ W (A_{0}) so that (P_{2}^{y}, A_{2}^{y}) is semi-standard. Using Lemma 3.11 we know that ^{∗}P_{1} = M_{1} ∩ P_{2}^{y} and ^{∗}P_{2} = M_{2} ∩ P_{1}^{y^{−1}} are cuspidal parabolic subgroups of M_{1} and M_{2} respectively.

Let E′(y) denote the space of linear functionals ϕ on E = V_{1} ⊗ V_{2} such that

δ_{1}(p)^{\frac{1}{2}} δ_{2}(p^{y^{−1}})^{\frac{1}{2}} < ϕ, σ_{1}(p)v_{1} ⊗ σ_{2}(p^{y^{−1}})v_{2} >= δ_{P_{1}∩P_{2}^{y}}(p) < ϕ, v_{1} ⊗ v_{2} >

for all p ∈ P_{1} ∩ P_{2}^{y}, v_{i} ∈ V_{i}, i = 1, 2.

Lemma 3.12. Let m ∈ ^{∗}M_{1} = M_{1} ∩ M_{2}^{y}. Then

δ_{P_{1}∩P_{2}^{y}}(m) = δ_{∗1}(m)^{\frac{1}{2}} δ_{∗2}(m^{y^{−1}})^{\frac{1}{2}} δ_{1}(m)^{\frac{1}{2}} δ_{2}(m^{y^{−1}})^{\frac{1}{2}}

where δ_{∗i} denotes the modular function for ^{∗}P_{i} and δ_{i} the modular function for P_{i}, i = 1, 2.

Proof: Define the homomorphism δ : ^{∗}M_{1} → R_{+}^{∗} by

δ(m) = δ_{∗1}(m)^{\frac{1}{2}} δ_{∗2}(m^{y^{−1}})^{\frac{1}{2}} δ_{1}(m)^{\frac{1}{2}} δ_{2}(m^{y^{−1}})^{\frac{1}{2}} δ_{P_{1}∩P_{2}^{y}}(m)^{−1}, m ∈ ^{∗}M_{1}.

By [13, §2.5.2], the restriction of δ to ^{∗}M_{1} ∩ G^{0} is trivial. Now since the quotient of ∗M_{1} by ^{∗}M_{1} ∩ G^{0} is a finite group, δ = 1 on all of ^{∗}M_{1}.

The proof of the following lemma is the same as that of Lemma 2.5.1 in [13].

Lemma 3.13. If ϕ ∈ E′(y), then ϕ vanishes on V_{1}( ^{∗}P_{1}) ⊗ V_{2} + V_{1} ⊗ V_{2}( ^{∗}P_{2}). Let m ∈ ^{∗}M_{1}, v_{1} ⊗ v_{2} ∈ E, and ϕ ∈ E′(y). Then

< ϕ, σ_{1}(m)v_{1} ⊗ σ_{2}(m^{y^{−1}})v_{2} >= δ_{∗1}(m)^{\frac{1}{2}} δ_{∗2}(m^{y^{−1}})^{\frac{1}{2}} < ϕ, v_{1} ⊗ v_{2} > .

Corollary 3.14. If A_{1} = A_{2}^{y}, then

< ϕ, σ_{1}(m)v_{1} ⊗ σ_{2}(m^{y^{−1}})v_{2} >=< ϕ, v_{1} ⊗ v_{2} >

for all m ∈ ^{∗}M_{1}, v_{1} ⊗ v_{2} ∈ E, and ϕ ∈ E′(y).

Proof: As in [13, §2.5.4], A_{1} = A_{2}^{y} implies that ^{∗}N_{1} = ^{∗}N_{2} = {1} so that ^{∗}P_{1} and ∗P_{2} are reductive and δ_{∗1} = δ_{∗2} = 1.

Corollary 3.15. Assume that σ_{1} and σ_{2} are either both supercuspidal or both discrete series. Then E′(y) ≠ {0} implies that A_{1} = A_{2}^{y}.

Proof: This follows as in [13, §§2.5.3, 2.5.5] using Proposition 2.17 and Lemma 3.6.

The following Lemma is now proven as in [13, §2.5.7].

Lemma 3.16. Assume that σ_{1} and σ_{2} are either both supercuspidal or both discrete series. Let O be an orbit in G of dimension d. Then dim T (O)/T_{d−1} = 0 unless there exists y ∈ O such that A_{1} = A_{2}^{y}. If A_{1} = A_{2}^{y} for some y ∈ O, then

dim(T (O)/T_{d−1}) ≤ I(σ_{1}, σ_{2}^{y}).

Proof of Theorem 3.1. Combine equations (3.1) and (3.2) to obtain

I(π_{1}, π_{2}) ≤ dim T ≤ ∑_{O} dim(T (O)/T_{d(O)−1}).

Now using Lemma 3.16, if A_{1} and A_{2} are not conjugate, dim(T (O)/T_{d(O)−1}) = 0 for every orbit O so that I(π_{1}, π_{2}) = 0. If A_{1} and A_{2} are conjugate we have, again using Lemma 3.16,

I(π_{1}, π_{2}) ≤ dim T ≤ ∑_{s∈W} dim(T (O_{s})/T_{d(O_{s})−1}) ≤ ∑_{s∈W} I(σ_{1}, σ_{2}^{y_{s}}).

To complete the proof note that J(π_{1}, π_{2}) = I(π̃_{1}, π_{2}), J(σ_{1}, σ_{2}^{y_{s}}) = I(σ̃_{1}, σ_{2}^{y_{s}}) and π̃_{1} = Ind_{P_{1}}^{G}(σ̃_{1}). Thus the statement with dimensions of spaces of intertwining operators rather than intertwining forms follows by substituting π̃_{1} for π_{1}.

4. R-groups for Ind_{P^{0}}^{G}(σ_{0})

In this section we will study representations of G which are induced from a parabolic subgroup P^{0} = M^{0}N of G^{0}. Because in this section we will only work with parabolic subgroups of G^{0}, we will simplify notation by dropping the superscripts on P^{0} and M^{0}.

Let P = MN be a parabolic subgroup of G^{0} and fix an irreducible discrete series representation (σ, V) of M. Define

H_{P} (σ) = \{f ∈ C^{∞}(G, V) : f(xmn) = δ_{P}^{− \frac{1}{2}} (m)σ(m)^{−1}f(x) for all x ∈ G, m ∈ M, n ∈ N\}

where δ_{P} is the modular function on P. Then G acts by left translation on H_{P} (σ) and we call this induced representation I_{P} (σ). We will also need to consider H_{P}^{0} (σ) =

\{ϕ ∈ C^{∞}(G^{0}, V) : ϕ(xmn) = δ_{P}^{− \frac{1}{2}} (m)σ(m)^{−1}ϕ(x) for all x ∈ G^{0}, m ∈ M, n ∈ N\}.

G^{0} acts by left translation on H_{P}^{0} (σ) and we call this induced representation I_{P}^{0} (σ).

It is well known that the equivalence class of I_{P}^{0} (σ) = Ind_{P}^{G^{0}}(σ) is independent of P. But

I_{P} (σ) = Ind_{P}^{G}(σ) ≃ Ind_{G^{0}}^{G} Ind_{P}^{G^{0}}(σ),

so that the equivalence class of I_{P} (σ) is also independent of P. We denote the equivalence classes of I_{P} (σ) and I_{P}^{0} (σ) by i_{G,M}(σ) and i_{G^{0},M}(σ) respectively. If π is a representation of G^{0} we will also write i_{G,G^{0}}(π) for the equivalence class of the induced representation Ind_{G^{0}}^{G}(π).

We first want to compare the dimensions of the intertwining algebras of i_{G,M}(σ) and i_{G^{0},M}(σ) . For this we need the results of Gelbart and Knapp summarized in §2 and the following facts.

Lemma 4.1. Suppose π_{1} and π_{2} are irreducible representations of G^{0}. Then i_{G,G^{0}}(π_{1}) and i_{G,G^{0}}(π_{2}) have an irreducible constituent in common if and only if π_{2} ≃ π_{1}^{g} for some g ∈ G. In this case they are equivalent.

Proof: For i = 1, 2, write

i_{G,G^{0}}(π_{i}) = r_{i} ∑_{χ∈X/X(Π_{i})} Π_{i} ⊗ χ

as in Lemma 2.13. Suppose that i_{G,G^{0}}(π_{1}) and i_{G,G^{0}}(π_{2}) have an irreducible constituent in common. Then Π_{1} ⊗ χ_{1} ≃ Π_{2} ⊗ χ_{2} for some χ_{1}, χ_{2} ∈ X. Now

π_{2} ⊂ Π_{2} ⊗ χ_{2}|_{G^{0}} ≃ Π_{1} ⊗ χ_{1}|_{G^{0}} ⊂ i_{G,G^{0}}(π_{1})|_{G^{0}} ≃ ∑_{g∈G/G^{0}} π_{1}^{g}.

Thus π_{2} ≃ π_{1}^{g} for some g ∈ G. Conversely, if π_{2} ≃ π_{1}^{g} for some g ∈ G, then clearly

i_{G,G^{0}}(π_{2}) ≃ i_{G,G^{0}}(π_{1}^{g}) ≃ i_{G,G^{0}}(π_{1}).

Lemma 4.2. Suppose that for some g ∈ G, both π and π^{g} are irreducible constituents of i_{G^{0},M}(σ). Then there is x_{0} ∈ G^{0} such that gx_{0} ∈ N_{G}(σ) = {x ∈ N_{G}(M) : σ^{x} ≃ σ}. Conversely, if π is an irreducible constituent of i_{G^{0},M}(σ) and if g ∈ N_{G}(σ)G^{0}, then π^{g} is also an irreducible constituent of i_{G^{0},M}(σ) and π and π^{g} occur with the same multiplicities.

Proof: Suppose that π, π^{g} ⊂ i_{G^{0},M}(σ). Then since π^{g} ⊂ i_{G^{0},M^{g}}(σ^{g}), we see that i_{G^{0},M}(σ) and i_{G^{0},M^{g}}(σ^{g}) have an irreducible constituent in common. Thus there is x_{0} ∈ G^{0} such that M^{gx_{0}} = M and σ^{gx_{0}} ≃ σ, i.e. gx_{0} ∈ N_{G}(σ). Conversely, if g ∈ N_{G}(σ)G^{0}, then the multiplicity of π in i_{G^{0},M}(σ) is equal to the multiplicity of π^{g} in i_{G^{0},M}(σ)^{g} ≃ i_{G^{0},M^{g}}(σ^{g}) ≃ i_{G^{0},M}(σ).

Lemma 4.3. Let π be an irreducible constituent of i_{G^{0},M}(σ) and let G_{π} = {x ∈ G : π^{x} ≃ π}. Then

[G_{π}/G^{0}] = [N_{G_{π}}(σ)/N_{G^{0}}(σ)]

where N_{G^{0}}(σ) = N_{G}(σ) ∩ G^{0} and N_{G_{π}}(σ) = N_{G}(σ) ∩ G_{π}.

Proof: Consider the mapping from N_{G_{π}}(σ) to G_{π}/G^{0} given by g ↦ gG^{0}. Its kernel is N_{G_{π}}(σ) ∩ G^{0} = N_{G^{0}}(σ). Further, given g ∈ G_{π}, π^{g} ≃ π occurs in i_{G^{0},M}(σ), so by Lemma 4.1 there is x_{0} ∈ G_{0} such that gx_{0} ∈ N_{G}(σ). But π^{gx_{0}} ≃ π^{g} ≃ π so that gx_{0} ∈ G_{π} ∩ N_{G}(σ) = N_{G_{π}}(σ) and gx_{0}G^{0} = gG^{0}. Thus the mapping is surjective.

Let C(σ) denote the algebra of G-intertwining operators for i_{G,M}(σ) and let C^{0}(σ) denote the algebra of G^{0}-intertwining operators for i_{G^{0},M}(σ).

Lemma 4.4. dim C(σ) = dim C^{0}(σ)[N_{G}(σ)/N_{G^{0}}(σ)].

Proof: Let

i_{G^{0},M}(σ) = ∑_{π∈S(σ)} m_{π}π

be the decomposition of i_{G^{0},M}(σ) into irreducible constituents. For π_{1}, π_{2} ∈ S(σ), we will say that π_{1} ∼ π_{2} if there is g ∈ G such that π_{2} ≃ π_{1}^{g}. Then using Lemma 4.2, we have π_{1} ∼ π_{2} if and only if π_{2} ≃ π_{1}^{g} for some g ∈ N_{G}(σ). We can write

i_{G^{0},M}(σ) = ∑_{π∈S(σ)/∼} m_{π} ∑_{g∈N_{G}(σ)/N_{Gπ} (σ)} π^{g}.

Now

i_{G,M}(σ) ≃ i_{G,G^{0}}(i_{G^{0},M}(σ)) ≃ ∑_{π∈S(σ)/∼} m_{π} ∑_{g∈N_{G}(σ)/N_{Gπ} (σ)} i_{G,G^{0}}(π^{g})

= ∑_{π∈S(σ)/∼} m_{π}[N_{G}(σ)/N_{G_{π}}(σ)]r_{π} ∑_{χ∈X/X(Π_{π})} Π_{π} ⊗ χ

where Π_{π} is an irreducible representation of G such that π ⊂ Π_{π}|_{G^{0}}. Because of Lemma 4.1, the representations Π_{π} ⊗ χ are pairwise inequivalent as π ranges over S(σ)/ ∼ and χ ranges over X/X(Π_{π}). Thus

dim C(σ) = ∑_{π∈S(σ)/∼} m_{π}^{2}[N_{G}(σ)/N_{G_{π}}(σ)]^{2}r_{π}^{2}[X/X(Π_{π})].

But by Lemmas 2.13 and 4.3,

r_{π}^{2}[X/X(Π_{π})] = [G_{π}/G^{0}] = [N_{G_{π}}(σ)/N_{G^{0}}(σ)].

Thus

dim C(σ) = ∑_{π∈S(σ)/∼} m_{π}^{2}[N_{G}(σ)/N_{G_{π}}(σ)]^{2}[N_{G_{π}}(σ)/N_{G^{0}}(σ)]

= [N_{G}(σ)/N_{G^{0}}(σ)] ∑_{π∈S(σ)/∼} m_{π}^{2}[N_{G}(σ)/N_{G_{π}}(σ)]

= [N_{G}(σ)/N_{G^{0}}(σ)] dim C^{0}(σ).

We now want to find a basis for C(σ). We proceed as in the connected case. Let A be the split component of the center of M and write \underline{a}_{C}^{∗} for the dual of its complex Lie algebra. Each ν ∈ \underline{a}_{C}^{∗} determines a one-dimensional character χ_{ν} of M which is defined by

χ_{ν}(m) = q^{<H_{P} (m),ν>}, m ∈ M.

We write (I_{P} (σ : ν), H_{P} (σ : ν)) and (I_{P}^{0} (σ : ν), H_{P}^{0} (σ : ν)) for the induced representations of G and G^{0} as above corresponding to σ_{ν} = σ ⊗ χ_{ν}. Let K be a good maximal compact subgroup of G^{0} with respect to a minimal parabolic pair (P_{0}, A_{0}) of G^{0} such that P_{0} ⊂ P, A ⊆ A_{0}. Then G^{0} = KP and we also have the usual compact realization of I_{P}^{0} (σ) on H_{P}^{K}(σ) =

{f_{K} ∈ C^{∞}(K, V) : f_{K}(kmn) = σ^{−1}(m)f_{K}(k) for all k ∈ K, m ∈ M ∩ K, n ∈ N ∩ K}.

The intertwining operators between H_{P}^{0} (σ, ν) and H_{P}^{K}(σ) are given by

F_{P}^{K}(ν) : H_{P}^{0} (σ : ν) → H_{P}^{K}(σ),
F_{P}^{K}(ν)ϕ(k) = ϕ(k), ϕ ∈ H_{P}^{0} (σ : ν), k ∈ K.

For all f_{K} ∈ H_{P}^{K}(σ), x ∈ G^{0},

F_{P}^{K}(ν)^{−1}f_{K}(x) = δ_{P}^{− \frac{1}{2}} (μ(x))σ_{ν}^{−1}(μ(x))f_{K}(κ(x)).

Here for any x ∈ G^{0}, κ(x) ∈ K, μ(x) ∈ M are chosen so that x ∈ κ(x)μ(x)N.

Since we don’t know whether there is a “good” maximal compact subgroup for G, we don’t have a single compact realization for I_{P} (σ). However we can proceed one coset at a time as follows. Write G as a disjoint union of cosets, G = ∪_{i=1}^{k}x_{i}G^{0}. For any f ∈ H_{P} (σ, ν), 1 ≤ i ≤ k, we can define

f_{i}(x) = \{ \begin{matrix}f(x), & if x ∈ x_{i}G^{0}; \\ 0, & otherwise. \end{matrix}

Then f_{i} ∈ H_{P} (σ : ν) for each i and f = ∑_{i=1}^{k} f_{i}. For each 1 ≤ i ≤ k, define

F_{P}^{i} (ν) : H_{P} (σ : ν) → H_{P}^{K}(σ)

by

F_{P}^{i} (ν)f(k) = f(x_{i}k), f ∈ H_{P} (σ : ν), k ∈ K.

Define

F_{P}^{i} (ν)^{−1} : H_{P}^{K}(σ) → H_{P} (σ : ν)

by

F_{P}^{i} (ν)^{−1}f_{K}(x) = \{ \begin{matrix}δ_{P}^{− \frac{1}{2}} (μ(x_{0}))σ_{ν}^{−1}(μ(x_{0}))f_{K}(κ(x_{0})), & if x = x_{i}x_{0}, x_{0} ∈ G^{0}; \\ 0, & otherwise. \end{matrix}

Then F_{P}^{i} (ν)F_{P}^{i} (ν)^{−1}f_{K} = f_{K} for all f_{K} ∈ H_{P}^{K}(σ) and F_{P}^{i} (ν)^{−1}F_{P}^{i} (ν)f = f_{i} for all f ∈ H_{P} (σ : ν).

If P = MN and P′ = MN′ are two parabolic subgroups of G^{0} with Levi component M, then we have the formal intertwining operators

J^{0}(P′ : P : σ : ν) : H_{P}^{0} (σ : ν) → H_{P′}^{0}(σ : ν)

given by the standard integral formula

J^{0}(P′ : P : σ : ν)ϕ(x) = ∫_{\overline{N}∩N′} ϕ(x\overline{n})d\overline{n}, x ∈ G^{0}.

Here M\overline{N} is the opposite parabolic to P and d\overline{n} is normalized Haar measure on \overline{N} ∩N′. We can define

J(P′ : P : σ : ν) : H_{P} (σ : ν) → H_{P′}(σ : ν)

by the same formula

J(P′ : P : σ : ν)f(x) = ∫_{\overline{N}∩N′} f(x\overline{n})d\overline{n}, x ∈ G.

In order to talk rigorously about holomorphicity and analytic continuation of these operators we transfer them to the compact realizations. Thus we have

J_{K}^{0}(P′ : P : σ : ν) : H_{P}^{K}(σ) → H_{P}^{K}(σ)

defined for f_{K} ∈ H_{P}^{K}(σ), k ∈ K, by

J_{K}^{0}(P′ : P : σ : ν)f_{K}(k) = F_{P′}^{K}(ν)J^{0}(P′ : P : σ : ν)F_{P}^{K}(ν)^{−1}f_{K}(k) =
∫_{\overline{N}∩N′} δ_{P}^{− \frac{1}{2}} (μ(k\overline{n}))σ_{ν}^{−1}(μ(k\overline{n}))f_{K}(κ(k\overline{n}))d\overline{n}.

Theorem 4.5. (Harish-Chandra [7,13]) Suppose σ is an irreducible discrete series representation of M. There is a chamber \underline{a}_{C}^{∗}(P′ : P) in \underline{a}_{C}^{∗} such that for ν ∈ \underline{a}_{C}^{∗}(P′ : P) the formal intertwining operator J^{0}(P′ : P : σ : ν) converges and defines a bounded operator. For a fixed f_{K} ∈ H_{P}^{K}(σ), the mapping

ν ↦ J_{K}^{0}(P′ : P : σ : ν)f_{K}

from \underline{a}_{C}^{∗}(P′ : P) to H_{P}^{K}(σ) is holomorphic. Further, it extends to a meromorphic function on all of \underline{a}_{C}^{∗}.

Because of Harish-Chandra’s theorem we can define J^{0}(P′ : P : σ : ν) for all ν ∈ \underline{a}_{C}^{∗} by

J^{0}(P′ : P : σ : ν) = F_{P′}^{K}(ν)^{−1}J_{K}^{0}(P′ : P : σ : ν)F_{P}^{K}(ν).

Corollary 4.6. Suppose σ is an irreducible discrete series representation of M. Then the formal intertwining operator J(P′ : P : σ : ν) converges and defines a bounded operator for ν ∈ \underline{a}_{C}^{∗}(P′ : P). Further, for all ν ∈ \underline{a}_{C}^{∗}(P′ : P), we have

J(P′ : P : σ : ν) = ∑_{i=1}^{k} F_{P′}^{i}(ν)^{−1}J_{K}^{0}(P′ : P : σ : ν)F_{P}^{i} (ν).

Proof: It is clear from the definitions that for ν ∈ \underline{a}_{C}^{∗}(P′ : P), f ∈ H_{P} (σ : ν), x ∈ G, x_{0} ∈ G^{0},

J(P′ : P : σ : ν)f(xx_{0}) = J^{0}(P′ : P : σ : ν)ϕ(x_{0})

where ϕ = l(x^{−1})f|_{G^{0}} is the restriction to G^{0} of the left translate of f by x^{−1}. Thus the integral converges. Fix 1 ≤ i ≤ k. An elementary calculation shows that

F_{P′}^{i}(ν)J(P′ : P : σ : ν)F_{P}^{i} (ν)^{−1} = J_{K}^{0}(P′ : P : σ : ν).

Thus for any f ∈ H_{P} (σ), since clearly J(P′ : P : σ : ν)(f_{i}) = (J(P′ : P : σ : ν)f)_{i} we have

F_{P′}^{i}(ν)^{−1}J_{K}^{0}(P′ : P : σ : ν)F_{P}^{i} (ν)f
= F_{P′}^{i}(ν)^{−1}F_{P′}^{i}(ν)J(P′ : P : σ : ν)F_{P}^{i} (ν)^{−1}F_{P}^{i} (ν)ϕ = (J(P′ : P : σ : ν)f)_{i}.

Because of Corollary 4.6 we can define J(P′ : P : σ : ν) for all ν ∈ \underline{a}_{C}^{∗} by the formula

J(P′ : P : σ : ν) = ∑_{i=1}^{k} F_{P′}^{i}(ν)^{−1}J_{K}^{0}(P′ : P : σ : ν)F_{P}^{i} (ν).

Corollary 4.7. Let ν ∈ \underline{a}_{C}^{∗}, f ∈ H_{P} (σ : ν), x ∈ G, x_{0} ∈ G^{0}. Then

J(P′ : P : σ : ν)f(xx_{0}) = J^{0}(P′ : P : σ : ν)ϕ(x_{0})

where ϕ = l(x^{−1})f|_{G^{0}}.

Proof: Suppose that 1 ≤ j ≤ k. Because J^{0}(P′ : P : σ : ν) commutes with the left action of G^{0}, it is enough to prove the result for x = x_{j}. For any ν ∈ \underline{a}_{C}^{∗}, f ∈ H_{P} (σ : ν), x_{0} ∈ G^{0}, J(P′ : P : σ : ν)f(x_{j}x_{0}) =

∑_{i=1}^{k} F_{P′}^{i}(ν)^{−1}J_{K}^{0}(P′ : P : σ : ν)F_{P}^{i} (ν)f(x_{j}x_{0}) = F_{P′}^{j}(ν)^{−1}J_{K}^{0}(P′ : P : σ : ν)F_{P}^{j} (ν)f(x_{j}x_{0})

= F_{P′}^{K}(ν)^{−1}J_{K}^{0}(P′ : P : σ : ν)F_{P}^{K}(ν)ϕ_{j}(x_{0})

where ϕ_{j} = L(x_{j}^{−1})f|_{G^{0}}.

Fix scalar normalizing factors r(P′ : P : σ : ν) as in [2] used to define the normalized intertwining operators

R^{0}(P′ : P : σ) = r(P′ : P : σ : 0)^{−1}J^{0}(P′ : P : σ : 0).

Using Corollary 4.7, the fact that r(P′ : P : σ : ν)^{−1}J^{0}(P′ : P : σ : ν) is holomorphic and non-zero at ν = 0 will imply that r(P′ : P : σ : ν)^{−1}J(P′ : P : σ : ν) is also holomorphic and non-zero at ν = 0. Thus we can use the same normalizing factors to define

R(P′ : P : σ) = r(P′ : P : σ : 0)^{−1}J(P′ : P : σ : 0).

For the normalized intertwining operators we will also have the formula

R(P′ : P : σ)f(xx_{0}) = R^{0}(P′ : P : σ)ϕ(x_{0})

where notation is as in Corollary 4.7.

Lemma 4.8. Suppose P_{1}, P_{2}, and P_{3} are parabolic subgroups of G^{0} with Levi component M. Then R(P_{1} : P_{3} : σ) = R(P_{1} : P_{2} : σ)R(P_{2} : P_{3} : σ).

Proof: This follows easily using Corollary 4.7 from the corresponding formula for the connected case.

Lemma 4.9. Suppose P_{1} and P_{2} are parabolic subgroups of G^{0} with Levi component M. Then R(P_{2} : P_{1} : σ) is an equivalence from H_{P_{1}}(σ) onto H_{P_{2}}(σ).

Proof: It follows from Lemma 4.8 that

R(P_{1} : P_{2} : σ)R(P_{2} : P_{1} : σ) = R(P_{1} : P_{1} : σ).

But it follows from the integral formula that J(P_{1} : P_{1} : σ : ν) is the identity operator for ν in the region of convergence, and hence for all ν. Thus R(P_{1} : P_{1} : σ) is a non-zero constant times the identity operator and so R(P_{2} : P_{1} : σ) is invertible.

Let x ∈ N_{G}(M). Then if P_{1} = MN_{1} is a parabolic subgroup of G^{0} with Levi component M, so is P_{2} = xP x^{−1} = xMx^{−1}xN_{1}x^{−1} = MN_{2}. Let dn_{1} and dn_{2} denote normalized Haar measure on N_{1} and N_{2} respectively. (Thus dn_{i} assigns measure one to K ∩ N_{i}, i = 1, 2.) Define α_{P_{1}}(x) ∈ R^{+} by

∫_{N_{2}} ϕ(n_{2})dn_{2} = α_{P_{1}}(x) ∫_{N_{1}} ϕ(xn_{1}x^{−1})dn_{1}, ϕ ∈ C_{c}^{∞}(N_{2}).

For all m ∈ M,

α_{P_{1}}(m) = δ_{P_{1}}(m).

Further, if x ∈ N_{G}(K) ∩ N_{G}(M), then α_{P_{1}}(x) = 1. The following lemma is an easy consequence of the definition.

Lemma 4.10. Let x, y ∈ N_{G}(M). Then

α_{P} (yx) = α_{xP x^{−1}}(y)α_{P} (x).

Moreover if m ∈ M, y ∈ N_{G}(M), then

α_{P} (ym) = α_{P} (y)δ_{P} (m).

Now let N_{G}(σ) = {g ∈ N_{G}(M) : σ^{g} ≃ σ} and let W_{G}(σ) = N_{G}(σ)/M. Let W_{G^{0}}(σ) = (N_{G}(σ) ∩ G^{0})/M. If w ∈ W_{G}(σ), σ can be extended to a representation of the group M_{w} generated by M and any representative n_{w} for w in N_{G}(σ). Fix such an extension and denote it by σ_{w}. Now we can define an intertwining operator

A_{P} (w) : H_{w^{−1}P w}(σ) → H_{P} (σ)

by

(A_{P} (w)f)(x) = σ_{w}(n_{w})α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} f(xn_{w}).

The α_{w^{−1}P w} term is not used in the connected case because coset representatives n_{w} can be chosen in K where α_{w^{−1}P w} = 1. In the general case we don’t know if there is a natural choice of coset representatives with α_{w^{−1}P w} = 1. Thus we add the α_{w^{−1}P w} term so that A_{P} (w) is independent of the coset representative n_{w} for w. A_{P} (w) does however depend on the choice of the extension σ_{w}.

Lemma 4.11. The intertwining operator A_{P} (w) is independent of the choice of coset representative n_{w} for w ∈ W_{G}(σ). For w_{1}, w_{2} ∈ W_{G}(σ) there is a non-zero complex constant c_{P} (w_{1}, w_{2}) so that

A_{P} (w_{1}w_{2}) = c_{P} (w_{1}, w_{2})A_{P} (w_{1})A_{w_{1}^{−1}P w_{1}}(w_{2}).

Proof: Using Lemma 4.10 we have for any m ∈ M, f ∈ H_{w^{−1}P w}(σ),

σ_{w}(n_{w}m)α_{w^{−1}P w}(n_{w}m)^{\frac{1}{2}} f(xn_{w}m)
= σ_{w}(n_{w})σ(m)α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} δ_{w^{−1}P w}(m)^{\frac{1}{2}} σ(m)^{−1}δ_{w^{−1}P w}(m)^{− \frac{1}{2}} f(xn_{w})
= σ_{w}(n_{w})α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} f(xn_{w}).

Thus the intertwining operator is independent of the choice of coset representative.

Let w_{1}, w_{2} ∈ W_{G}(σ) and fix coset representatives n_{1}, n_{2} for w_{1}, w_{2} respectively. Then we can use n_{1}n_{2} as a coset representative for w_{1}w_{2}. We will also write P_{1} = w_{1}^{−1}P w_{1} and P_{12} = w_{2}^{−1}w_{1}^{−1}P w_{1}w_{2}. For any w ∈ W_{G}(σ) and representative n_{w} for w we have

σ_{w}(n_{w})σ(m)σ_{w}(n_{w})^{−1} = σ(n_{w}mn_{w}^{−1})

for all m ∈ M. Thus

σ_{w_{1}w_{2}}(n_{1}n_{2})σ_{w_{2}}(n_{2})^{−1}σ_{w_{1}}(n_{1})^{−1}

is a non-zero self-intertwining operator for σ and hence there is a non-zero constant c so that

σ_{w_{1}w_{2}}(n_{1}n_{2}) = cσ_{w_{1}}(n_{1})σ_{w_{2}}(n_{2}).

Further, from Lemma 4.10 we have

α_{P_{12}}(n_{1}n_{2}) = α_{P_{1}}(n_{1})α_{P_{12}}(n_{2}).

By definition, for any x ∈ G, f ∈ H_{P_{12}}(σ),

A_{P} (w_{1}w_{2})f(x) = σ_{w_{1}w_{2}}(n_{1}n_{2})α_{P_{12}}(n_{1}n_{2})^{\frac{1}{2}} f(xn_{1}n_{2})
= cσ_{w_{1}}(n_{1})α_{P_{1}}(n_{1})^{\frac{1}{2}} σ_{w_{2}}(n_{2})α_{P_{12}}(n_{2})^{\frac{1}{2}} f(xn_{1}n_{2})
= cA_{P} (w_{1})A_{w_{1}^{−1}P w_{1}}(w_{2})f(x).

If w ∈ W_{G^{0}}(σ) and n_{w} is chosen to be in K, then we also have

A^{0}(w) : H_{w^{−1}P w}^{0}(σ) → H_{P}^{0} (σ)

given by

(A^{0}(w)ϕ)(x) = σ_{w}(n_{w})ϕ(xn_{w}).

The compositions

R^{0}(w_{0}, σ) = A^{0}(w_{0})R^{0}(w_{0}^{−1}P w_{0} : P : σ), w_{0} ∈ W_{G^{0}}(σ),
R(w, σ) = A_{P} (w)R(w^{−1}P w : P : σ), w ∈ W_{G}(σ),

give self-intertwining operators for I_{P}^{0} (σ) and I_{P} (σ) respectively.

Lemma 4.12. There is a cocycle η so that

R(w_{1}w_{2}, σ) = η(w_{1}, w_{2})R(w_{1}, σ)R(w_{2}, σ)

for all w_{1}, w_{2} ∈ W_{G}(σ).

Proof: Using Lemmas 4.8 and 4.11, we have

R(w_{1}w_{2}, σ) = A_{P} (w_{1}w_{2})R(w_{2}^{−1}w_{1}^{−1}P w_{1}w_{2} : P : σ)

= c_{P} (w_{1}, w_{2})A_{P} (w_{1})A_{w_{1}^{−1}P w_{1}}(w_{2})R(w_{2}^{−1}w_{1}^{−1}P w_{1}w_{2} : w_{2}^{−1}P w_{2} : σ)R(w_{2}^{−1}P w_{2} : P : σ).

We will show that there is a non-zero constant c′_{P} (w_{1}, w_{2}) so that

A_{w_{1}^{−1}P w_{1}}(w_{2})R(w_{2}^{−1}w_{1}^{−1}P w_{1}w_{2} : w_{2}^{−1}P w_{2} : σ)A_{P} (w_{2})^{−1}
= c′_{P} (w_{1}, w_{2})R(w_{1}^{−1}P w_{1} : P : σ).

When this is established we will have R(w_{1}w_{2}, σ) =

c_{P} (w_{1}, w_{2})c′_{P} (w_{1}, w_{2})A_{P} (w_{1})R(w_{1}^{−1}P w_{1} : P : σ)A_{P} (w_{2})R(w_{2}^{−1}P w_{2} : P : σ)
= c_{P} (w_{1}, w_{2})c′_{P} (w_{1}, w_{2})R(w_{1}, σ)R(w_{2}, σ).

Thus η can be defined by η(w_{1}, w_{2}) = c_{P} (w_{1}, w_{2})c′_{P} (w_{1}, w_{2}). It is immediate from the formulas for the composition of the operators R(w_{i}, σ) that η is a 2–cocycle.

In order to prove the above identity we need to go back to the original definition of the standard intertwining operators. W_{G}(σ) acts on \underline{a}_{C}^{∗} and χ_{ν}(n_{w}^{−1}mn_{w}) = χ_{wν}(m) if n_{w} is any representative of w ∈ W_{G}(σ). Now for any ν ∈ \underline{a}_{C}^{∗}, f ∈ H_{w^{−1}P w}(σ, ν), x ∈ G, m ∈ M, n ∈ N, if A_{P} (w) is defined exactly as above, we have

A_{P} (w)f(xmn) = σ_{w}(n_{w})α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} f(xmnn_{w})

= σ_{w}(n_{w})α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} δ_{w^{−1}P w}(n_{w}^{−1}m^{−1}n_{w})^{− \frac{1}{2}} σ(n_{w}^{−1}m^{−1}n_{w})χ_{ν}(n_{w}^{−1}m^{−1}n_{w})f(xn_{w})

= δ_{P} (m)^{− \frac{1}{2}} σ(m)^{−1}χ_{wν}(m)^{−1}σ_{w}(n_{w})α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} f(xn_{w})
= δ_{P} (m)^{− \frac{1}{2}} σ(m)^{−1}χ_{wν}(m)^{−1}A_{P} (w)f(x).

Thus A_{P} (w) maps H_{w^{−1}P w}(σ, ν) to H_{P} (σ, wν). Now write P_{1} = w_{1}^{−1}P w_{1}, P_{2} = w_{2}^{−1}P w_{2}, P_{12} = w_{2}^{−1}w_{1}^{−1}P w_{1}w_{2} and consider the composition

A_{P_{1}}(w_{2})J(P_{12} : P_{2} : σ : w_{2}^{−1}ν)A_{P} (w_{2})^{−1}

It maps

H_{P} (σ, ν) → H_{P_{2}}(σ, w_{2}^{−1}ν) → H_{P_{12}}(σ, w_{2}^{−1}ν) → H_{P_{1}}(σ, ν).

If ν is in the region of convergence for the integral formula for J(P_{12} : P_{2} : σ : w_{2}^{−1}ν) and d\overline{n}_{2} denotes normalized Haar measure on \overline{N}_{2} ∩ N_{12}, then we have

A_{P_{1}}(w_{2})J(P_{12} : P_{2} : σ : w_{2}^{−1}ν)A_{P} (w_{2})^{−1}f(x)
= σ_{w_{2}}(n_{w_{2}})α_{P_{12}}(n_{w_{2}})^{\frac{1}{2}} J(P_{12} : P_{2} : σ : w_{2}^{−1}ν)A_{P} (w_{2})^{−1}f(xn_{w_{2}})
= σ_{w_{2}}(n_{w_{2}})α_{P_{12}}(n_{w_{2}})^{\frac{1}{2}} ∫_{\overline{N}_{2}∩N_{12}} A_{P} (w_{2})^{−1}f(xn_{w_{2}} \overline{n}_{2})d\overline{n}_{2}
= σ_{w_{2}}(n_{w_{2}})α_{P_{12}}(n_{w_{2}})^{\frac{1}{2}} ∫_{\overline{N}_{2}∩N_{12}} α_{P_{2}}(n_{w_{2}})^{− \frac{1}{2}} σ_{w_{2}}(n_{w_{2}})^{−1}f(xn_{w_{2}} \overline{n}_{2}n_{w_{2}}^{−1})d\overline{n}_{2}
= α_{P_{12}}(n_{w_{2}})^{\frac{1}{2}} α_{P_{2}}(n_{w_{2}})^{− \frac{1}{2}} ∫_{\overline{N}_{2}∩N_{12}} f(xn_{w_{2}} \overline{n}_{2}n_{w_{2}}^{−1})d\overline{n}_{2}.

But \overline{N}_{2} ∩N_{12} = w_{2}^{−1}(\overline{N} ∩N_{1})w_{2} so if d\overline{n} denotes normalized Haar measure on \overline{N} ∩N_{1}, there is a positive real number r so that

∫_{\overline{N}_{2}∩N_{12}} ϕ(n_{w_{2}} \overline{n}_{2}n_{w_{2}}^{−1})d\overline{n}_{2} = r ∫_{\overline{N}∩N_{1}} ϕ(\overline{n})d\overline{n}

for all ϕ ∈ C_{c}^{∞}(\overline{N} ∩ N_{1}). Thus

A_{P_{1}}(w_{2})J(P_{12} : P_{2} : σ : w_{2}^{−1}ν)A_{P} (w_{2})^{−1}f(x)
= α_{P_{12}}(n_{w_{2}})^{\frac{1}{2}} α_{P_{2}}(n_{w_{2}})^{− \frac{1}{2}} r ∫_{\overline{N}∩N_{1}} f(x\overline{n})d\overline{n}
= α_{P_{12}}(n_{w_{2}})^{\frac{1}{2}} α_{P_{2}}(n_{w_{2}})^{− \frac{1}{2}} rJ(P_{1} : P : σ : ν)f(x).

Setting

c′′_{P} (w_{1}, w_{2}) = α_{P_{12}}(n_{w_{2}})^{\frac{1}{2}} α_{P_{2}}(n_{w_{2}})^{− \frac{1}{2}} r

we have

A_{P_{1}}(w_{2})J(P_{12} : P_{2} : σ : w_{2}^{−1}ν)A_{P} (w_{2})^{−1} = c′′_{P} (w_{1}, w_{2})J(P_{1} : P : σ : ν)

for all ν in the region of convergence for the integral formula for J(P_{12} : P_{2} : σ : w_{2}^{−1}ν). By analytic continuation, the identity is valid for all ν. Now divide both sides of the equation by r(P_{12} : P_{2} : σ : w_{2}^{−1}ν) and evaluate at ν = 0. We obtain

A_{P_{1}}(w_{2})R(P_{12} : P_{2} : σ)A_{P} (w_{2})^{−1} = c′_{P} (w_{1}, w_{2})R(P_{1} : P : σ)

where

c′_{P} (w_{1}, w_{2}) = c′′_{P} (w_{1}, w_{2})r(P_{1} : P : σ : 0)r(P_{12} : P_{2} : σ : 0)^{−1}.

For ϕ ∈ H_{P}^{0} (σ) define f = Φ(ϕ) ∈ H_{P} (σ) such that f(x) = 0 if x ∉ G^{0} and f(x_{0}) = ϕ(x_{0}) if x_{0} ∈ G^{0}.

Lemma 4.13. Let f = Φ(ϕ) as above. Then for all w, w_{1} ∈ W_{G}(σ), x_{0} ∈ G^{0},

R(w, σ)f(x_{0}n_{w_{1}}^{−1}) = 0

unless w = w_{1}w_{0}, w_{0} ∈ W_{G^{0}}(σ). If w = w_{1}w_{0}, w_{0} ∈ W_{G^{0}}(σ), then R(w, σ)f(x_{0}n_{w_{1}}^{−1}) =

η(w_{1}, w_{0})σ_{w_{1}}(n_{w_{1}})α_{w_{1}^{−1}P w_{1}}(n_{w_{1}})^{\frac{1}{2}} R^{0}(w_{1}^{−1}P w_{1} : P : σ)R^{0}(w_{0} : σ)ϕ(x_{0}).

Proof: For any x ∈ G, w ∈ W_{G}(σ), using Corollary 4.7, R(w^{−1}P w : P : σ)f(x) = R^{0}(w^{−1}P w : P : σ)ϕ′(1) where ϕ′ is the restriction to G^{0} of l(x^{−1})f. Now since f = Φ(ϕ) is supported on G^{0}, ϕ′ = 0 unless x ∈ G^{0}. If x_{0} ∈ G^{0}, then R(w^{−1}P w : P : σ)f(x_{0}) = R^{0}(w^{−1}P w : P : σ)ϕ(x_{0}).

Now by definition, R(w, σ)f(x_{0}n_{w_{1}}^{−1}) =

σ_{w}(n_{w})α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} R(w^{−1}P w : P : σ)f(x_{0}n_{w_{1}}^{−1}n_{w}).

By the above this is zero unless n_{w_{1}}^{−1}n_{w} ∈ G^{0}, that is unless w = w_{1}w_{0}, w_{0} ∈ W_{G^{0}}(σ). In this case, using Lemma 4.12,

R(w_{1}w_{0}, σ)f(x_{0}n_{w_{1}}^{−1}) = η(w_{1}, w_{0})R(w_{1} : σ)R(w_{0} : σ)f(x_{0}n_{w_{1}}^{−1})
= η(w_{1}, w_{0})σ_{w_{1}}(n_{w_{1}})α_{w_{1}^{−1}P w_{1}}(n_{w_{1}})^{\frac{1}{2}} R(w_{1}P w_{1}^{−1} : P : σ)R(w_{0} : σ)f(x_{0})
= η(w_{1}, w_{0})σ_{w_{1}}(n_{w_{1}})α_{w_{1}^{−1}P w_{1}}(n_{w_{1}})^{\frac{1}{2}} R^{0}(w_{1}P w_{1}^{−1} : P : σ)R^{0}(w_{0} : σ)ϕ(x_{0}).

Recall that if we write W_{G^{0}}^{0}(σ) for the subgroup of elements w ∈ W_{G^{0}}(σ) such that R^{0}(w, σ) is scalar, then W_{G^{0}}^{0}(σ) = W (Φ_{1}) is generated by reflections in a set Φ_{1} of reduced roots of (G, A). Let Φ^{+} be the positive system of reduced roots of (G, A) determined by P and let Φ_{1}^{+} = Φ_{1} ∩ Φ^{+}. If we define

R_{σ}^{0} = {w ∈ W_{G^{0}}(σ) : wβ ∈ Φ^{+} for all β ∈ Φ_{1}^{+}},

then W_{G^{0}}(σ) is the semidirect product of R_{σ}^{0} and W (Φ_{1}). R_{σ}^{0} is called the R-group for I_{P}^{0} (σ) and the operators

{R^{0}(r, σ), r ∈ R_{σ}^{0}}

form a basis for the algebra of intertwining operators of I_{P}^{0} (σ). We will define

R_{σ} = {w ∈ W_{G}(σ) : wβ ∈ Φ^{+} for all β ∈ Φ_{1}^{+}}.

Clearly R_{σ} ∩ G^{0} = R_{σ}^{0}.

Lemma 4.14. R(w, σ) is scalar if w ∈ W (Φ_{1}), and W_{G}(σ) is the semidirect product of W (Φ_{1}) and R_{σ}.

Proof: Fix w ∈ W (Φ_{1}), n_{w} ∈ K a representative for w, and f ∈ H_{P} (σ). Then for all x ∈ G,

R(w, σ)f(x) = σ_{w}(n_{w})R(w^{−1}P w : P : σ)f(xn_{w}) = σ_{w}(n_{w})R^{0}(w^{−1}P w : P : σ)ϕ(n_{w}) =

where ϕ(x_{0}) = f(xx_{0}) for all x_{0} ∈ G^{0}. Thus

R(w, σ)f(x) = R^{0}(w, σ)ϕ(1).

But since w ∈ W (Φ_{1}) there is a constant c_{w} such that R^{0}(w, σ)ϕ = c_{w}ϕ for all ϕ ∈ H_{P}^{0} (σ). Thus

R(w, σ)f(x) = c_{w}ϕ(1) = c_{w}f(x)

so R(w, σ) is scalar.

We must show that for any w ∈ W_{G}(σ), wΦ_{1} = Φ_{1} so that

R_{σ} = {w ∈ W_{G}(σ) : wΦ_{1}^{+} = Φ_{1}^{+}}.

Then as in the connected case it will be clear that W_{G}(σ) is the semidirect product of R_{σ} and W (Φ_{1}). Let w ∈ W_{G}(σ), α ∈ Φ_{1}, and let s_{α} ∈ W (Φ_{1}) denote the reflection corresponding to α. Then ws_{α}w^{−1} = s_{wα} ∈ W_{G^{0}}(σ). But R(s_{α} : σ) is scalar so that using Lemma 4.12, so is R(s_{wα}, σ) = R(ws_{α}w^{−1} : σ). This implies as above that R^{0}(s_{wα}, σ) is scalar. Hence s_{wα} ∈ W (Φ_{1}) and wα ∈ Φ_{1}.

Lemma 4.15. The dimension of C(σ) is equal to [R_{σ}].

Proof: By Lemma 4.4,

dim C(σ) = [N_{G}(σ)/N_{G^{0}}(σ)] dim C^{0}(σ) = [W_{G}(σ)/W_{G^{0}}(σ)][R_{σ}^{0}]
= [W_{G}(σ)/W_{G^{0}}(σ)][W_{G^{0}}(σ)/W (Φ_{1})] = [W_{G}(σ)/W (Φ_{1})] = [R_{σ}].

Theorem 4.16. The operators {R(r, σ), r ∈ R_{σ}} form a basis for the algebra of intertwining operators of I_{P} (σ).

Proof: By Lemma 4.15 it suffices to show that the operators are linearly independent. Suppose that c_{w}, w ∈ R_{σ}, are constants so that

∑_{w∈R_{σ}} c_{w}R(w, σ)f(x) = 0

for all f ∈ H_{P} (σ), x ∈ G. Fix w_{1} ∈ R_{σ}. Then for all f = Φ(ϕ) with ϕ ∈ H_{P}^{0} (σ) and all x_{0} ∈ G^{0}, we have

∑_{w∈R_{σ}} c_{w}R(w, σ)f(x_{0}n_{w_{1}}^{−1}) = 0.

Now by Lemma 4.13, R(w, σ)f(x_{0}n_{w_{1}}^{−1}) = 0 unless w = w_{1}w_{0} where w_{0} ∈ W_{G^{0}}(σ) ∩ R_{σ} = R_{σ}^{0}. Now again using Lemma 4.13,

0 = ∑_{w_{0}∈R_{σ}^{0}} c_{w_{1}w_{0}}R(w_{1}w_{0}, σ)f(x_{0}n_{w_{1}}^{−1})

= ∑_{w_{0}∈R_{σ}^{0}} c_{w_{1}w_{0}}η(w_{1}, w_{0})σ_{w_{1}}(n_{w_{1}})α_{w_{1}^{−1}P w_{1}}(n_{w_{1}})^{\frac{1}{2}} R^{0}(w_{1}^{−1}P w_{1} : P : σ)R^{0}(w_{0} : σ)ϕ(x_{0})

= σ_{w_{1}}(n_{w_{1}})α_{w_{1}^{−1}P w_{1}}(n_{w_{1}})^{\frac{1}{2}} R^{0}(w_{1}^{−1}P w_{1} : P : σ) ∑_{w_{0}∈R_{σ}^{0}} c_{w_{1}w_{0}}η(w_{1}, w_{0})R^{0}(w_{0} : σ)ϕ(x_{0}).

Thus

∑_{w_{0}∈R_{σ}^{0}} c_{w_{1}w_{0}}η(w_{1}, w_{0})R^{0}(w_{0} : σ)ϕ(x_{0}) = 0

for all ϕ ∈ H_{P}^{0} (σ) and all x_{0} ∈ G^{0}. Now since we know that the operators R^{0}(w_{0} : σ) are linearly independent on H_{P}^{0} (σ), we can conclude that the c_{w_{1}w_{0}}η(w_{1}, w_{0}) and hence the c_{w_{1}w_{0}} are all zero.

As in Arthur [2] we now have to deal with the cocycle η of Lemma 4.12. Fix a finite central extension

1 → Z_{σ} → R̃_{σ} → R_{σ} → 1

over which η splits. Also define the functions ξ_{σ} : R̃_{σ} → C^{∗} and the character χ_{σ} of Z_{σ} as in Arthur [2]. Then we obtain a homomorphism

R̃(r, σ) = ξ_{σ}^{−1}(r)R(r, σ), r ∈ R̃_{σ},

of R̃_{σ} into the group of unitary intertwining operators for I_{P} (σ) which transforms by

R̃(zr, σ) = χ_{σ}(z)^{−1}R̃(r, σ), z ∈ Z_{σ}, r ∈ R̃_{σ}.

Now we can define a representation R of R̃_{σ} × G on H_{P} (σ) given by

R(r, x) = R̃(r, σ)I_{P} (σ, x), r ∈ R̃_{σ}, x ∈ G.

Let Π(R̃_{σ}, χ_{σ}) denote the set of irreducible representations of R̃_{σ} with Z_{σ} central character χ_{σ} and let Π_{σ}(G) denote the set of irreducible constituents of I_{P} (σ).

Theorem 4.17. There is a bijection ρ ↦ π_{ρ} of Π(R̃_{σ}, χ_{σ}) onto Π_{σ}(G) such that

R = ⊕_{ρ∈Π(R̃_{σ},χ_{σ})} (ρ^{∨} ⊗ π_{ρ}).

Proof: Write the decomposition of R into irreducibles as

R = ∑_{ρ,π} m_{ρ,π} (ρ^{∨} ⊗ π)

where ρ runs over Π(R̃_{σ}, χ_{σ}), π runs over Π_{σ}(G), and each m_{ρ,π} ≥ 0. This corresponds to a decomposition

H_{P} (σ) ≃ ∑_{ρ,π}(V_{ρ^{∨}} ⊗ W_{π})^{m_{ρ,π}}

where V_{ρ^{∨}} and W_{π} denote the representation spaces for the irreducible representations ρ^{∨} and π. For each ρ, we also write

H_{ρ} ≃ V_{ρ^{∨}} ⊗ ∑_{π} W_{π}^{m_{ρ,π}}

for the ρ-isotypic component of H_{P} (σ). Each subspace H_{ρ} is invariant under the action of R, in particular by all of the intertwining operators R̃(r, σ), r ∈ R̃_{σ}. Since these intertwining operators span the space C(σ) of G-intertwining operators for H_{P} (σ), each T ∈ C(σ) must satisfy T (H_{ρ}) ⊂ H_{ρ} for every ρ.

We will first show that given π, there is at most one ρ such that m_{ρ,π} > 0. So fix π and suppose that there are ρ_{1} and ρ_{2} such that m_{ρ_{i},π} > 0, i = 1, 2. Then W_{π} occurs as a G-summand of both Hρ_{1} and Hρ_{2}. Thus there is T_{12} ≠ 0 in Hom_{G}(Hρ_{1}, Hρ_{2}). We can extend T_{12} to an element T of C(σ) by setting T = T_{12} on H_{ρ_{1}} and T = 0 on H_{ρ}, ρ ≠ ρ_{1}. Thus there is T in C(σ) such that T (H_{ρ_{1}}) ⊂ H_{ρ_{2}}. But by the remark in the previous paragraph, T (H_{ρ_{1}}) ⊂ H_{ρ_{1}}. Now since T (H_{ρ_{1}}) ≠ 0, we must have ρ_{1} ≃ ρ_{2}.

Now fix ρ and look at H_{ρ} ≃ V_{ρ^{∨}} ⊗ W where W = ∑_{π} W_{π}^{m_{ρ,π}}. We will show that W is irreducible as a G-module. Thus suppose that W = W_{1} ⊕ W_{2} where W_{1}, W_{2} are G-submodules of W. Then we can define T ∈ C(σ) by T (v ⊗ (w_{1} + w_{2})) = v ⊗ w_{1} if v ∈ V_{ρ^{∨}}, w_{1} ∈ W_{1}, w_{2} ∈ W_{2}, and T = 0 on H_{ρ′} if ρ′ ≄ ρ. Now since T ∈ C(σ), we can write T = ∑_{r} c_{r}R̃(r, σ) where r runs over R̃_{σ}. Thus for all v ∈ V_{ρ^{∨}}, w_{1} ∈ W_{1}, w_{2} ∈ W_{2}, we have

v ⊗ w_{1} = T (v ⊗ (w_{1} + w_{2})) = (∑_{r} c_{r}ρ^{∨}(r)v) ⊗ w_{1} + (∑_{r} c_{r}ρ^{∨}(r)v) ⊗ w_{2}.

Suppose W_{2} ≠ 0. This implies that ∑_{r} c_{r}ρ^{∨}(r)v = 0 for all v so that v ⊗ w_{1} = 0 for all w_{1}. Thus W_{1} = {0} and hence W is irreducible. But this implies that m_{ρ,π} ≤ 1 for all π and that there is at most one π such that m_{ρ,π} = 1.

Define

Π_{1} = {ρ ∈ Π(R̃_{σ}, χ_{σ}) : m_{ρ,π} = 1 for some π}.

For each ρ ∈ Π_{1} we have shown that the representation π such that m_{ρ,π} = 1 is unique. Thus we will call it π_{ρ}. Further, we have shown that π_{ρ_{1}} ≃ π_{ρ_{2}} just in case ρ_{1} ≃ ρ_{2}.

Further, by definition of Π_{σ}(G), each π ∈ Π_{σ}(G) occurs in H_{P} (σ) and so must be of the form π_{ρ} for some ρ ∈ Π_{1}. Thus to complete the proof of the theorem we need only show that Π_{1} = Π(R̃_{σ}, χ_{σ}).

Since

H_{P} (σ) ≃ ∑_{ρ∈Π_{1}}(V_{ρ^{∨}} ⊗ W_{π_{ρ}}),

we must have dim C(σ) = ∑_{ρ∈Π_{1}}(deg ρ)^{2}. But since the R(r, σ), r ∈ R_{σ}, form a basis for C(σ), we know that

dim C(σ) = [R_{σ}] = ∑_{ρ∈Π(R̃_{σ},χ_{σ})}(deg ρ)^{2}.

Thus Π_{1} = Π(R̃_{σ}, χ_{σ}).

Remark 4.18 Suppose now that G/G^{0} is cyclic. Then in [1], Arthur predicts a dual group construction of a group R_{ψ,σ}, in terms of the conjectural parameter ψ for the L–packet of σ, which should also describe the components of Ind_{P}^{G}(σ). In particular, R_{ψ,σ} = W_{ψ,σ}/W_{ψ,σ}^{0}, where these groups are defined in terms of centralizers of the image of ψ. Furthermore, Arthur has conjecturally identified W_{ψ,σ}^{0} with W (Φ_{1}) and W_{ψ,σ} with W_{G}(σ). Thus, if the conjectural parameterization exists in the connected case, and Shelstad’s Theorem [12] extends to the p-adic case, then it must be the case that R_{σ} ≃ R_{ψ,σ}. That is, we have shown that there is a group side construction of Arthur’s R–group, if such an object exists. (For more details see [1] and [6], particularly Sections 1 and 4.)

5. R-groups for Ind_{P}^{G}(σ)

In this section we will study representations of G which are induced from discrete series representations of a parabolic subgroup P of G. Thus we revert to the notation that parabolic subgroups of G^{0} are denoted by P^{0}.

Let P = MN be a cuspidal parabolic subgroup of G. Let σ be an irreducible discrete series representation of M and let σ_{0} be an irreducible constituent of the restriction of σ to M^{0}. We want to find a basis for the intertwining algebra C(σ) of the induced representation Ind_{P}^{G}(σ). Since σ is contained in Ind_{M^{0}}^{M}(σ_{0}) we know Ind_{P}^{G}(σ) is contained in Ind_{P}^{G}(Ind_{M^{0}}^{M}(σ_{0})) ≃ Ind_{P^{0}}^{G}(σ_{0}). In §4 we found a basis for the intertwining algebra C(σ_{0}) of Ind_{P^{0}}^{G}(σ_{0}). We will see how to obtain a basis for C(σ) by restricting the intertwining operators defined in §4.

We first need to embed σ in a family σ_{ν}, ν ∈ \underline{a}_{C}^{∗}, where \underline{a} is the real Lie algebra of the split component A of M. Write X(M), X(A) for the groups of rational characters of M, A respectively. Let

r : X(M) ⊗_{Z} R → X(A) ⊗_{Z} R

be the map given by restriction. That is r(χ ⊗ t) = χ|_{A} ⊗ t for χ ∈ X(M), t ∈ R.

Lemma 5.1. The homomorphism r : X(M) ⊗_{Z} R → X(A) ⊗_{Z} R is surjective.

Proof: Since G is a linear group we have an embedding of G in L = GL(V), where V is a finite dimensional F –vector space. Since A is a split torus, the action of A on V can be diagonalized. For any χ ∈ X(A) let V (χ) = {v ∈ V : av = χ(a)v for all a ∈ A}. Let χ_{i}, 1 ≤ i ≤ k, denote the distinct elements of X(A) such that V_{i} = V (χ_{i}) ≠ {0}. We can identify a ∈ A with the block diagonal matrix with diagonal entries χ_{i}(a)I_{d_{i}} where d_{i} = dim V_{i} and I_{d_{i}} denotes the identity matrix of size d_{i}, 1 ≤ i ≤ k.

Since M = C_{G}(A), we have M ⊂ C_{L}(A) ≃ GL(V_{1}) × GL(V_{2}) × ... × GL(V_{k}). For each 1 ≤ i ≤ k we can define det_{i} ∈ X(M) by det_{i}(m_{1}, ..., m_{k}) = det m_{i}. Now det_{i} ⊗d_{i}^{−1} ∈ X(M) ⊗_{Z} R and for a ∈ A, det_{i} ⊗d_{i}^{−1}(a) = χ_{i}(a)^{d_{i}} ⊗ d_{i}^{−1} = χ_{i}(a) ⊗ 1. Thus r(det_{i} ⊗d_{i}^{−1}) = χ_{i} ⊗ 1. The χ_{i}, 1 ≤ i ≤ k, are generators of X(A), although they need not be independent. Thus r is surjective.

Let X_{0}(M) = {χ ∈ X(M) : χ|_{M^{0}} = 1}.

Lemma 5.2. The kernel of r is X_{0}(M) ⊗_{Z} R.

Proof: Suppose χ ⊗ t is in the kernel of r where χ ∈ X(M) and t ∈ R. If t = 0 then χ ⊗ t is the identity element. Assume t ≠ 0. Then |χ(a)|_{F}^{t} = 1 for all a ∈ A implies that |χ(a)|_{F} = 1 for all a ∈ A. Since χ|_{A} is a rational character of a split torus this implies that χ|_{A} = 1. But restriction from X(M^{0}) to X(A) is injective [13, Lemma 0.4.1], so that χ|_{M^{0}} = 1. Thus χ ∈ X_{0}(M).

Recall the homomorphism H_{M^{0}} : M^{0} → Hom(X(M^{0}), Z) defined by

< H_{M^{0}}(m), χ >= log_{q}|χ(m)|_{F} , m ∈ M^{0}, χ ∈ X(M^{0}).

Define an analogous homomorphism H_{M} : M → Hom(X(M), Z) by

< H_{M}(m), χ >= log_{q}|χ(m)|_{F} , m ∈ M, χ ∈ X(M).

Lemma 5.3. Suppose that χ ∈ X_{0}(M). Then < H_{M}(m), χ >= 0 for all m ∈ M.

.

Proof: Let χ ∈ X_{0}(M). Thus χ(m_{0}) = 1 for all m_{0} ∈ M^{0}. Let d be the index of M^{0} in M. Thus m^{d} ∈ M^{0} for all m ∈ M so that χ(m^{d}) = 1 for all m ∈ M. Thus χ(m) is a d^{th} root of unity and |χ(m)|_{F} = 1 for all m ∈ M. Thus < H_{M}(m), χ >= log_{q}|χ(m)|_{F} = 0 for all m ∈ M.

Recall that Hom(X(M^{0}), Z)⊗_{Z}R ≃ Hom(X(A), Z)⊗_{Z}R = \underline{a} is the real Lie algebra of A, \underline{a}^{∗} = X(A) ⊗_{Z} R is its real dual, and \underline{a}_{C}^{∗} = \underline{a}^{∗} ⊗_{R} C is its complex dual. For each ν ∈ \underline{a}_{C}^{∗} we have a character χ_{ν}^{0} of M^{0} defined by χ_{ν}^{0}(m) = q^{<H_{M^{0}}(m),ν>}, m ∈ M^{0}. By Lemmas 5.1 and 5.2 the mapping r above induces an isomomorphism

r_{∗} : \frac{X(M) ⊗_{Z} C}{X_{0}(M) ⊗_{Z} C} ≃ X(A) ⊗_{Z} C ≃ \underline{a}_{C}^{∗}.

By Lemma 5.3, for each m ∈ M, H_{M}(m) is an element of the complex dual of \frac{X(M)⊗_{Z}C}{X_{0}(M)⊗_{Z}C}. Thus for each ν ∈ \underline{a}_{C}^{∗}, we can define a character χ_{ν} of M by

χ_{ν}(m) = q^{<H_{M} (m),r_{∗}^{−1}(ν)>}, m ∈ M.

Lemma 5.4. For all ν ∈ \underline{a}_{C}^{∗}, the restriction of χ_{ν} to M^{0} is χ_{ν}^{0}.

Proof: For m_{0} ∈ M^{0}, χ ∈ X(M) we have

< H_{M}(m_{0}), χ >= log_{q}|χ(m_{0})|_{F} = log_{q}|χ_{0}(m_{0})|_{F} =< H_{M_{0}}(m_{0}), χ_{0} >

where χ_{0} denotes the restriction of χ to M^{0}. Since the isomorphism r_{∗} comes from the restriction map it is easy to see that < H_{M}(m_{0}), r_{∗}^{−1}(ν) >=< H_{M^{0}}(m_{0}), ν > for all m_{0} ∈ M^{0}, ν ∈ \underline{a}_{C}^{∗}.

As above, let σ be an irreducible discrete series representation of M and let σ_{0} be an irreducible constituent of the restriction of σ to M^{0}. Let V_{0} be the representation space for σ_{0} and let

W = {f : M → V_{0} : f(mm_{0}) = σ_{0}(m_{0})^{−1}f(m) for all m ∈ M, m_{0} ∈ M^{0}}.

Then M acts on W by left translation and we will call this induced representation (I_{M}, W). Let V denote the representation space for σ and fix a non-zero intertwining operator S : V → W so that Sσ(m) = I_{M}(m)S for all m ∈ M. Since I_{M} is unitary, we can also define a projection operator P : W → V so that P I_{M}(m) = σ(m)P for all m ∈ M and P Sv = v for all v ∈ V. We also define representation spaces

H_{P} (σ) = \{ϕ ∈ C^{∞}(G, V) : ϕ(xmn) = δ_{P}^{− \frac{1}{2}} (m)σ(m)^{−1}ϕ(x) for all x ∈ G, m ∈ M, n ∈ N\};

H_{P^{0}}(σ_{0}) =

\{ψ ∈ C^{∞}(G, V_{0}) : ψ(xm_{0}n) = δ_{P^{0}}^{− \frac{1}{2}} (m_{0})σ_{0}(m_{0})^{−1}ψ(x) for all x ∈ G, m_{0} ∈ M^{0}, n ∈ N\};

H_{P} (I_{M}) =

\{ψ ∈ C^{∞}(G, W) : ψ(xmn) = δ_{P}^{− \frac{1}{2}} (m)I_{M}(m)^{−1}ψ(x) for all x ∈ G, m ∈ M, n ∈ N\}.

In each case G acts on the representation space by left translations and we call the induced representations I_{P} (σ), I_{P^{0}}(σ_{0}), and I_{P} (I_{M}) respectively. They are the representations Ind_{P}^{G}(σ), Ind_{P^{0}}^{G}(σ_{0}), and Ind_{P}^{G}(Ind_{M^{0}}^{M}(σ_{0})) respectively.

The intertwining operators S : V → W and P : W → V induce intertwining operators S^{∗} from (I_{P} (σ), H_{P} (σ)) to (I_{P} (I_{M}), H_{P} (I_{M})) and P^{∗} from (I_{P} (I_{M}), H_{P} (I_{M})) to (I_{P} (σ), H_{P} (σ)) given by

(S^{∗}ϕ)(x) = Sϕ(x) for all ϕ ∈ H_{P} (σ), x ∈ G;
(P^{∗}ψ)(x) = P ψ(x) for all ψ ∈ H_{P} (I_{M}), x ∈ G.

There is also an equivalence T between (I_{P} (I_{M}), H_{P} (I_{M})) and (I_{P^{0}}(σ_{0}), H_{P^{0}}(σ_{0})) given by

(T ψ)(x) = ψ(x)(1) for all ψ ∈ H_{P} (I_{M}), x ∈ G.

Its inverse is given by

(T^{−1}ψ′)(x)(m) = δP^{\frac{1}{2}} (m)ψ′(xm) for all ψ′ ∈ H_{P^{0}}(σ_{0}), x ∈ G, m ∈ M.

Recall for each ν ∈ \underline{a}_{C}^{∗} we have defined characters χ_{ν}^{0} of M^{0} and χ_{ν} of M such that χ_{ν}^{0} is the restriction of χ_{ν} to M^{0}. We use these characters to define representations σ(ν) = σ ⊗χ_{ν} and I_{M}(ν) = I_{M} ⊗χ_{ν} of M and σ_{0}(ν) = σ_{0} ⊗χ_{0}(ν) of M^{0}. As above we use these to form induced representation spaces H_{P} (σ, ν) = H_{P} (σ(ν)), H_{P} (I_{M}, ν) = H_{P} (I_{M}(ν)), and H_{P^{0}}(σ, ν) = H_{P^{0}}(σ_{0}(ν)). The intertwining operators S : V → W and P : W → V also intertwine σ(ν) and I_{M}(ν) and so as above define induced intertwining operators S_{ν}^{∗} : H_{P} (σ, ν) → H_{P} (I_{M}, ν) and P_{ν}^{∗} : H_{P} (I_{M}, ν) → H_{P} (σ, ν). There are also equivalences T_{ν} : H_{P} (I_{M}, ν) → H_{P^{0}}(σ_{0}, ν) given by

(T_{ν}ψ)(x) = ψ(x)(1) for all ψ ∈ H_{P} (I_{M}, ν), x ∈ G.

The inverses are given by

(T_{ν}^{−1}ψ′)(x)(m) = δP^{\frac{1}{2}} (m)χ_{ν}(m)ψ′(xm) for all ψ′ ∈ H_{P^{0}}(σ_{0}, ν), x ∈ G, m ∈ M.

Suppose P_{1} = MN_{1} and P_{2} = MN_{2} are two cuspidal parabolic subgroups of G with Levi component M. In §4 we defined a meromorphic family of intertwining operators

J(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν) : H_{P_{1}^{0}}(σ_{0}, ν) → H_{P_{2}^{0}}(σ_{0}, ν).

We can transfer these intertwining operators to the equivalent spaces H_{P_{i}}(I_{M}, ν), i = 1, 2, by means of the equivalences T_{ν,P_{i}}. Thus we define

J(P_{2} : P_{1} : I_{M} : ν) : H_{P_{1}}(I_{M}, ν) → H_{P_{2}}(I_{M}, ν)

by

J(P_{2} : P_{1} : I_{M} : ν) = T_{ν,P_{2}}^{−1}J(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν)T_{ν,P_{1}}.

We can also define

J(P_{2} : P_{1} : σ : ν) : H_{P_{1}}(σ, ν) → H_{P_{2}}(σ, ν)

by

J(P_{2} : P_{1} : σ : ν) = P_{ν,P_{2}}^{∗}J(P_{2} : P_{1} : I_{M} : ν)S_{ν,P_{1}}^{∗}.

Lemma 5.5. Suppose that ν ∈ \underline{a}_{C}^{∗}(P_{2}^{0} : P_{1}^{0}) so that J(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν) is given by the convergent integral

J(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν)ψ′(x) = ∫_{\overline{N}_{1}∩N_{2}} ψ′(x\overline{n})d\overline{n}, x ∈ G, ψ′ ∈ H_{P_{1}^{0}}(σ_{0}, ν).

Then J(P_{2} : P_{1} : I_{M} : ν) is given by the convergent integral

J(P_{2} : P_{1} : I_{M} : ν)ψ(x) = ∫_{\overline{N}_{1}∩N_{2}} ψ(x\overline{n})d\overline{n}, x ∈ G, ψ ∈ H_{P_{1}}(I_{M}, ν)

and J(P_{2} : P_{1} : σ : ν) is also given by the convergent integral

J(P_{2} : P_{1} : σ : ν)ϕ(x) = ∫_{\overline{N}_{1}∩N_{2}} ϕ(x\overline{n})d\overline{n}, x ∈ G, ϕ ∈ H_{P_{1}}(σ, ν).

Proof: Using the definitions of the operators and the transformation property of the representation space H_{P_{1}}(I_{M}, ν), we have for all x ∈ G, m ∈ M, ψ ∈ H_{P_{1}}(I_{M}, ν),

J(P_{2} : P_{1} : I_{M} : ν)ψ(x)(m) = T_{ν,P_{2}}^{−1}J(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν)T_{ν,P_{1}}ψ(x)(m)

= χ_{ν}(m)δP_{2}^{\frac{1}{2}}(m)(J(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν)T_{ν,P_{1}}ψ)(xm)

= χ_{ν}(m)δP_{2}^{\frac{1}{2}}(m) ∫_{\overline{N}_{1}∩N_{2}}(T_{ν,P_{1}}ψ)(xm\overline{n})d\overline{n}

= χ_{ν}(m)δP_{2}^{\frac{1}{2}}(m) ∫_{\overline{N}_{1}∩N_{2}} ψ(xm\overline{n})(1)d\overline{n}

= δP_{2}^{\frac{1}{2}}(m) ∫_{\overline{N}_{1}∩N_{2}} δ_{P_{1}}^{− \frac{1}{2}} (m)ψ(xm\overline{n}m^{−1})(m)d\overline{n}.

Now there is a homomorphism β : M → R^{+} so that for all m ∈ M, f ∈ C_{c}^{∞}(\overline{N}_{1} ∩ N_{2}),

∫_{\overline{N}_{1}∩N_{2}} f(m\overline{n}m^{−1})d\overline{n} = β(m) ∫_{\overline{N}_{1}∩N_{2}} f(\overline{n})d\overline{n}.

For m ∈ M^{0} we know that

β(m) = δP_{1}^{\frac{1}{2}}(m)δ_{P_{2}}^{− \frac{1}{2}} (m).

Since βδ_{P_{1}}^{− \frac{1}{2}} δP_{2}^{\frac{1}{2}} is a homomorphism from the finite group M/M^{0} into R^{+}, it must be identically one. Hence for all m ∈ M we have

∫_{\overline{N}_{1}∩N_{2}} ψ(xm\overline{n}m^{−1})(m)d\overline{n} = δP_{1}^{\frac{1}{2}}(m)δ_{P_{2}}^{− \frac{1}{2}} (m) ∫_{\overline{N}_{1}∩N_{2}} ψ(x\overline{n})(m)d\overline{n}.

Thus

J(P_{2} : P_{1} : I_{M} : ν)ψ(x)(m) = ∫_{\overline{N}_{1}∩N_{2}} ψ(x\overline{n})(m)d\overline{n}.

Now for x ∈ G, m ∈ M, ϕ ∈ H_{P_{1}}(σ, ν), we have

J(P_{2} : P_{1} : σ : ν)ϕ(x) = [P_{ν,P_{2}}^{∗}J(P_{2} : P_{1} : I_{M} : ν)S_{ν,P_{1}}^{∗}ϕ](x)
= P · [J(P_{2} : P_{1} : I_{M} : ν)S_{ν,P_{1}}^{∗}ϕ](x)
= P · ∫_{\overline{N}_{1}∩N_{2}}[S_{ν,P_{1}}^{∗}ϕ](x\overline{n})d\overline{n}
= P S · ∫_{\overline{N}_{1}∩N_{2}} ϕ(x\overline{n})d\overline{n}
= ∫_{\overline{N}_{1}∩N_{2}} ϕ(x\overline{n})d\overline{n}.

Let r(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν) be the scalar normalizing factors used in §4 to define the normalized intertwining operators

R(P_{2}^{0} : P_{1}^{0} : σ_{0}) = r(P_{2}^{0} : P_{1}^{0} : σ_{0} : 0)^{−1}J(P_{2}^{0} : P_{1}^{0} : σ_{0} : 0).

The fact that

r(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν)^{−1}J(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν)

is holomorphic and non-zero at ν = 0 and that T_{ν,P_{i}}, i = 1, 2, are equivalences, will imply that

r(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν)^{−1}J(P_{2} : P_{1} : I_{M} : ν)

is also holomorphic and non-zero at ν = 0. Thus we can define

R(P_{2} : P_{1} : I_{M}) = r(P_{2}^{0} : P_{1}^{0} : σ_{0} : 0)^{−1}J(P_{2} : P_{1} : I_{M} : 0).

We also define

R(P_{2} : P_{1} : σ) = r(P_{2}^{0} : P_{1}^{0} : σ_{0} : 0)^{−1}J(P_{2} : P_{1} : σ : 0) = P^{∗}R(P_{2} : P_{1} : I_{M})S^{∗}.

Lemma 5.6. Let ϕ ∈ H_{P_{1}}(σ). Then R(P_{2} : P_{1} : I_{M})S^{∗}ϕ(x) ∈ S(V) for all x ∈ G.

Proof: For every ν we have an intertwining operator J(P_{2} : P_{1} : I_{M} : ν)S_{ν}^{∗} : H_{P_{1}}(σ, ν) → H_{P_{2}}(I_{M} : ν). In order to carry out arguments using the integral formula and meromorphic extension of the intertwining operator we want a compact realization of the representation. Since we do not know if there is a maximal compact subgroup of G which meets every connected component, we proceed one coset at a time. Let G_{M} = G^{0}M and write G = ∪_{i=1}^{k}x_{i}G_{M}. Then P ⊂ G_{M} for any parabolic subgroup P of G with Levi component M. Let K^{0} be a good maximal compact subgroup of G^{0} so that G^{0} = K^{0}P^{0}. Thus G_{M} = K^{0}P. Let H_{K^{0}}(σ) =

{f_{K} ∈ C^{∞}(K^{0}, V) : f_{K}(kmn) = σ^{−1}(m)f_{K}(k) for all m ∈ K^{0}∩M, n ∈ K^{0}∩N, k ∈ K^{0}}.

For any ϕ ∈ H_{P} (σ : ν) and 1 ≤ i ≤ k we can define

ϕ_{i}(x) = \{ \begin{matrix}ϕ(x), & if x ∈ x_{i}G_{M}; \\ 0, & otherwise. \end{matrix}

Then ϕ_{i} ∈ H_{P} (σ : ν) for each i and ϕ = ∑_{i=1}^{k} ϕ_{i}. Thus every element of H_{P} (σ : ν) is a sum of elements supported on a single coset of G_{M} in G and so it is enough to prove the lemma for ϕ ∈ H_{P_{1}}(σ) supported on a single coset of G_{M} in G.

Fix 1 ≤ i ≤ k and define

F_{i}(ν) : H_{P_{1}}(σ : ν) → H_{K^{0}}(σ)

by F_{i}(ν)ϕ(k) = ϕ(x_{i}k) for all k ∈ K^{0}. Define

F_{i}^{−1}(ν) : H_{K^{0}}(σ) → H_{P_{1}}(σ : ν)

by

F_{i}^{−1}(ν)f_{K}(x) = \{ \begin{matrix}δ_{P_{1}}^{− \frac{1}{2}} (m)σ^{−1}(m)χ_{ν}^{−1}(m)f_{K}(k), & if x = x_{i}kmn, k ∈ K^{0}, m ∈ M, n ∈ N; \\ 0, & otherwise. \end{matrix}

Then F_{i}(ν)F_{i}^{−1}(ν)f_{K} = f_{K} for all f_{K} ∈ H_{K^{0}}(σ) and F_{i}^{−1}(ν)F_{i}(ν)ϕ = ϕ_{i} for all ϕ ∈ H_{P_{1}}(σ : ν).

Fix 1 ≤ i ≤ k and ϕ ∈ H_{P_{1}}(σ) = H_{P_{1}}(σ : 0) such that ϕ = ϕ_{i} is supported on x_{i}G_{M}. Let f_{K} = F_{i}(0)ϕ ∈ H_{K^{0}}(σ) and for each ν define ϕ_{i}(ν) ∈ H_{P_{1}}(σ : ν) by ϕ_{i}(ν : x) = F_{i}^{−1}(ν)f_{K}(x), x ∈ G. We have ϕ_{i}(0) = F_{i}^{−1}(0)F_{i}(0)ϕ = ϕ_{i} = ϕ.

Fix w^{∗} ∈ S(V)^{⊥} and define Φ(ν : x) =< J(P_{2} : P_{1} : I_{M} : ν)S_{ν}^{∗}ϕ_{i}(ν : x), w^{∗} >. Then ν ↦ Φ(ν : x) is a meromorphic function of ν ∈ \underline{a}_{C}^{∗} for each x ∈ G. If ν ∈ \underline{a}_{C}^{∗}(P_{2}^{0} : P_{1}^{0}), by Lemma 5.5 we have

J(P_{2} : P_{1} : I_{M} : ν)S_{ν}^{∗}ϕ_{i}(ν : x) = ∫_{\overline{N}_{1}∩N_{2}} Sϕ_{i}(ν : x\overline{n})d\overline{n} ∈ S(V).

Thus Φ(x : ν) = 0 for all ν ∈ \underline{a}_{C}^{∗}(P_{2}^{0} : P_{1}^{0}) and hence for all ν. Thus

J(P_{2} : P_{1} : I_{M} : ν)S_{ν}^{∗}ϕ_{i}(ν : x) ∈ S(V)

for all ν and so

r(P_{2}^{0} : P_{1}^{0} : σ_{0} : ν)^{−1}J(P_{2} : P_{1} : I_{M} : ν)S_{ν}^{∗}ϕ_{i}(ν : x) ∈ S(V)

for all ν ∈ \underline{a}_{C}^{∗}, x ∈ G. In particular for ν = 0 we have R(P_{2} : P_{1} : I_{M})S^{∗}ϕ(x) ∈ S(V) for all x ∈ G.

Corollary 5.7. Let ϕ ∈ H_{P_{1}}(σ). Then R(P_{2} : P_{1} : I_{M})S^{∗}ϕ = S^{∗}R(P_{2} : P_{1} : σ)ϕ.

Proof: This follows from Lemma 5.6 since SP is the identity on S(V).

Lemma 5.8. Suppose P_{1}, P_{2}, and P_{3} are cuspidal parabolic subgroups of G with Levi component M. Then

R(P_{1} : P_{3} : I_{M}) = R(P_{1} : P_{2} : I_{M})R(P_{2} : P_{3} : I_{M})

and

R(P_{1} : P_{3} : σ) = R(P_{1} : P_{2} : σ)R(P_{2} : P_{3} : σ).

Proof: The statement for I_{M} follows easily from Lemma 4.8 since the intertwining operators T_{ν} are equivalences. Now using Corollary 5.7 we have

R(P_{1} : P_{3} : σ) = P^{∗}R(P_{1} : P_{3} : I_{M})S^{∗} = P^{∗}R(P_{1} : P_{2} : I_{M})R(P_{2} : P_{3} : I_{M})S^{∗}
P^{∗}R(P_{1} : P_{2} : I_{M})S^{∗}R(P_{2} : P_{3} : σ) = R(P_{1} : P_{2} : σ)R(P_{2} : P_{3} : σ).

Lemma 5.9. Suppose P_{1} and P_{2} are parabolic subgroups of G with Levi component M. Then R(P_{2} : P_{1} : σ) is an equivalence from H_{P_{1}}(σ) onto H_{P_{2}}(σ).

Proof: The proof is the same as that of Lemma 4.9.

Now as above we define N_{G}(σ_{0}) = {g ∈ N_{G}(A) : σ_{0}^{g} ≃ σ_{0}}. If w ∈ W_{G}(σ_{0}) = N_{G}(σ_{0})/M^{0}, σ_{0} can be extended to a representation σ_{0,w} of the group M_{w}^{0} generated by M^{0} and any representative n_{w} for w. Define T (n_{w}) : W → W by

T (n_{w})f(m) = σ_{0,w}(n_{w})f(n_{w}^{−1}mn_{w}), m ∈ M.

It is easy to check that

T (n_{w})I_{M}(m) = I_{M}(n_{w}mn_{w}^{−1})T (n_{w}), m ∈ M.

Next we define intertwining operators B_{P} (w) : H_{w^{−1}P w}(I_{M}) → H_{P} (I_{M}) by

B_{P} (w)ψ(g) = α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} T (n_{w})ψ(gn_{w}), g ∈ G.

Finally we define self-intertwining operators R(w, I_{M}) : H_{P} (I_{M}) → H_{P} (I_{M}) by

R(w, I_{M}) = B_{P} (w)R(w^{−1}P w : P : I_{M}).

Recall that in §4 we defined intertwining operators

A_{P} (w) : H_{w^{−1}P^{0}w}(σ_{0}) → H_{P^{0}}(σ_{0}) and R(w, σ_{0}) : H_{P^{0}}(σ_{0}) → H_{P^{0}}(σ_{0}).

It is easy to check that

B_{P} (w) = T_{P}^{−1}A_{P} (w)T_{w^{−1}P w} and R(w, I_{M}) = T_{P}^{−1}R(w, σ_{0})T_{P}

where T_{P} : H_{P} (I_{M}) → H_{P^{0}}(σ_{0}) is the equivalence defined above. In particular this implies that B_{P} (w) and R(w, I_{M}) are independent of the coset representatives chosen.

Define N_{G}(σ) = {g ∈ N_{G}(M) : σ^{g} ≃ σ}. If w ∈ W_{G}(σ) = N_{G}(σ)/M, σ can be extended to a representation of the group M_{w} generated by M and any representative n_{w} for w. Denote such an extension by σ_{w} and define A′_{P} (w) : H_{w^{−1}P w}(σ) → H_{P} (σ) by

(A′_{P} (w)ϕ)(x) = σ_{w}(n_{w})α_{w^{−1}P w}(n_{w})^{\frac{1}{2}} ϕ(xn_{w}).

Lemma 5.10. The intertwining operator A′_{P} (w) is independent of the choice of coset representative n_{w} for w ∈ W_{G}(σ). For w_{1}, w_{2} ∈ W_{G}(σ) there is a non-zero constant c_{P} (w_{1}, w_{2}) so that

A′_{P} (w_{1}, w_{2}) = c_{P} (w_{1}, w_{2})A′_{P} (w_{1})A′_{w_{1}^{−1}P w_{1}}(w_{2}).

Proof: The proof is exactly the same as that of Lemma 4.11.

Finally, for w ∈ W_{G}(σ), we define R(w, σ) : H_{P} (σ) → H_{P} (σ) by

R(w, σ) = A′_{P} (w)R(w^{−1}P w : P : σ).

Note that for u ∈ W_{G}(σ_{0}) we could also have defined an intertwining operator

R′(u, σ) : H_{P} (σ) → H_{P} (σ)

by

R′(u, σ) = P^{∗}R(u, I_{M})S^{∗}.

We want to relate these two definitions. Let

W_{G}(σ_{0}, σ) = [N_{G}(σ_{0}) ∩ N_{G}(σ)]/M^{0} ⊂ W_{G}(σ_{0}).

Suppose that x ∈ N_{G}(σ_{0}) ∩ N_{G}(σ). Then x represents an element xM^{0} ∈ W_{G}(σ_{0}, σ) and an element xM ∈ W_{G}(σ). Let

p : W_{G}(σ_{0}, σ) → W_{G}(σ)

be given by p(xM^{0}) = xM, x ∈ N_{G}(σ_{0}) ∩ N_{G}(σ).

Lemma 5.11. The mapping p is surjective. Its kernel is W_{M}(σ_{0}) = N_{M}(σ_{0})/M^{0}.

Proof: Let w ∈ W_{G}(σ) and let x ∈ N_{G}(σ) be a representative for w. Then σ_{0}^{x} is contained in the restriction of σ^{x} ≃ σ to M^{0} so that there is m ∈ M such that σ_{0}^{x} ≃ σ_{0}^{m}. Hence w has a representative xm^{−1} ∈ N_{G}(σ)∩N_{G}(σ_{0}). Thus p is surjective. Clearly p(xM^{0}) = M just in case x ∈ N_{G}(σ_{0}) ∩ N_{G}(σ) ∩ M = N_{M}(σ_{0}).

Lemma 5.12. Suppose that u ∈ W_{G}(σ_{0}) is in the complement of W_{G}(σ_{0}, σ). Then

R′(u, σ) = 0.

If u ∈ W_{G}(σ_{0}, σ), then there is a complex constant c so that

R′(u, σ) = cR(p(u), σ).

Proof: Using Corollary 5.7, for any u ∈ W_{G}(σ_{0}),

R′(u, σ) = P^{∗}R(u, I_{M})S^{∗} = P^{∗}B_{P} (u)R(u^{−1}P u : P : I_{M})S^{∗}
= P^{∗}B_{P} (u)S^{∗}R(u^{−1}P u : P : σ).

But for any g ∈ G, ϕ ∈ H_{u^{−1}P u}(σ), if x ∈ N_{G}(σ_{0}) is a representative for u,

P^{∗}B_{P} (u)S^{∗}ϕ(g) = α_{u^{−1}P u}(x)^{\frac{1}{2}} (P T (x)S)ϕ(gx).

Since T (x) intertwines I_{M} and I_{M}^{x}, we see that P T (x)S intertwines σ and σ^{x} ≃ σ^{u}. Thus P T (x)S = 0 and hence R′(u, σ) = 0 unless σ^{u} ≃ σ.

Suppose u ∈ W_{G}(σ_{0}, σ). Write u = u_{x}, w = w_{x} = p(u), x ∈ N_{G}(σ_{0}) ∩ N_{G}(σ). Then P T (x)S and σ_{w}(x) both intertwine σ and σ^{x}, and σ_{w}(x) ≠ 0. Thus there is a complex constant c′ so that P T (x)S = c′σ_{w}(x). Thus for any g ∈ G,

R′(u, σ)ϕ(g) = α_{u^{−1}P u}(x)^{\frac{1}{2}} (P T (x)S)[R(u^{−1}P u : P : σ)ϕ](gx)
= c′α_{u^{−1}P u}(x)^{\frac{1}{2}} σ_{w}(x)[R(u^{−1}P u : P : σ)ϕ](gx)
= c′A′_{P} (w)R(w^{−1}P w : P : σ)ϕ(g) = c′R(w, σ)ϕ(g).

Lemma 5.13. The R(w, σ), w ∈ W_{G}(σ), span the algebra C(σ) of self-intertwining operators on H_{P} (σ).

Proof: Let R be a self-intertwining operator for H_{P} (σ). Then S^{∗}RP^{∗} is a selfintertwining operator for H_{P} (I_{M}), hence in the span of the R(u, I_{M}), u ∈ W_{G}(σ_{0}). But then R = P^{∗}S^{∗}RP^{∗}S^{∗} is in the span of the P^{∗}R(u, I_{M})S^{∗} = R′(u, σ), u ∈ W_{G}(σ_{0}). But by Lemma 5.12, each R′(u, σ) is either zero or a multiple of one of the operators R(w, σ), w ∈ W_{G}(σ).

Lemma 5.14. Let u ∈ W_{G}(σ_{0}) and suppose that R(u, I_{M}) is scalar. Then u ∈ W_{G}(σ_{0}, σ) and R(p(u), σ) is scalar.

Proof: Suppose that there is a constant s ∈ C such R(u, I_{M})ψ = sψ for all ψ ∈ H_{P} (I_{M}). Since R(u, I_{M}) ≠ 0, s ≠ 0. Now for all ϕ ∈ H_{P} (σ),

R′(u, σ)ϕ = P^{∗}R(u, I_{M})S^{∗}ϕ = sP^{∗}S^{∗}ϕ = sϕ.

Thus R′(u, σ) is scalar and non-zero. Thus by Lemma 5.12 we have u ∈ W_{G}(σ_{0}, σ). Further, by Lemma 5.12, there is a constant c so that R′(u, σ) = cR(p(u), σ). Since R′(u, σ) ≠ 0, c ≠ 0. Thus R(p(u), σ) = c^{−1}R′(u, σ) is scalar.

Lemma 5.15. There is a cocycle η so that

R(w_{1}w_{2}, σ) = η(w_{1}, w_{2})R(w_{1}, σ)R(w_{2}, σ)

for all w_{1}, w_{2} ∈ W_{G}(σ).

Proof: The proof is similar to Lemma 4.12.

As in §4, if W_{G^{0}}^{0}(σ_{0}) is the subgroup of elements u ∈ W_{G^{0}}(σ_{0}) such that R^{0}(u, σ_{0}) is scalar, then W_{G^{0}}^{0}(σ_{0}) = W (Φ_{1}) is generated by reflections in a set Φ_{1} of reduced roots of (G, A). Let Φ^{+}, Φ_{1}^{+} be defined as in §4. Since M centralizes A, W_{G}(σ) ⊂ N_{G}(A)/M acts on roots of A and we can define

R_{σ} = {w ∈ W_{G}(σ) : wβ ∈ Φ^{+} for all β ∈ Φ_{1}^{+}}.

We want to prove the following.

Theorem 5.16. The R(w, σ), w ∈ R_{σ}, form a basis for the algebra of intertwining operators of I_{P} (σ).

In order to prove Theorem 5.16, we will first compute the dimension of C(σ) using our knowledge of the dimension of C(σ_{0}). We denote the equivalence classes of Ind_{P}^{G}(σ) and Ind_{P^{0}}^{G}(σ_{0}) by i_{G,M}(σ) and i_{G,M^{0}}(σ_{0}) respectively. Let X and Y denote the groups of unitary characters of G/G^{0} and M/M^{0} respectively. For χ ∈ X, let χ_{M} ∈ Y denote the restriction of χ to M. Define

X(σ) = {χ ∈ X : χ_{M} ⊗ σ ≃ σ};
X_{1}(σ) = {χ ∈ X : χ ⊗ i_{G,M}(σ) ≃ i_{G,M}(σ)};
Y (σ) = {η ∈ Y : σ ⊗ η ≃ σ}.

Let s denote the multiplicity of σ_{0} in the restriction of σ to M^{0}.

Lemma 5.17.

dim C(σ_{0}) = dim C(σ)s^{2}[X/X(σ)][X_{1}(σ)/X(σ)].

Proof: Using Lemma 2.13,

i_{M,M^{0}}(σ_{0}) = s ∑_{η∈Y/Y (σ)} σ ⊗ η.

This implies that

i_{G,M}(i_{M,M^{0}}(σ_{0})) = s ∑_{η∈Y/Y (σ)} i_{G,M}(σ ⊗ η).

Since both G/G^{0} and M/M^{0} are finite abelian, it is clear that the map χ ↦ χ_{M} induces an isomorphism between X/X(σ) and Y/Y (σ). Thus we can rewrite

i_{G,M}(i_{M,M^{0}}(σ_{0})) = s ∑_{χ∈X/X(σ)} i_{G,M}(σ ⊗ χ_{M}).

But by Corollary 3.3 the induced representations i_{G,M}(σ ⊗ χ_{M}) are either disjoint or equal. Further,

i_{G,M}(σ ⊗ χ_{M}) = i_{G,M}(σ) ⊗ χ = i_{G,M}(σ)

just in case χ ∈ X_{1}(σ). Thus we have

i_{G,M}(i_{M,M^{0}}(σ_{0})) = s[X_{1}(σ)/X(σ)] ∑_{χ∈X/X_{1}(σ)} i_{G,M}(σ) ⊗ χ

where the representations i_{G,M}(σ) ⊗ χ are disjoint for χ ∈ X/X_{1}(σ). Thus

dim C(σ_{0}) = dim C(σ)s^{2}[X_{1}(σ)/X(σ)]^{2}[X/X_{1}(σ)]
= dim C(σ)s^{2}[X/X(σ)][X_{1}(σ)/X(σ)].

Lemma 5.18.

s^{2}[X/X(σ)][X_{1}(σ)/X(σ)] = [W_{G}(σ_{0})]/[W_{G}(σ)].

Proof: First, using Lemma 2.13 we have

s^{2}[X/X(σ)] = s^{2}[Y/Y (σ)] = [N_{M}(σ_{0})/M^{0}] = [W_{M}(σ_{0})].

We claim that

[X_{1}(σ)/X(σ)] = [W_{G}(σ_{0})/W_{G}(σ_{0}, σ)].

This would establish the lemma since by Lemma 5.11 we have

[W_{G}(σ)] = [W_{G}(σ_{0}, σ)/W_{M}(σ_{0})].

We will define a bijection between X_{1}(σ)/X(σ) and

(N_{G}(σ_{0}) ∩ N_{G}(σ))\N_{G}(σ_{0}) ≃ W_{G}(σ_{0}, σ)\W_{G}(σ_{0}).

Let χ ∈ X_{1}(σ). The equivalence class of σ ⊗ χ_{M} depends only on the coset \overline{χ} of χ in X_{1}(σ)/X(σ). Further, by definition of X_{1}(σ), we have

i_{G,M}(σ ⊗ χ_{M}) = i_{G,M}(σ).

By Corollary 3.2 there is x ∈ N_{G}(A) such that σ ⊗ χ_{M} ≃ σ^{x}. Thus σ|_{M^{0}} ≃ σ^{x}|_{M^{0}}. Thus σ_{0}^{x} occurs in σ|_{M^{0}} and so there is m ∈ M such that σ_{0}^{x} ≃ σ_{0}^{m}. Then y = xm^{−1} ∈ N_{G}(σ_{0}). Although y ∈ N_{G}(σ_{0}) is not uniquely determined by χ,

σ ⊗ χ_{M} ≃ σ^{y_{1}} ≃ σ^{y_{2}}

if and only if y_{1}y_{2}^{−1} ∈ N_{G}(σ_{0}) ∩ N_{G}(σ). Thus for each \overline{χ} ∈ X_{1}(σ)/X(σ) there is a unique coset \overline{x}(\overline{χ}) = (N_{G}(σ_{0}) ∩ N_{G}(σ))x in (N_{G}(σ_{0}) ∩ N_{G}(σ))\N_{G}(σ_{0}) such that χ_{M} ⊗ σ ≃ σ^{x}. Finally, given x ∈ N_{G}(σ_{0}), σ^{x} is a constituent of i_{M,M^{0}}(σ_{0}^{x}) ≃ i_{M,M^{0}}(σ_{0}) so that there is η ∈ Y such that σ^{x} ≃ σ ⊗ η. Now let χ ∈ X such that χ_{M} = η. Then σ ⊗ χ_{M} ≃ σ^{x} so that \overline{x} = \overline{x}(\overline{χ}).

Recall from §4 that W_{G}(σ_{0}) is the semidirect product of subgroups R_{σ_{0}} and W (Φ_{1}) where R(w, σ_{0}) is scalar for w ∈ W (Φ_{1}) and the R(r, σ_{0}), r ∈ R_{σ_{0}}, give a basis for C(σ_{0}).

Lemma 5.19.

dim C(σ) = [W_{G}(σ)]/[W (Φ_{1})].

Proof: Combining Lemmas 5.17 and 5.18 we have

dim C(σ_{0}) = dim C(σ) · \frac{[W_{G}(σ_{0})]}{[W_{G}(σ)]}.

But from Lemma 4.15, dim C(σ_{0}) = [R_{σ_{0}}] = [W_{G}(σ_{0})]/[W (Φ_{1})].

Since W (Φ_{1}) ⊂ N_{G^{0}}(A)/M^{0}, it can be naturally embedded in W_{G}(A) = N_{G}(A)/M.

Lemma 5.20. W_{G}(σ) is the semidirect product of W (Φ_{1}) and R_{σ}. For w ∈ W_{G}(σ), R(w, σ) is scalar if and only if w ∈ W (Φ_{1}).

Proof: If x ∈ N_{G^{0}}(σ_{0}) represents an element of W (Φ_{1}), then by Lemma 4.14 R(u_{x}, I_{M}) is scalar. Thus by Lemma 5.14, w_{x} ∈ W_{G}(σ) and R(w_{x}, σ) is scalar. Let W_{G}^{0}(σ) denote the set of all w ∈ W_{G}(σ) such that R(w, σ) is scalar. By the above W (Φ_{1}) ⊂ W_{G}^{0}(σ). Using Lemmas 5.13 and 5.19 we see that

[W_{G}(σ)]/[W (Φ_{1})] = dim C(σ) ≤ [W_{G}(σ)]/[W_{G}^{0}(σ)].

Thus W (Φ_{1}) = W_{G}^{0}(σ).

Now as in the proof Lemma 4.14, W (Φ_{1}) is a normal subgroup of W_{G}(σ) and so wΦ_{1} = Φ_{1} for all w ∈ W_{G}(σ). This implies that

R_{σ} = {w ∈ W_{G}(σ) : wΦ_{1}^{+} = Φ_{1}^{+}}

which yields the semidirect product decomposition.

Proof of Theorem 5.16. It follows from Lemmas 5.13,5.15, and 5.20 that the R(w, σ), w ∈ R_{σ}, span the algebra C(σ). Further, by Lemmas 5.19 and 5.20,

dim C(σ) = [W_{G}(σ)]/W (Φ_{1})] = [R_{σ}].

Let η be the cocycle of Lemma 5.15. Exactly as in §4 we can fix a finite central extension

1 → Z_{σ} → R̃_{σ} → R_{σ} → 1

over which η splits, a character χ_{σ} of Z_{σ}, and a representation R of R̃_{σ} ×G on H_{P} (σ). Let Π(R̃_{σ}, χ_{σ}) denote the set of irreducible representations of R̃_{σ} with Z_{σ} central character χ_{σ}, and let Π_{σ}(G) denote the set of irreducible constituents of I_{P} (σ).

Theorem 5.21. There is a bijection ρ ↦ π_{ρ} of Π(R̃_{σ}, χ_{σ}) onto Π_{σ}(G) such that

R = ⊕_{ρ∈Π(R̃_{σ},χ_{σ})} (ρ^{∨} ⊗ π_{ρ}).

Proof: The proof is exactly the same as that of Theorem 4.17.

6. Examples

For applications involving comparisons of representations between groups and twisted trace formulas it is customary to use the following definition of parabolic subgroup. Let P^{0} be a parabolic subgroup of G^{0}. Then P = N_{G}(P^{0}) is a parabolic subgroup of G. Thus, using this definition, parabolic subgroups of G are in one to one correspondence with parabolic subgroups of G^{0}. We will show that the parabolic subgroups obtained using this definition are also parabolic subgroups using the definition of §2. However they are not cuspidal in general. Indeed, recall from Proposition 2.10 that if P^{0} is a parabolic subgroup of G, then the corresponding cuspidal parabolic subgroup is the smallest parabolic subgroup of G lying over P^{0}. On the other hand, if P is any subgroup of G with P ∩ G^{0} = P^{0}, then P ⊂ N_{G}(P^{0}). Thus N_{G}(P^{0}) will be the largest parabolic subgroup of G lying over P^{0}. We will give examples to show that this class of parabolic subgroups, which we call N-parabolic subgroups (N for normalizer), do not yield a nice theory of parabolically induced representations of G.

Lemma 6.1. Let P^{0} be a parabolic subgroup of G^{0}. Then P = N_{G}(P^{0}) is a parabolic subgroup of G. It is the largest parabolic subgroup lying over P^{0}. If M^{0} is a Levi component for P^{0}, then M = N_{G}(M^{0}) ∩ P is a Levi component for P.

Proof: Let P^{0} = M^{0}N be a Levi decomposition of P^{0} and define M = N_{G}(M^{0}) ∩ P. Then M ∩ G^{0} = M^{0} and MN ⊂ P. Let x ∈ P = N_{G}(P^{0}). Then xM^{0}x^{−1} is a Levi subgroup of P^{0} and so there is n ∈ N such that xM^{0}x^{−1} = nM^{0}n^{−1}. Now n^{−1}x ∈ N_{G}(M^{0}) ∩ N_{G}(P^{0}) = M and so x ∈ NM = MN. Thus P = MN.

Let A be the split component of M^{0}. Then M normalizes A and we define a Weyl group W = M/C_{M}(A) where C_{M}(A) denotes the centralizer of A in M. Since M^{0} ⊂ C_{M}(A), we know that W is a finite abelian group. The split component of M is A′ = {a ∈ A : xax^{−1} = a for all x ∈ M} = {a ∈ A : wa = a for all a ∈ W }. Let M′ = C_{G}(A′). If we can show that A′ is the split component of M′, then A′ is a special vector subgroup.

Let Φ^{+} = Φ(P^{0}, A) denote the set of roots of A in P^{0}, \underline{a} the real Lie algebra of A, and \underline{a}^{+} the positive chamber of \underline{a} with respect to Φ^{+}. Fix w ∈ W and define \underline{a}_{w} = \{H ∈ \underline{a} : wH = H\}. Since M normalizes P^{0}, we have wΦ^{+} = Φ^{+} and w\underline{a}^{+} = \underline{a}^{+}. Let k be the order of w. Then since \underline{a}^{+} is convex, for any H ∈ \underline{a}^{+} we have H_{w} = H + wH + w^{2}H + ... + w^{k−1}H ∈ \underline{a}^{+} with wH_{w} = H_{w}. Thus \underline{a}_{w}^{+} = \underline{a}_{w} ∩ \underline{a}^{+} ≠ ∅ and so for any α ∈ Φ^{+}, the restriction of α to \underline{a}_{w} is non-zero. Since W is a finite abelian group, an easy induction argument shows in fact that the restriction of α to \underline{a}′, the real Lie algebra of A′, is non-zero for every α ∈ Φ^{+}. Thus M′ ∩G^{0} = C_{G^{0}}(A′) = C_{G^{0}}(A) = M^{0}.

Let A′′ be the split component of M′. Thus A′ ⊂ A′′. But since M′ ∩ G^{0} = M^{0} and M ⊂ M′, we have A′′ = {a ∈ A : xax^{−1} = a for all x ∈ M′} ⊂ A′ . Thus A′ = A′′ is the split component of M′. This implies that A′ is a special vector subgroup and that M′ = C_{G}(A′) is a Levi subgroup of G. But since the restriction of α to \underline{a}′, is non-zero for every α ∈ Φ^{+}, we can choose a set of positive roots (Φ′)^{+} of L(G) with respect to L(A′) by restricting the roots in Φ^{+}. With this choice of positive roots, we obtain a parabolic subgroup P′ = M′N′ of G with N′ = N. Thus M′ normalizes N. It also normalizes M^{0} since M′ ∩ G^{0} = M^{0}. Thus M′ ⊂ N_{G}(M^{0}) ∩ N_{G}(P^{0}) = M. Now M′ = M so P′ = MN = P.

Let (P_{0}^{0}, A_{0}) be a minimal p-pair in G^{0} and let Δ denote the set of simple roots of A_{0} in P_{0}^{0}. Then as usual the standard parabolic subgroups of G^{0} are indexed by subsets Θ of Δ. Write (P_{Θ}^{0}, A_{Θ}) for the standard parabolic pair of G^{0} corresponding to Θ ⊂ Δ and write P_{Θ} = N_{G}(P_{Θ}^{0}), the N-parabolic subgroup of G lying over P_{Θ}^{0}. Let N_{G}(P_{0}^{0}, A_{0}) be the set of elements in G that normalize both A_{0} and P_{0}^{0}. Clearly N_{G}(P_{0}^{0}, A_{0}) ∩ G^{0} = P_{0}^{0} ∩ N_{G^{0}}(A_{0}) = C_{G^{0}}(A_{0}) = M_{0}^{0}. Write W_{G}(P_{0}^{0}, A_{0}) = N_{G}(P_{0}^{0}, A_{0})/M_{0}^{0}. Then W_{G}(P_{0}^{0}, A_{0}) acts on Δ and for each Θ ⊂ Δ we write

W (Θ) = {w ∈ W_{G}(P_{0}^{0}, A_{0}) : wΘ = Θ}.

Lemma 6.2. For all Θ ⊂ Δ,

P_{Θ} = ∪_{w∈W (Θ)}wP_{Θ}^{0}.

Proof: This follows from Lemma 3.8.

Now that we have a simple method of computing the groups P_{Θ}, we will give examples to show the following unpleasant facts.

Fact 6.3. Let P_{1}^{0} and P_{2}^{0} be parabolic subgroups of G^{0} and let P_{i} = N_{G}(P_{i}^{0}), i = 1,2. Then P_{1}^{0} ⊂ P_{2}^{0} does not imply that P_{1} ⊂ P_{2}.

Example 6.4. Let G = O(2n), n ≥ 2. Then G^{0} = SO(2n) and the minimal parabolic subgroup B^{0} of G^{0} is the group of upper triangular matrices in SO(2n) with A_{0} the subgroup of diagonal matrices. The simple roots are

Δ = {e_{1} − e_{2}, e_{2} − e_{3}, ..., e_{n−1} − e_{n}, e_{n−1} + e_{n}}.

The Weyl group W_{G}(B^{0}, A_{0}) has order two and is generated by the sign change c_{n} : e_{n} ↦ −e_{n} that interchanges e_{n−1} − e_{n} and e_{n−1} + e_{n}. Thus a subset Θ of Δ is stable under c_{n} just in case neither or both of e_{n−1} ± e_{n} belong to Θ. So for example if Θ = {e_{n−1} − e_{n}}, then B^{0} ⊂ P_{Θ}^{0}, but B = B^{0} ∪ c_{n}B^{0} is not contained in P_{Θ} = P_{Θ}^{0}.

We call P_{0} a minimal N-parabolic subgroup of G if given any N-parabolic subgroup P of G there is x ∈ G such that P_{0} ⊂ xP x^{−1}.

Fact 6.5. G need not have a minimal N-parabolic subgroup.

Suppose P_{0} is a minimal N-parabolic subgroup of G. Then it is easy to see that P_{0}^{0} = P_{0} ∩ G^{0} is a minimal parabolic subgroup of G^{0}. Now by Lemma 6.2, P_{0} meets every connected component of G. But as Example 6.4 shows, there are N-parabolic subgroups of O(2n) which are contained in the identity component SO(2n). Thus no conjugate could contain the minimal N-parabolic subgroup.

If P^{0} = M^{0}N is a Levi decomposition for P^{0}, then we obtained a Levi decomposition P = MN for P = N_{G}(P^{0}) by defining M = N_{G}(M^{0}) ∩ P. Thus M depends on both M^{0} and P.

Fact 6.6. Suppose that P_{1} = M_{1}N_{1} and P_{2} = M_{2}N_{2} are N-parabolic subgroups of G such that M_{1}^{0} = M_{2}^{0}, ie. P_{1}^{0} and P_{2}^{0} have the same Levi subgroup. Then it need not be true that M_{1} = M_{2}, or even that M_{1} and M_{2} have the same number of connected components.

Example 6.7. Let G = O(8) as in Example 6.4. Let P_{1} = P_{Θ_{1}} where Θ_{1} = {e_{3} − e_{4}} and let P′_{2} = P_{Θ_{2}} where Θ_{2} = {e_{1} − e_{2}}. Then P_{1} = P_{1}^{0} is connected and P′_{2} = (P′_{2})^{0} ∪ c_{4}(P′_{2})^{0} meets both components of G. Let w = (13)(24) ∈ N_{G}(A_{0})/A_{0} be the Weyl group element that permutes the pairs (e_{1}, e_{3}) and (e_{2}, e_{4}) and define P_{2} = wP′_{2}w^{−1}. Then wA_{Θ_{2}}w^{−1} = A_{Θ_{1}} and so P_{2}^{0} = w(P′_{2})^{0}w^{−1} and P_{1}^{0} both have Levi component M_{1}^{0} = C_{G^{0}}(A_{Θ_{1}}). However M_{1} = M_{1}^{0} is connected and M_{2} = M_{2}^{0} ∪ c_{2}M_{2}^{0} meets both components of G.

In addition to structural problems, the class of N-parabolic subgroups does not yield a nice theory of parabolic induction. One of the basic cornerstones of representation theory in the connected case is that every irreducible admissible representation is contained in a representation which is parabolically induced from a supercuspidal representation and every tempered representation is a subrepresentation of a representation which is parabolically induced from a discrete series representation. But if supercuspidal and discrete series representations are defined as in the connected case and in §2, then the Levi component M of a parabolic subgroup P has no supercuspidal or discrete series representations unless P is cuspidal. Thus we will not in general be able to obtain all irreducible admissible or tempered representations of G via induction from supercuspidal or discrete series representations of N-parabolic subgroups.

Example 6.8. Define G = O(2) = SO(2) ∪ wSO(2) as in Remark 2.2. The only parabolic subgroup of G^{0} = SO(2) ≃ F^{×} is itself. Thus the only N-parabolic subgroup of G is N_{G}(G^{0}) = G itself which has split component Z = {1}. But G has no representations with compactly supported matrix coefficients, hence no supercuspidal representations. It also has no representations with square-integrable matrix coefficients, hence no discrete series representations.

As can be seen in Example 6.8, the problem with using the standard definitions for supercuspidal and discrete series representations with N-parabolic subgroups P = MN is that the split component of M may be smaller than the split component of M^{0}. In order to guarantee the existence of enough supercuspidal and discrete series representations we could define a representation of M to be supercuspidal (respectively discrete series) just in case its restriction to M^{0} is supercuspidal (respectively discrete series). Then it would be easy to prove as in Theorem 2.18 that every irreducible admissible (respectively tempered) representation of G is contained in a representation which is induced from a supercuspidal (respectively discrete series) representation of an N-parabolic subgroup.

Another basic property of parabolic induction in the connected case is the following. Suppose that P_{1} = M_{1}N_{1} and P_{2} = M_{2}N_{2} are parabolic subgroups and σ_{i}, i = 1, 2, are irreducible representations of M_{i} which are both either supercuspidal or discrete series. Then if the induced representations Ind_{P_{i}}^{G}(σ_{i}) are not disjoint, then the pairs (M_{1}, σ_{1}) and (M_{2}, σ_{2}) are conjugate. Further, in the discrete series case, the induced representations are equivalent. These properties fail in the disconnected case when the P_{i} are N-parabolic subgroups of G and supercuspidal and discrete series representations are defined as above.

Example 6.9. Let G = O(8) and define P_{1} and P_{2} as in Example 6.7. Recall in this case that M_{1} = M_{1}^{0} ≃ GL(2)×GL(1)^{2} while M_{2} = M_{2}^{0}∪c_{2}M_{2}^{0} ≃ GL(2)×GL(1)×O(2) with M_{2}^{0} = GL(2) × GL(1) × SO(2) = M_{1}^{0}. Let σ_{0} = ρ ⊗ χ_{1} ⊗ χ_{2} be an irreducible unitary supercuspidal representation of M_{1}^{0} = M_{2}^{0} where ρ is an irreducible unitary supercuspidal representation of GL(2), χ_{1} is a unitary character of GL(1), and χ_{2} is a non-trivial unitary character of GL(1) with χ_{2}^{2} = 1. Then σ_{0}^{c_{2}} = ρ ⊗ χ_{1} ⊗ χ_{2}^{−1} = σ_{0} so there is an irreducible representation σ_{2} of M_{2} which extends σ_{0}. Further,

Ind_{M_{2}^{0}}^{M_{2}}(σ_{0}) = σ_{2} ⊕ (σ_{2} ⊗ η)

where η is the non-trivial character of M_{2}/M_{2}^{0}. The representations σ_{2} and σ_{2} ⊗ η of M_{2} would both be supercuspidal (and discrete series) since they both restrict to σ_{0} on M_{2}^{0}. Now

Ind_{P_{2}^{0}}^{G}(σ_{0}) ≃ Ind_{P_{2}}^{G} Ind_{M_{2}^{0}}^{M_{2}}(σ_{0}) = Ind_{P_{2}}^{G}(σ_{2}) ⊕ Ind_{P_{2}}^{G}(σ_{2} ⊗ η).

Since P_{1}^{0} and P_{2}^{0} are parabolic subgroups of G^{0} with the same Levi component M_{1}^{0} = M_{2}^{0} we have

Ind_{P_{1}^{0}}^{G^{0}}(σ_{0}) ≃ Ind_{P_{2}^{0}}^{G^{0}}(σ_{0}).

But σ_{1} = σ_{0} is an irreducible supercuspidal (and discrete series) representation of M_{1} = M_{1}^{0} and

Ind_{P_{1}}^{G}(σ_{1}) = Ind_{P_{1}^{0}}^{G}(σ_{0}) ≃ Ind_{G^{0}}^{G} Ind_{P_{1}^{0}}^{G^{0}}(σ_{0})
≃ Ind_{G^{0}}^{G} Ind_{P_{2}^{0}}^{G^{0}}(σ_{0}) ≃ Ind_{P_{2}^{0}}^{G}(σ_{0}) ≃ Ind_{P_{2}}^{G}(σ_{2}) ⊕ Ind_{P_{2}}^{G}(σ_{2} ⊗ η).

Thus we have irreducible supercuspidal (and discrete series) representations σ_{1} of M_{1} and σ_{2} of M_{2} so that

Ind_{P_{1}}^{G}(σ_{1}) ≃ Ind_{P_{2}}^{G}(σ_{2}) ⊕ Ind_{P_{2}}^{G}(σ_{2} ⊗ η).

Clearly M_{1} and M_{2} cannot be conjugate in G since M_{1} is connected while M_{2} is not. Further, the representations Ind_{P_{i}}^{G}(σ_{i}) have a nontrivial intertwining, but are not equivalent.

A final nice property of parabolic induction in the connected case is the theory of R-groups. If P = MN is a parabolic subgroup and σ is an irreducible discrete series representation of M, then R is a subgroup of W_{G}(σ), the group of Weyl group elements fixing σ, which determines the reducibility of Ind_{P}^{G}(σ). Its most basic property is that the dimension of the algebra of self-intertwining operators of Ind_{P}^{G}(σ) is equal to the number of elements in R. The following example shows that there could not be such a simple R-group theory in the disconnected case for N-parabolic subgroups.

Example 6.10. Let G^{0} = SL_{2}(F) × SL_{2}(F) = {(x, y) : x, y ∈ SL_{2}(F)}. Let G = G^{0} ∪ γG_{0} where γ(x, y)γ^{−1} = (y, x). Let B_{1} = A_{1}N_{1} denote the usual Borel subgroup of G_{1} = SL_{2}(F) where A_{1} is the subgroup of diagonal elements and N_{1} is the subgroup of upper triangular matrices with ones on the diagonal. Then B^{0} = B_{1} ×B_{1} is a Borel subgroup of G^{0} and B = N_{G}(B^{0}) = B^{0} ∪ γB^{0}. B^{0} and B have Levi decompositions B^{0} = M^{0}N and B = MN where M^{0} = A^{0} = A_{1} × A_{1}, M = M^{0} ∪ γM^{0}, and N = N_{1} × N_{1}. We have Weyl groups W (G^{0}, A^{0}) = W (G_{1}, A_{1}) × W (G_{1}, A_{1}) ≃ Z_{2} × Z_{2} and W (G, A^{0}) = W (G^{0}, A^{0}) ∪ γW (G^{0}, A^{0}) ≃ D_{8}, the dihedral group of order 8.

Let χ_{1} be a non-trivial character of A_{1} ≃ F^{×} of order two so that Ind_{B_{1}}^{G_{1}}(χ_{1}) = π_{1} ⊕ π_{s} is a reducible principal series representation. Note that the R-group here is R_{1} = W (G_{1}, A_{1}) ≃ Z_{2} and we denote the irreducible constituents of the induced representation by π_{1}, π_{s} to indicate that they are parameterized by the trivial and sign characters ρ_{1} and ρ_{s} of R_{1} respectively. Now let χ = χ_{1} ⊗ χ_{1}. Then

Ind_{B^{0}}^{G^{0}}(χ) ≃ (π_{1} ⊕ π_{s}) ⊗ (π_{1} ⊕ π_{s}) = π_{11} ⊕ π_{1s} ⊕ π_{s1} ⊕ π_{ss}

where for i, j ∈ {1, s}, we write π_{ij} = π_{i} ⊗ π_{j}. Note that π_{ij}^{γ} ≃ π_{ji}, so that π_{ij}^{γ} ≃ π_{ij} if and only if i = j. Let Π_{ij} be an irreducible representation of G such that π_{ij} is contained in the restriction of Π_{ij} to G^{0}. Then if i = j we have

Ind_{G^{0}}^{G}(π_{ij}) = Π_{ij} ⊕ (Π_{ij} ⊗ η)

where η is the non-trivial character of G/G^{0} and Π_{ij} ⊗ η ≄ Π_{ij}. If i ≠ j we have

Ind_{G^{0}}^{G}(π_{ij}) = Π_{ij} ≃ Π_{ji} = Ind_{G^{0}}^{G}(π_{ji}).

In this case we also have Π_{ij} ⊗ η ≃ Π_{ij}. Thus we have

Ind_{B^{0}}^{G}(χ) = Ind_{G^{0}}^{G}(π_{11} ⊕ π_{1s} ⊕ π_{s1} ⊕ π_{ss})
= Π_{11} ⊕ (Π_{11} ⊗ η) ⊕ Π_{ss} ⊕ (Π_{ss} ⊗ η) ⊕ 2Π_{1s}.

Note that the above decompositions are reflected in the R-groups as follows. First, the R-group for Ind_{B^{0}}^{G^{0}}(χ) is given by R^{0} = R_{1} × R_{1} = W (G^{0}, A^{0}) ≃ Z_{2} × Z_{2}. It has 4 characters ρ_{ij}^{0} = ρ_{i} ⊗ ρ_{j}, i, j ∈ {1, s} corresponding to the irreducible representations π_{ij}. The R-group for Ind_{B^{0}}^{G}(χ) is R = W (G, A_{0}) = R^{0} ∪γR^{0}. By abuse of notation we will denote the non-trivial character of R/R^{0} by the same letter η as used above for the non-trivial character of G/G^{0} and below for the non-trivial character of M/M^{0}. The irreducible representations of R are the characters ρ_{ii} and ρ_{ii} ⊗ η where

Ind_{R^{0}}^{R}(ρ_{ii}^{0}) = ρ_{ii} ⊕ (ρ_{ii} ⊗ η)

and the two-dimensional irreducible representation

Ind_{R^{0}}^{R}(ρ_{1s}^{0}) = ρ_{1s} ≃ Ind_{R^{0}}^{R}(ρ_{s1}^{0}).

Thus we have the irreducible constituents of Ind_{B^{0}}^{G}(χ) parameterized by the irreducible representations of R and occurring with multiplicities given by the degrees of the corresponding representations.

Now we consider Ind_{M^{0}}^{M}(χ). Since M = M^{0} ∪ γM^{0} and χ^{γ} = χ, we have

Ind_{M^{0}}^{M}(χ) = σ ⊕ (σ ⊗ η)

where σ, σ⊗η are distinct one-dimensional unitary representations of M which restrict to χ on M^{0}. Now, using transitivity of induction and properties of tensor products, we can write

Ind_{B^{0}}^{G}(χ) ≃ Ind_{B}^{G} Ind_{M^{0}}^{M}(χ)
≃ Ind_{B}^{G}(σ) ⊕ Ind_{B}^{G}(σ ⊗ η).

Now since Ind_{B}^{G}(σ ⊗ η) ≃ Ind_{B}^{G}(σ) ⊗ η, we see that

Ind_{B}^{G}(σ) ⊕ (Ind_{B}^{G}(σ) ⊗ η) ≃
Π_{11} ⊕ (Π_{11} ⊗ η) ⊕ Π_{ss} ⊕ (Π_{ss} ⊗ η) ⊕ 2Π_{1s}.

Thus we can assume that Π_{11}, Π_{ss} were chosen so that

Ind_{B}^{G}(σ) ≃ Π_{11} ⊕ Π_{ss} ⊕ Π_{1s}

and

Ind_{B}^{G}(σ ⊗ η) ≃ (Π_{11} ⊗ η) ⊕ (Π_{ss} ⊗ η) ⊕ Π_{1s}.

This example exhibits a number of unpleasant features. First, we have irreducible discrete series representations σ_{1} = σ and σ_{2} = σ ⊗ η of B such that Ind_{B}^{G}(σ_{1}) and Ind_{B}^{G}(σ_{2}) have a non-trivial intertwining, but are not equivalent. Second, Ind_{B}^{G}(σ) has 3 inequivalent irreducible subrepresentations, each occuring with multiplicity one, so that the dimension of its space of intertwining operators is 3. There are no subgroups of any possible Weyl groups here with order 3.

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David Goldberg, Department of Mathematics, Purdue University, West Lafayette, IN 47907

  E-mail address: goldberg@math.purdue.edu

Rebecca A. Herb, Department of Mathematics, University of Maryland, College Park, MD 20742

  E-mail address: rah@math.umd.edu