EXCEPTIONAL Θ-CORRESPONDENCES I

Kay Magaard and Gordan Savin

Abstract. Let G be a split simply laced group defined over a p-adic field F. In this paper we study the restriction of the minimal representation of G to various dual pairs in G. For example, the restriction of the minimal representation of E_{7} to the dual pair G_{2} × Sp(6) gives the non-endoscopic Langlands lift of irreducible representations of G_{2} to Sp(6).

Introduction

Let F be a p-adic field and G be a split simple group of type D_{n} or E_{n}, (n = 6, 7, 8) over F. If the type is D_{n} then G is SO(2n). In [KS] the minimal representation V of G was introduced and in [S] some further properties of it were studied. It is an analogue of the Weil representation of Sp(2n) and in some ways it is better behaved then the Weil representation. Therefore it makes sense to ask if one can use it to obtain dual pair correspondences.

We have two types of dual pairs in G. In SO(2n + 2) we have a dual pair

SO(2n − 1) × SO(3).

If G is exceptional then we have a dual pair

G_{2}(F) × H_{D}

with H_{D} of type A_{2}, C_{3} or F_{4}, respectively.

Let A × B be a dual pair in G and X an irreducible representation of A. We say that an irreducible representation Y of B is a Θ-lift of X if X ⊗ Y is a quotient of V (see [H1]). Let Θ(X) be the set of all such Y.

In this paper we compute Jacquet functors V_{U} of V where U is a unipotent radical of a maximal parabolic subgroup of SO(3) or H_{D}, with U commutative, or a unipotent radical of a maximal parabolic subgroup of G_{2} such that U is a Heisenberg group. The results are reminiscent of those of Kudla [Ku] and Rallis [Ra] in the classical case. We then use the knowledge of Jacquet functors to obtain dual pair correspondences. For example, we compute Θ-lifts of tempered spherical representations of SO(3), A_{2}, C_{3} and G_{2}. The lifts from A_{2} to G_{2} and from G_{2} to C_{3} obtained by restricting the minimal representation of E_{6} and E_{7} respectively, are Langlands correspondences.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 22E35 and 50, Secondary 11F70.

Research at MSRI is supported in part by NSF grant DMS-9022140

In a sequel to this paper we plan to compute Jacquet functors V_{U} for unipotent radicals of remaining maximal parabolic subgroups. Once this is done, it will be possible to compute lifts of most Langlands quotients and check the conjectural answer for the exceptional correspondences as given by B. Gross [G2]. Local computations are, in a way, a preparation for global correspondences (i.e. correspondences of automorphic forms). So it is worth mentioning that in a forthcoming work, D. Ginzburg, S. Rallis and D. Soudry are studying a global variant of exceptional correspondences. Another possible global application has recently been initiated by B. Gross in connection to a realization of a G_{2}-motive [G1].

The paper is organized as follows. In Section 1 we recall the definition and few properties of the minimal representation from [S]. In Section 2 we study the dual pair SO(3)×SO(2n− 1). It is the simpliest case and as such it is a good introduction to exceptional dual pairs which form a more interesting part of this work. In Sections 3 and 4 we study the dual pair G_{2} × A_{2}. We compute Θ-lifts of spherical tempered representations of A_{2}. In Section 5 we study the dual pair G_{2} × C_{3}. We compute Θ-lifts of spherical tempered representations of C_{3}. In Section 6 we recall some facts about simple Jordan algebras or rank 3. Exceptional groups can be described nicely in terms of Jordan algebras. We use this description in Section 7 to compute V_{U} where U is the unipotent radical of the ”Heisenberg” maximal parabolic subgroup of G_{2} in all three cases. We finish the paper by computing Θ-lifts of spherical tempered representations of G_{2} in all three cases.

1. Minimal representation

Let G be a simple, split, group of type A_{2n−1}, D_{n} or E_{n}. Let ^{L}G be the complex L-group of G (see [B]). It is well known that conjugacy classes of unipotent elements in LG correspond to conjugacy classes of homomorphisms

φ : SL_{2}(C) → ^{L}G.

Assume now that φ corresponds to the subregular unipotent orbit. Let q be the order of the residual field of F. Let

s = φ ( \begin{matrix}q^{\frac{1}{2}} & 0 \\ 0 & q^{− \frac{1}{2}} \end{matrix} ) .

Then V is the spherical representation of G with the Satake parameter s (see [Ca]).

Let 𝔤 be the Lie algebra of G. Let < ·, · > be the Killing form on 𝔤. Fix ψ : F → ℂ^{×}, a non-trivial character. Let C_{c}^{∞}(𝔤) denote the space of locally constant, compactly supported functions on 𝔤. Define the Fourier transform on C_{c}^{∞}(𝔤) by

f̂(y) = ∫_{𝔤} f(x)ψ(< x, y >)dx.

Let O_{min} be the unique minimal nilpotent G-orbit in 𝔤 and μ_{O_{min}} a G-invariant measure on O_{min} normalized as in [MW]. Let 𝔤_{ℤ} be the ℤ-span of a Chevaley basis of 𝔤. Let R be the ring of integers of F and ϖ the uniformizing element. Let

𝔤_{n} = 𝔤_{ℤ} ⊗_{ℤ} ϖ^{n}R.

In [S] we have shown that there exists m > 0 such that

tr ∫_{𝔤} f(x)π(exp x)dx = ∫ f̂μ_{O_{min}}

for any f ∈ C_{c}^{∞}(𝔤) supported on 𝔤_{m} and such that f̂(0) = 0.

This implies ”smallness” of the N-spectrum of V, where N is a unipotent radical of a maximal parabolic subgroup of G. More precisely, let Δ be the Dynkin diagram of G. Mark the diagram Δ as follows. Attach 0 to the unique branch vertex (or to the middle vertex of Δ if the type of G is A_{2n−1}) and 2 to all other. This marking corresponds to the subregular nilpotent orbit [D]. Let P = MN be a maximal parabolic subgroup of G. Let Δ_{M} be the Dynkin diagram of M. Assume that we are in the following favorable situation:

(1) N is a commutative group.

(2) The marking of Δ corresponding to the subregular nilpotent orbit of G restricts to

the marking of Δ_{M} corresponding to the subregular nilpotent orbit of M. The possible cases are given by the following table:

\begin{matrix}G & M & N \\ D_{n+1} & D_{n} & F^{2n} \\ E_{6} & D_{5} & F^{16} \\ E_{7} & E_{6} & F^{27} \end{matrix}

Here F^{2n} is the standard representation of SO(2n), F^{16} is a spin-representation of Spin_{10} and F^{27} is isomorphic to the exceptional Jordan algebra. We say that a point in N is singular if it is a highest weight vector for a Borel subgroup of M. Let ω and \overline{ω} be the sets of singular vectors in N and \overline{N}. Note that ω is the smallest non-trivial M-orbit in N. If G = SO(2n) then ω is the null-cone of the invariant quadratic form (with 0 excluded).

Theorem 1.1. The minimal representation V of G has a P -invariant filtration

0 → C_{c}^{∞}(\overline{ω}) → V → V_{N} → 0.

Here C_{c}^{∞}(\overline{ω}) denotes the space of locally constant, compactly supported functions on \overline{ω} and V_{N} is the space of N-coinvariants of V (Jacquet functor).

(1) Let f ∈ C_{c}^{∞}(\overline{ω}). The action of P is given by

π(n)f(x) = ψ(< x, n >)f(x), n ∈ N

and

π(m)f(x) = | det(m)|^{\frac{s}{d}} f(m^{−1}x), m ∈ M.

(2)

V_{N} ≅ V_{M} ⊗ | det |^{\frac{t}{d}} + | det |^{\frac{s}{d}}

where V_{M} is the minimal representation of M (center acting trivially).

In the above formulas < ·, · > is the F -valued pairing between N and \overline{N} induced by the Killing form on 𝔤, det is determinant of the representation of M on N, d is the dimension of N and the values of s and t are given in the following table.

\begin{matrix}G & s & t \\ D_{n+1} & n − 1 & 1 \\ E_{6} & 4 & 2 \\ E_{7} & 6 & 3 \end{matrix}

Proof. This is just Theorem 6.5 in [S] if G is E_{7}. The other two cases also satisfy conditions of Proposition 4.1 in [S]. Hence the proof carries over with no changes. The proof given in [S], however, is valid only when the residual characterisitic is odd. It remains to discuss the case when the residual characteristic is 2.

Let x ∈ \overline{N}, and define ψ_{x}(n) = ψ(< x, n >). Let V_{N,ψ_{x}} be the quotient of V by the space spanned by the elements {π(n)v −ψ_{x}(n)v|n ∈ N, v ∈ V }. The key point in the proof of Theorem 6.5 in [S] is to show that

V_{N,ψ_{x}} = 0

for x ≠ 0 and not in \overline{ω}, i.e. the N-spectrum of V is concentrated on the closure of the smallest M-orbit in \overline{N}. This follows from the work of Moeglin and Waldspurger [MW], but only if the residual characteristic of F is odd.

Let k be a number field and 𝔸 its ring of adeles. Ginzburg, Rallis and Soudry [GR] have constructed a square integrable automorphic form Π = ⊗_{v}Π_{v} on G_{𝔸} isomorphic to V at every finite place. But arguing exactly as Howe ([H2] Lemma 2.4), one shows that if the N-spectrum is concentrated on the closure of \overline{ω} at one place v, it has to be concentrated on the closure of \overline{ω} at all places. This completes the proof of the theorem.

2. Dual pair SO(2n − 1) × SO(3)

Let G = SO(2n + 2). Let e, h, f be the standard basis for sl(2), the Lie algebra of SO(3) = P GL_{2}(F). Let P = MN be the maximal parabolic of G as in Section 1. Let 𝔤 be the Lie algebra of G and let

𝔤 = \overline{𝔫} + 𝔪 + 𝔫

be the corresponding decomposition of 𝔤. The embedding SO(3) ⊆ SO(2n+2) corresponds to sl(2) ⊆ 𝔤 such that

(1) \overline{𝔫} = \{x ∈ 𝔤 | [h, x] = −2x\}.

(2) 𝔪 = {x ∈ 𝔤 | [h, x] = 0}.

(3) 𝔫 = {x ∈ 𝔤 | [h, x] = 2x}. Note that e ∈ 𝔫, f ∈ \overline{𝔫} and SO(2n − 1) is the centralizer of e in M.

Let Q = LU = P ∩ SO(3) be a Borel subgroup of SO(3).

Proposition 2.1. Let ( \begin{matrix}a \\ & b \end{matrix} ) ∈ L ⊂ P GL_{2}(F) = SO(3).

(1)

π(( \begin{matrix}a \\ & b \end{matrix} ))f(x) = |\frac{a}{b}|^{n−1}f(\frac{a}{b}x), f ∈ C_{c}^{∞}(\overline{ω}).

(2) The eigenvalues of π(( \begin{matrix}a \\ & b \end{matrix} )) on V_{N} are |\frac{a}{b}| and |\frac{a}{b}|^{n−1}.

Proof. This is just Theorem 1.1.

Let χ be a multiplicative character of F. Let ρχ denote the character of L defined by

ρχ(( \begin{matrix}a \\ & b \end{matrix} )) = |\frac{a}{b}|^{\frac{1}{2}} χ(\frac{a}{b}).

Let τ_{χ} = Ind_{Q}^{SO(3)} ρχ. If χ is unitary than τ_{χ} is an irreducible tempered spherical representation of SO(3). Let σ ∈ Θ(τ_{χ}). By Frobenius reciprocity

Hom_{SO(2n−1)×SO(3)}(V, σ ⊗ τ_{χ}) = Hom_{SO(2n−1)×L}(V_{U}, σ ⊗ ρχ).

Therefore, σ ∈ Θ(τ_{χ}) is a quotient of V if and only if σ ⊗ ρχ is a quotient of V_{U}. Since

0 → C_{c}^{∞}(\overline{ω})_{U} → V_{U} → V_{N} → 0

we need to understand C_{c}^{∞}(\overline{ω})_{U}. Let NN and \overline{NN} be the complements of \overline{U} and U in N and \overline{N}. Let \overline{ω}, as before, be the set of singular vectors in \overline{N}. Put

\overline{ωω} = \overline{ω} ∩ \overline{NN}.

Lemma 2.2.

C_{c}^{∞}(\overline{ω})_{U} = C_{c}^{∞}(\overline{ωω}).

Proof. Let us recall few known facts about Jacquet functors. Let E be a U-module. Then E_{U} = E/E(U) where E(U) can be defined either as the space spanned by the elements {π(u)v − v|u ∈ U, v ∈ E} or the space of all v such that

∫_{U_{c}} π(u)vdu = 0

for some open compact subgroup U_{c} ⊂ U depending on v (2.33 [BZ]).

Obviously, C_{c}^{∞}(\overline{ωω}) is a quotient of C_{c}^{∞}(\overline{ω}) and by Theorem 1.1 (1), U acts trivially on C_{c}^{∞}(\overline{ωω}). Let f ∈ C_{c}^{∞}(\overline{ω}) such that f|\overline{ωω} = 0. To prove the lemma, we need to find an open compact subgroup U_{c} such that

∫_{U_{c}} ψ(< x, u >)f(x)du = 0

for all x ∈ \overline{ω}. Let x be such that f(x) ≠ 0. Since x is not in \overline{NN}, there exists an open compact neighborhood O of x and an open compact subgroup U_{O} such that ψ(< y, u >) is a non-trivial character of U_{O} for any y ∈ O. Since the support of f is compact, a finite collection of O covers the support of f. The union of the corresponding U_{O} is the desired U_{c}. The lemma follows.

We can, therefore, summarize the situation with the following proposition.

Proposition 2.3. V_{U} has a filtration with two quotients:

C_{c}^{∞}(\overline{ωω}), and V_{N}.

As SO(2n − 1) × L-modules:

(1) the action on C_{c}^{∞}(\overline{ωω}) is geometric, twisted with |\frac{a}{b}|^{n−1} for ( \begin{matrix}a \\ & b \end{matrix} ) ∈ L.

(2)

V_{N} ≅ V_{M} ⊗ |\frac{a}{b}| + 1 ⊗ |\frac{a}{b}|^{n−1}

where V_{M} is the minimal representation of M (center acting trivially).

Note that \overline{NN} = F^{2n−1} and \overline{ωω} is the null-cone of the SO(2n − 1)-invariant quadratic form (with 0 excluded). Let C^{∞}(\overline{ωω}) be the space of locally constant functions on \overline{ωω}. We can define degenerate principal series representations σ_{χ} by

σ_{χ} = \{f ∈ C^{∞}(\overline{ωω}) | f(cx) = χ(c)|c|^{\frac{3}{2} −n}f(x)\}.

Analogously, σ_{χ} can be defined as a quotient of C_{c}^{∞}(\overline{ωω}) consisting of \overline{f} such that

\overline{f}(cx) = χ(c)|c|^{\frac{3}{2} −n}\overline{f}(x).

If χ is unramified and unitary than σ_{χ} is an irreducible unitarizable spherical representation by a result of Tadić [T2], Theorem 9.2. We are now ready to state and prove the main result of this section.

Proposition 2.4. Let χ be an unramified, unitary multiplicative character. Then

Θ(τ_{χ}) = {σ_{χ}}.

Proof. Let σ ∈ Θ(τ_{χ}). Then σ × ρχ is a quotient of V_{U}. Since ρχ is different from | · |^{n−1} and | · | if χ is unitary, it follows that σ × ρχ is quotient of C_{c}^{∞}(\overline{ωω}). Therefore σ must be isomorphic to σ_{χ}. To finish the theorem, we have to show that σ_{χ} ∈ Θ(τ_{χ}). By Frobenius reciprocity, it suffices to show that σ_{χ} × ρχ is a quotient of V_{U}. We know that it is a quotient of C_{c}^{∞}(\overline{ωω}). We need the following lemma.

Lemma 2.5. Let H be a group and

0 → V_{1} → V_{2} → V_{3} → 0

a sequence of H-modules. Let W be a H-invariant quotient of V_{1}. Assume that there exists a H-invariant linear map T on V_{2} reducing to scalar multiplications by λ and μ on W and V_{3}. If λ ≠ μ then W is a quotient of V_{3}.

Proof. Let V_{0} be such that

0 → V_{0} → V_{1} → W → 0.

Then

0 → W → V_{2}/V_{0} → V_{3} → 0

Since the eigenvalues of T are different, the last sequence splits and the lemma follows.

Let ϖ be the uniformizing element of F. Put

T = π ( \begin{matrix}ϖ \\ & 1 \end{matrix} ) .

If χ is unitary, T satisfies the conditions of Lemma, hence σ_{χ} × ρχ is a quotient of V_{U} and the theorem is proved.

3. Group E_{6}

Let J be the 27-dimensional exceptional Jordan algebra and (x, y, z) be the Dickson trilinear form on J (see [J2]). Let G_{S} be the isometry group of the Dickson form. It is a split, simply-connected groups of type E_{6} [A1].

Let 𝔤_{S} be the Lie algebra of G_{S}. Then

𝔤_{S} = 𝔫̄ + 𝔪_{S} + 𝔫

where [𝔪_{S}, 𝔪_{S}] = so(10) and 𝔫̄, 𝔫 are two spin-representations.

Let P_{S} = M_{S}N be the corresponding maximal parabolic subgroup in G_{S}. Then [M_{S}, M_{S}] = Spin_{10} and Spin_{10} can be defined as the subgroup of G_{S} stabilizing the 10-dimensional subspace J_{10} of J consisting of matrices in J such that the coefficients in the third row and the third column are zero and fixing the vector

d = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

In particular, Q_{d}(x, y) = (x, y, d) is the invariant quadratic form on J_{10} and N (one of the spin-modules) can be identified with the space in J consisting of matrices

( \begin{matrix}0 & 0 & y \\ 0 & 0 & z \\ \overline{y} & \overline{z} & 0 \end{matrix} ) .

The group G_{S} has a unique smallest, non-trivial orbit Ω on J. It is the orbit of d and its dimension is 17. On the other hand, M_{S} has two non-trivial orbits on N and the smaller (call it ω) can be identified with Ω ∩ N [A1].

4. Dual pair G_{2}(F) × P GL_{3}(F)

Let D be a nondegenerate subalgebra of the Octonions and let J_{D} denote the subalgebra of J obtained by restricting the off-diagonal entries of the matrices of J to D. When dim(D) = 1, i.e. D =< 1_{𝕆} >= F then J_{F} is the algebra of symmetric 3 × 3 matrices over F and C_{G_{S}}(J_{F}) = G_{2}(F), the split group of type G_{2}. The centralizer of G_{2}(F) in G_{S} is SL_{3}(F) [A1].

Let G, G_{S} ⊂ G, be a reductive group of type E_{6} appearing as a Levi factor of a maximal parabolic subgroup of a split, simply-connected group of type E_{7}. Then C_{G}(G_{2}(F)) = GL_{3}(F). The action of G_{S} on J can be extended to G. Indeed, the inclusion GL_{3} ⊂ G induces an isomorphism GL_{3}/[GL_{3}, GL_{3}] ≅ G/[G, G]. Since [G, G] = G_{S}, it suffices to extend the action to GL_{3} ⊂ SL_{3}. This is done by the formula

gxg^{t}

where x ∈ J, g ∈ GL_{3}(F) and g^{t} is the transpose of g.

Let 𝔤 be the Lie algebra of G. Then, as in Section 3,

𝔤 = 𝔫̄ + 𝔪 + 𝔫

with 𝔪_{S} ⊂ 𝔪. Let P = MN be the corresponding maximal parabolic subgroup of G. Then Q = LU = P ∩GL_{3}(F) is a maximal parabolic subgroup of GL_{3} with L = GL_{2}(F)× GL_{1}(F). One can write

gl(3) = 𝔲̄ + 𝔩 + 𝔲

where 𝔩 ⊂ 𝔪, \overline{𝔲} ⊂ \overline{𝔫} and 𝔲 ⊂ 𝔫.

Proposition 4.1. Let V be the minimal representation of G, the center acting trivially. Identify \overline{N} with the set of pairs of Octonions (y, z). Let f ∈ C_{c}^{∞}(\overline{ω}). Then:

(1)

π(g)f((y, z)) = f((g^{−1}y, g^{−1}z)), g ∈ G_{2}(F).

(2)

π(l_{2} × l_{1})f((y, z)) = \frac{| det l_{2}|^{2}}{|l_{1}|^{4}} f(l_{1}^{−1}(y, z)l_{2}), l_{2} × l_{1} ∈ GL_{2}(F) × GL_{1}(F)

where det denotes the usual determinant of 2 × 2 matrices.

Proof. This is just Theorem 1.1.

Let Y be an irreducible representations of GL_{3}(F) and Ỹ an irreducible representation of G_{2}(F) such that Ỹ ∈ Θ(Y). Assume that Y is a submodule of Ind_{Q}^{GL_{3}} X for some irreducible representation X of L. By Frobenius reciprocity

Hom_{G_{2}×GL_{3}}(V, Ỹ ⊗ Ind X) = Hom_{G_{2}×L}(V_{U}, Ỹ ⊗ X).

Therefore, Ỹ ⊗ Y is a quotient of V only if Ỹ ⊗ X is a quotient of V_{U}. Since

0 → C_{c}^{∞}(\overline{ω})_{U} → V_{U} → V_{N} → 0

we need to understand C_{c}^{∞}(\overline{ω})_{U}. Let NN and \overline{NN} be the orthogonal complements of \overline{U} and U in N and \overline{N}. Since U can be identified with the space in J consisting of matrices

( \begin{matrix}0 & 0 & y \\ 0 & 0 & z \\ y & z & 0 \end{matrix} )

where y, z ∈ F, it follows that

NN = \{( \begin{matrix}0 & 0 & y \\ 0 & 0 & z \\ \overline{y} & \overline{z} & 0 \end{matrix} ) | \overline{y} = −y and \overline{z} = −z\}.

Let \overline{ωω} = \overline{ω} ∩ \overline{NN}. As in Lemma 2.2 C_{c}^{∞}(\overline{ω})_{U} = C_{c}^{∞}(\overline{ωω}), and we have to understand the structure of G_{2}(F)×L orbits on \overline{ωω} which is, of course, the same as the structure of orbits on ωω = ω ∩ NN.

Proposition 4.2.

(1) \overline{ωω} = \{(y, z) ≠ (0, 0)|\overline{y} = −y, \overline{z} = −z and y^{2} = z^{2} = yz = 0\}.

(2) Let AA and BB be the subsets of \overline{ωω} consisting all pairs (y, z) such that the space F y + F z has dimension 2 and 1 respectively. Then AA and BB are G_{2}(F) × GL_{2}(F)-orbits.

Proof. Let us prove the corresponding statement for ωω. Let

n = ( \begin{matrix}0 & 0 & y \\ 0 & 0 & z \\ −y & −z & 0 \end{matrix} ) ∈ ωω.

Since n is a traceless matrix in J it will be singular iff n^{2} = 0. But this is equivalent to y^{2} = z^{2} = yz = 0. The first part of the proposition is proved.

We go on to observe that G_{2} has three orbits on the set of spaces of traceless Octonions with the property that the Octonion multiplication is trivial. These are characterized by their dimension; the possible choices being 0, 1, 2. The stabilizers of the nontrivial spaces are the maximal parabolics of G_{2}.

Let x be a traceless Octonion such that x^{2} = 0. Let P_{1} be the maximal parabolic subgroup of G_{2}(F) stabilizing the line F x. The Levi factor of P_{1} is ”spanned” by a long root. Consider

B = {(ax, 0)|a ∈ F and a ≠ 0}.

Let QQ be the maximal parabolic subgroup of GL_{2}(F) stabilizing B. Then P_{1} × QQ acts transitively on B and

BB = (G_{2}(F) × GL_{2}(F)) ×_{(P_{1}×QQ)} B.

Let y and z be two traceless and linearly independent Octonions such that y^{2} = z^{2} = yz = 0. Let P_{2} be the maximal parabolic subgroup of G_{2}(F) stabilizing the space F y +F z. The Levi factor of P_{2} is ”spanned” by a short root. Consider

A = {(ay + bz, cy + zd)|a, b, c, d ∈ F and ad − bc ≠ 0}.

Then P_{2} × GL(2) acts transitively on A and since G_{2}(F) acts transitively on the set of all two-dimensional spaces of traceless Octonions with trivial multiplication,

AA = G_{2}(F) ×_{P_{2}} A.

The proposition is proved.

We can now summarize the structure of V_{U} as a G_{2}(F)×GL_{2}(F)-module in the following theorem (compare [Ku]).

Theorem 4.3. V_{U} has a filtration with three quotients:

C_{c}^{∞}(AA), C_{c}^{∞}(BB), and V_{N}.

Moreover

(1)

C_{c}^{∞}(AA) = ind_{P_{2}}^{G_{2}}(C_{c}^{∞}(A)) ⊗ | det |^{2}

(2)

C_{c}^{∞}(BB) = ind_{P_{1}×QQ}^{G_{2}×GL_{2}}(C_{c}^{∞}(B)) ⊗ | det |^{2}

(3)

V_{N} ≅ V_{M} ⊗ | det | + 1 ⊗ | det |^{2}

as G_{2}(F) × GL_{2}(F)-modules.

We are now ready to state and prove the main theorem.

Theorem 4.4. Let Φ : SL_{3}(ℂ) → G_{2}(ℂ) be the standard inclusion of the L-groups of P GL_{3}(F) and G_{2}(F). Let Y be a tempered spherical representation of P GL_{3}(F). Let s ∈ SL_{3}(ℂ) be its Satake parameter. Let Ỹ be the tempered spherical representation of G_{2}(F) whose Satake parameter is Φ(s). Then

Θ(Y) = {Ỹ }.

Proof. Write Y = Ind_{Q}^{GL_{3}} X (note that there are up to three different choices for X). Then the restriction of X to GL_{2}(F) (a factor of L) is the irreducible spherical representation whose parameter is (χ| · |^{1/2}, μ| · |^{1/2}) where χ and μ are unitary characters. Let X̃ be the irreducible spherical representation of GL_{2}(F) whose parameter is (χ^{−1}| · |^{3/2}, μ^{−1}| · |^{3/2}). Note that C_{c}^{∞}(A) is the regular representation of GL(2). After taking into account the twist with | det |^{2} it follows that there is a unique quotient of C_{c}^{∞}(AA) isomorphic to X̃ ⊗X as a GL_{2}(F) × GL_{2}(F)-module. Here the first copy of GL_{2}(F) is the Levi factor of P_{2} and the second is the factor of L. Therefore,

Ind_{P_{2}}^{G_{2}}(X̃) ⊗ X

is a quotient of C_{c}^{∞}(AA). Write Ỹ = Ind_{P_{2}}^{G_{2}} X̃. Let U_{2} be the unipotent radical of P_{2}. It is Heisenberg group. Let Z be the center of U_{2}. The action of GL_{2}(F) (Levi factor of P_{2}) on Z is det and on U_{2}/Z it is isomorphic to S^{3}(F^{2}) ⊗ det^{−1}. It follows that ρ_{U_{2}} = | det |^{3}. Hence Ỹ is tempered. It is irreducible by a result of Keys [Ke]. We have to show that Ỹ ⊗ X is a quotient of V_{U} and that X does not appear as a quotient composition factors of V_{U} different from C_{c}^{∞}(AA).

We again use Lemma 2.5, but in this case T will be in the Bernstein center (see [BD]) of GL_{2}(F) ⊆ L. Recall that the component of the Bernstein center corresponding to representations generated by their Iwahori-fixed vectors is isomorphic to

ℂ[x, x^{−1}, y, y^{−1}]^{W}

where W = {1, w}, w(x) = y and w(y) = x is the Weyl group of GL_{2}. Let I be the Iwahori subgroup of GL_{2}(F). Let ϖ be the uniformizing element in F. Then any unramified character χ of F^{∗} is determined by its value on ϖ. If E is a subquotient of an induced representation with the parameter (χ_{1}, χ_{2}) then

(x + y) = χ_{1}(ϖ) + χ_{2}(ϖ) and xy = χ_{1}(ϖ)χ_{2}(ϖ)

on E. Let

T_{1} = q^{3/2}(x^{−1} + y^{−1}) − (xy)^{−1},

where q = |ϖ|^{−1}. On X T_{1} acts as the scalar

q^{2}(χ(ϖ)^{−1} + μ(ϖ)^{−1}) − qχ(ϖ)^{−1}μ(ϖ)^{−1},

which if real, is strictly less then 2q^{2}. On the other hand, C_{c}^{∞}(B) is the regular representation of GL_{1}, so C_{c}^{∞}(B) as a GL_{2}(F)-module consists of induced representations whose inducing parameters are (| · |^{3/2}, χ). It has an Iwahori-fixed vector only when χ is an unramified character. On such induced representations T_{1} acts as

q^{3} = q^{3/2}(q^{3/2} + χ(ϖ)^{−1}) − q^{3/2}χ(ϖ)^{−1}.

Hence T_{1} acts on C_{c}^{∞}(B)^{I} as the scalar q^{3}, which is always bigger then 2q^{2}. Therefore X can not be a quotient of C_{c}^{∞}(B).

Let

T_{2} = xy.

Then T_{2} acts on X as |ϖ|χ(ϖ)μ(ϖ) which is different from |ϖ|^{2} and |ϖ|^{4}, the eigenvalues of T_{2} on V_{N}^{I}. Therefore X can not be a quotient of V_{N}. Finally, Lemma 2.5 applied to T_{1} and T_{2} shows that Ỹ ⊗ X is a quotient of V_{U}.

To finish the proof we have to check that the correspondence Y → Ỹ is Langlands. Since Ỹ = Ind_{P_{2}}^{G_{2}} X̃ and P_{2} is spanned by a short root, the Satake parameter of Ỹ sits in a Levi factor of a parabolic subgroup of G_{2}(ℂ) spanned by a long root. Since SL_{3}(ℂ) is spanned by long roots of G_{2}, Ỹ must be a lift of a representation of P GL_{3}(F) induced from Q: Y or Y^{∗}. Note that replacing the pair (χ, μ) by (χ^{−1}, μ^{−1}) does not change Ỹ but replaces Y by Y^{∗}. The theorem is proved.

5. Dual pair G_{2}(F) × P GSp_{6}(F)

Let G be the simply connected group of type E_{7}. Let P = MN be the maximal parabolic subgroup of G such that M is the reductive group of type E_{6} discussed in the last section. The nilpotent radical N is commutative and isomorphic to J as an M-module. Let G_{2}(F) × GL_{3}(F) be the dual pair in M, described in Section 4. The centralizer of G_{2}(F) in G is Sp(6). This can be easily seen on the level of Lie algebras. Let 𝔤 be the Lie algebra of G. Then

𝔤 = 𝔫̄ + 𝔪 + 𝔫

and write

C_{𝔤}(G_{2}(F)) = 𝔲̄ + 𝔩 + 𝔲

where 𝔩 ⊂ 𝔪, \overline{𝔲} ⊂ \overline{𝔫} and 𝔲 ⊂ 𝔫. Then 𝔩 = gl(3) and 𝔲 ⊂ 𝔫 corresponds to the inclusion J_{F} ⊂ J since

J^{G_{2}(F)} = J_{F} .

Therefore C_{𝔤}(G_{2}(F)) = sp(6) whose Siegel parabolic subalgebra is 𝔩 + 𝔲. Note also that we have a distinguished SL_{2},

SL_{2}(F) ⊂ Sp_{6}(F) ⊂ G

such that if (e, h, f) is the standard basis of sl(2) (the Lie algebra of SL_{2}) then h ∈ 𝔪, f ∈ \overline{𝔫} and e ∈ 𝔫. Moreover e corresponds to the unit element of J under the identification of 𝔫 and J. Let G_{A} be the split adjoint group of type E_{7}, and let P_{A} = M_{A}N be the corresponding maximal parabolic subgroup. Then M_{A} can be defined as the group of isogenies of the Dickson form. We have a dual pair G_{2}(F) × P GSp_{6}(F) ⊂ G_{A}. Note that P GSp_{6}(F) is generated by the distinguished P GL_{2}(F) ⊂ P GSp_{6}(F) and the image of Sp_{6}(F).

Let Q = GL_{3}(F)U be the Siegel parabolic subgroup of Sp_{6}(F) and Q_{A} = LU be the corresponding maximal parabolic of P GSp_{6}(F). Let V be the minimal representation of G_{A}. We want to compute V_{U}. Since

0 → C_{c}^{∞}(\overline{ω})_{U} → V_{U} → V_{N} → 0

we need to understand C_{c}^{∞}(\overline{ω})_{U}. Let NN and \overline{NN} be the orthogonal complements of \overline{U} and U in N and \overline{N}. Since U can be identified with J_{F}, it follows that

NN = \{( \begin{matrix}0 & x & y \\ \overline{x} & 0 & z \\ \overline{y} & \overline{z} & 0 \end{matrix} ) | \overline{x} = −x, \overline{y} = −y and \overline{z} = −z\}.

Let \overline{ωω} = \overline{ω} ∩ \overline{NN}. As in Lemma 2.2 C_{c}^{∞}(\overline{ω})_{U} = C_{c}^{∞}(\overline{ωω}) and we have:

Proposition 5.1. Identify \overline{NN} with the set of triples of traceless Octonions (x, y, z). Let f ∈ C_{c}^{∞}(\overline{ωω}). The action of G_{2}(F) × L is given by

(1)

π(g)f((x, y, z)) = f((g^{−1}x, g^{−1}y, g^{−1}z)), g ∈ G_{2}(F).

(2)

π ( \begin{matrix}a \\ & b \end{matrix} ) f((x, y, z)) = |\frac{a}{b}|^{6}f(\frac{a}{b}(x, y, z)), ( \begin{matrix}a \\ & b \end{matrix} ) ∈ P GL_{2}(F) ⊂ P GSp_{6}(F).

(3)

π(g)f((x, y, z)) = | det g|^{4}f(det g(x, y, z)^{t}g^{−1}), g ∈ GL_{3}(F).

Proof. It follows from Theorem 1.1. Note, however, that in this case the formulas are describing the action on C_{c}^{∞}(\overline{ωω}).

To keep our notation simple we work in G. We have to understand the structure of G_{2}(F) × GL_{3}(F) orbits on \overline{ωω} which is, of course, the same as the structure of orbits on ωω = ω ∩ NN.

Proposition 5.2.

(1) \overline{ωω} = \{(x, y, z)|\overline{x} = −x, \overline{y} = −y, \overline{z} = −z and x^{2} = y^{2} = z^{2} = xy = xz = yz = 0\}.

(2) Let AA and BB be the subsets of \overline{ωω} consisting of all triples (x, y, z) such that the space F x + F y + F z has dimension 2 and 1 respectively. Then AA and BB are G_{2}(F) × GL_{3}(F)-orbits and

\overline{ωω} = AA ∪ BB.

Proof. Let us prove the corresponding statement for ωω. Let

n = ( \begin{matrix}0 & x & y \\ −x & 0 & z \\ −y & −z & 0 \end{matrix} ) ∈ ωω.

Since n is a traceless matrix in J it will be singular iff n^{2} = 0. But this is equivalent to x^{2} = y^{2} = z^{2} = xy = xz = yz = 0. The first part of the proposition is proved.

Again, recall that G_{2} has three orbits on the set of spaces of traceless Octonions with the property that the Octonion multiplication is trivial. These are characterized by their dimension; the possible choices being 0, 1, 2. The stabilizers of the nontrivial spaces are the maximal parabolics of G_{2}. It follows that x, y and z are linearly dependent. hence \overline{ωω} = AA ∪ BB. It remains to show that AA and BB are single orbits. The proof is analogous to the proof of Proposition 4.2.

Let x be a traceless Octonion such that x^{2} = 0. Let P_{1} be the maximal parabolic subgroup of G_{2}(F) stabilizing the line F x. Consider

B = {(ax, 0, 0)|a ∈ F and a ≠ 0}.

Let Q_{1} be the maximal parabolic of GL_{3}(F) stabilizing B. Then P_{1} × Q_{1} acts transitively on B and

BB = (G_{2}(F) × GL_{3}(F)) ×_{(P_{1}×Q_{1})} B.

Let y and z be two traceless and linearly independent Octonions such that y^{2} = z^{2} = yz = 0. Let P_{2} be the maximal parabolic subgroup of G_{2}(F) stabilizing the space F y + F z. Consider

A = {(ay + bz, cy + dz, 0)|a, b, c, d ∈ F and ad − bc ≠ 0}.

Let Q_{2} be the maximal parabolic subgroup of GL_{3}(F) stabilizing A. Then P_{2} × Q_{2} acts transitively on A and

AA = (G_{2}(F) × GL_{3}(F)) ×_{(P_{2}×Q_{2})} A.

The proposition is proved.

We can now summarize the structure of V_{U} as a G_{2}(F) × GL_{3}(F)-module.

Theorem 5.3. V_{U} has a filtration with three subquotients:

C_{c}^{∞}(AA), C_{c}^{∞}(BB), and V_{N}.

Moreover

(1)

C_{c}^{∞}(AA) = ind_{P_{2}×Q_{2}}^{G_{2}×GL_{3}}(C_{c}^{∞}(A)) ⊗ | det |^{4}

(2)

C_{c}^{∞}(BB) = ind_{P_{1}×Q_{1}}^{G_{2}×GL_{3}}(C_{c}^{∞}(B)) ⊗ | det |^{4}

(3)

V_{N} ≅ V_{M} ⊗ | det |^{2} + 1 ⊗ | det |^{4}

as G_{2}(F) × GL_{3}(F)-modules. Here det denotes the usual determinant of 3 × 3 matrices.

We are now ready to state and prove the main theorem.

Theorem 5.4. Let Φ : G_{2}(ℂ) → Spin_{7}(ℂ) be the standard inclusion of the L-groups of G_{2}(F) and P GSp_{6}(F); G_{2}(ℂ) fixes a non-zero vector in the 8-dimensional spin representation of Spin_{7}(ℂ). Let Y′ be a tempered spherical representation of P GSp_{6}(F). Then Θ(Y′) is not empty only if the Satake parameter of Y′ is Φ(s) for some s, a Satake parameter of a tempered spherical representation Ỹ of G_{2}(F). In that case

Θ(Y′) = {Ỹ }.

Proof. Let X′ be a spherical tempered representation of P GSp_{6}(F). Recall that every tempered spherical representation of P GSp_{6}(F) is fully induced (see [T1]). To simplify notation we work with Sp_{6}(F). The restriction of X′ to Sp_{6}(F) can be written as Ind_{Q}^{Sp_{6}} X ⊗| det |^{2} where X is a tempered spherical representation of GL_{3}(F) (note that ρ_{U} = | det |^{2}).

Let Y′ be a spherical tempered representation of P GSp_{6}(F) whose Satake parameter is Φ(s). This means that the restriction of Y′ to Sp_{6}(F) can be written as Ind_{Q}^{Sp_{6}}(Y ⊗| det |^{2}) where Y is a tempered spherical representation of P GL_{3}(F). Also Ỹ, the representation of G_{2}(F) whose parameter is s, is the lift of Y from P GL_{3}(F). By Theorem 5.3, V_{M} ⊗| det |^{2} is a quotient of V_{U} (M is a group of type E_{6}). It follows from the Frobenius reciprocity that Ỹ ∈ Θ(Y′).

The rest of the theorem follows form the knowledge of V_{U}. Indeed, let X′ be a tempered spherical representation of P GSp_{6}(F). Again, it is a fully induced representation so its restriction to Sp_{6}(F) can be written as Ind_{Q}^{Sp_{6}} X ⊗ | det |^{2} where X is a tempered spherical representation of GL_{3}(F). If Ỹ ⊗ X′ is a quotient of V for some irreducible representation Ỹ of G_{2}(F), then Ỹ ⊗ (X ⊗ | det |^{2}) is a quotient of V_{U}, i.e. it is a quotient of one of the three pieces in Theorem 5.3. For example, if it is a quotient of C_{c}^{∞}(AA) then

Ỹ ⊗ (X ⊗ | det |^{2}) = Ind_{P_{2}}^{G_{2}}(X̃) ⊗ (X ⊗ | det |^{2})

where the parameter of X̃ is (| · |^{3/2}χ^{−1}, | · |^{3/2}μ^{−1}) and the parameter of X is (χ, μ, χμ) for some unitary characters χ and μ. Since Ind X ⊗ | det |^{2} = Ind Y ⊗ | det |^{2}, where Y is a representation of P GL_{3}(F) with parameter (χ, μ, χ^{−1}μ^{−1}), we are back in the situation as in the beginning of the proof. The theorem is proved.

With only minor modifications of the preceeding proof one can easily show the following proposition.

Proposition 5.5. Let Y be a spherical representation of P GSp_{6}(F) which appears as a quotient of V, the minimal representation of E_{7}. Then the Satake parameter of Y is Φ(s) for some s ∈ G_{2}(ℂ).

6. Some facts about Jordan algebras

In this section we shall describe structure of exceptional groups using simple Jordan algebras of rank 3.

Let D be a non-degenerate subalgebra of Octonions. Let J_{D} ⊆ J be the corresponding Jordan algebra of rank 3. The Dickson form on J_{D} is given by

6(x, y, z) = 2tr(xyz) − tr(x)tr(yz) − tr(y)tr(zx) − tr(z)tr(xy) − tr(x)tr(y)tr(z).

Define

𝔩_{D} = {x ∈ gl(J_{D}) | (xv, v, v) = 0 for all v ∈ J_{D}}.

Now assume that D = F, F + F, M_{2}(F) (2 × 2-matrices) or 𝕆. Let G_{D} be a simply connected group of type F_{4} or E_{n}, n = 6, 7 or 8 respectively. Let 𝔤_{D} be its Lie algebra. Let Δ be the Dynkin diagram of 𝔤_{D}. We shall identify it with a set of simple roots. Let α̃ be the highest positive root. Let α be the unique simple root not perpendicular to α̃. Let 𝔪_{D} ⊂ 𝔤_{D} be the simple algebra whose Dynkin diagram is Δ\{α}. Extend Δ by adding −α̃. Let β be the unique simple root not perpendicular to α. Remove the vertex corresponding to the simple root β. The extended diagram breaks into several pieces, one of which is an A_{2} diagram corresponding to {α, −α̃}. Then 𝔩_{D} ⊂ 𝔤_{D} corresponds to the rest of the diagram. Consider the adjoint action of sl(3) + 𝔩_{D} on 𝔤_{D}. One has a decomposition

𝔤_{D} = sl(3) + 𝔩_{D} + V ⊗ J_{D} + V^{∗} ⊗ J_{D}^{∗}

where V is the standard 3-dimensional representation of sl(3) (see [HP]). Let M_{D} and L_{D} be the corresponding simply connected groups and let H_{D} be a subgroup of L_{D} centralizing the identity of J_{D}. The possible choices are given by the Freudenthal’s magic square:

\begin{matrix}G_{D} & M_{D} & L_{D} & H_{D} & J_{D} \\ F_{4} & Sp_{6} & SL_{3} & SO_{3} & S^{2}V \\ E_{6} & SL_{6} & SL_{3} × SL_{3} & SL_{3} & V ⊗ V^{∗} \\ E_{7} & E_{6} & SL_{6} & Sp_{6} & ∧^{2}F^{6} \\ E_{8} & E_{7} & E_{6} & F_{4} & F^{27} \end{matrix}

In the above table we have identified J_{D} with a ”usual” representation of L_{D} whenever possible. Since the centralizer of H_{D} in J_{D} is one-dimensional, it follows that the centralizer of H_{D} in 𝔤_{D} is (see [FH])

𝔤_{2} = sl(3) + V + V^{∗}.

Assume that D ≠ F. We shall need few facts about the action of H_{D} on J_{D}^{0}, the set of traceless elements in J_{D}. Recall that a point x ∈ J_{D}^{0} is singular if x^{2} = 0. The group H_{D} acts transitively on the set of singular vectors in J_{D}^{0}. Denote by Q_{1} a parabolic subgroup of H_{D} stabilizing a singular line in J_{D}^{0}.

Let F y + F z ⊂ J_{D}^{0} be a singular space, i.e. each point is singular. This is equivalent to

y^{2} = z^{2} = yz = 0.

We need to know H_{D}-orbits of singular two-dimensional subspaces of J_{D}^{0}. We have two different cases.

If dim D = 2 then J_{D} is isomorphic to the algebra of all 3 × 3-matrices with coefficients in F and H_{D} = SL_{3} acts by conjugation. The restriction of the Dickson form to J_{D} is given by

(x, x, x) = det(x)

where det is the determinant of 3 × 3 matrices. In this case, a singular point in J_{D}^{0} is a nilpotent rank-one matrix. There are two SL_{3}-orbits of singular two-dimensional spaces. Let F y +F z be a singular space. Then either the images or the kernels of y and z coincide. Denote by Q^{+} or Q^{−} the stabilizer of the singular space depending on the orbit. It is a maximal parabolic subgroup of SL_{3}.

In other cases, a stabilizer of a singular two-dimensional space is a parabolic subgroup only if the space is ”amber” ([A]).

Definition 6.1. Let S ⊂ J_{D}^{0} be a singular two-dimensional space. We say that S is amber if S ⊂ xJ_{D} for every x ∈ S.

Proposition 6.2. If dim D ≥ 4 then H_{D} acts transitively on the space of amber twodimensional subspaces of J_{D}^{0}.

Proof. If dim D = 8 this is a result of Aschbacher, 9.3-5 [A1]. Let F y + F z = S be an amber space. Since H_{D} acts transitively on the set of singular vectors, we can assume that z = x, where x is a fixed singular vector in J_{D}^{0}. Let Q_{1} be a parabolic subgroup of H_{D} stabilizing the line F x. Let L_{1} be a Levi factor of Q_{1}. Let I_{x} ⊂ J_{D} be an L_{1}-invariant space such that

xJ_{D} ∩ J_{D}^{0} = F x + I_{x}.

To finish the proof, we have to show that L_{1} acts transitively on the set of singular points in I_{x}.

If dim D = 4 then L_{1} = GL_{2} × SL_{2} and I_{x} can be identified with the space of 2 × 2matrices. If dim D = 8 then [L_{1}, L_{1}] = Spin_{7} and I_{x} is the corresponding 8-dimensional spin-representation. In both cases L_{1} has two non-trivial orbits in I_{x}, the smaller consisting of singular points. The proposition is proved.

Let Q_{2} be the stabilizer of an amber plane. We finish this section with a description of Q_{2} in both cases. Identify the amber space with F^{2}. Let L_{2} be a Levi factor of Q_{2}.

If dim D = 4 then Q_{2} is contained in a Siegel parabolic subgroup of Sp(6) and L_{2} = GL_{2} × GL_{1}. The action of L_{2} on the amber space is the standard action of GL_{2} × GL_{1} on the space of 2 × 1 matrices.

If dim D = 8. Then L_{2} is a quotient of GL_{3}×GL_{2} obtained by identifying centers of both factors (consider GL_{3}×GL_{2} acting on 3×2 matrices, for example). Let (l_{3}, l_{2}) ∈ GL_{3}×GL_{2}. The action of L_{2} on the amber space is given by

det(l_{3}) det(l_{2})l_{2}x x ∈ F^{2}.

In particular, the action of L_{2} on the amber space induces a non-split exact sequence

1 → SL_{3} → L_{2} → GL_{2} → 1.

We have [L_{2}, L_{2}] = SL_{3} ×SL_{2} where SL_{3} is spanned by long roots and SL_{2} by short roots.

7. Jacquet functor for G_{2} - Heisenberg parabolic

We continue to use the notation and definitions introduced in the previous section. Let

x = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & −1 \end{matrix} ) ∈ sl(3) ⊂ 𝔤_{D}.

Let 𝔤(k) = {y ∈ 𝔤_{D}|[x, y] = ky}. Then 𝔤(k) ≠ 0 for k = −2, −1, 0, 1, 2. Let 𝔭 = 𝔪+𝔫 be the maximal parabolic subalgebra with 𝔪 = 𝔤(0) and 𝔫 = 𝔤(1) + 𝔤(2). Note that [𝔪, 𝔪] = 𝔪_{D} and 𝔫 is a Heisenberg Lie algebra with center 𝔤(2). Let P = MN be the corresponding maximal parabolic subgroup. Let Z be the center of N and let N_{D} be the quotient of N by Z. Then N_{D} is an irreducible M_{D}-module. Since N_{D} ≅ 𝔤(1), the restriction to L_{D} is isomorphic to

F + J_{D} + J_{D}^{∗} + F.

The centralizer of H_{D} in G_{D} is G_{2}(F), and

G_{2}(F) ∩ P = P_{2} = GL_{2}(F)U_{2}

is the Heisenberg parabolic of G_{2}(F). Then Z ⊂ U_{2} and let U be the quotient of U_{2} by Z.

Let Ω be the set of singular vectors in N_{D}. Let \overline{N} denote the opposite nilpotent subgroup. Let \overline{N}_{D}, \overline{Z}... be the corresponding objects attached to \overline{N}. Note that the Killing form on 𝔤_{D}, the Lie algebra of G_{D}, defines a non-degenerate pairing < ·, · > between N_{D} and \overline{N}_{D}.

Theorem 7.1. Assume that the residual characteristic of F is odd. Let V be the minimal representation of G_{D}. Let Z be the center of N as above. Then V_{Z} has a P -invariant filtration

0 → C_{c}^{∞}(\overline{Ω}) → V_{Z} → V_{N} → 0

where C_{c}^{∞}(\overline{Ω}) denotes the space of locally constant, compactly supported functions on \overline{Ω}. (1) Let f ∈ C_{c}^{∞}(\overline{Ω}). Then

π(n)f(x) = ψ(< x, n >)f(x), n ∈ N

and

π(m)f(x) = | det(m)|^{\frac{s}{d}} f(m^{−1}x), m ∈ M.

(2)

V_{N} ≅ V_{M} ⊗ | det |^{\frac{t}{d}} + | det |^{\frac{s}{d}}

where V_{M} is the minimal representation of M (center acting trivially). In the above formulas det is the determinant of the representation of M on N_{D}, d is the dimension of N_{D} and the values of s and t are given in the following table.

\begin{matrix}G & s & t \\ E_{6} & 4 & 3 \\ E_{7} & 6 & 4 \\ E_{8} & 10 & 6 \end{matrix}

Proof. Part (2) is Prop. 4.1 [S]. Let W be the kernel of the projection of V_{Z} onto V_{N}. Let x ∈ \overline{N}_{D} and let

ψ_{x}(y) = ψ(< x, y >)

be a character of N. The theorem of Moeglin and Waldspurger [MW] implies that dim W_{N,ψ_{x}} = 0 or 1 and it is one if and only if x ∈ \overline{Ω}. Let x ∈ \overline{Ω}. Let M_{x} be the stabilizer of x in M and χ the character of M_{x} describing the action of M_{x} on W_{N,ψ_{x}}. By Frobenius reciprocity there exists a non-trivial P -homomorphism

T : W → Ind_{M_{x}N}^{P}(χ ⊗ ψ_{x}).

Let C^{∞}(\overline{Ω}) denote the space of locally constant functions on \overline{Ω}. Note that we have an inclusion

Ind_{M_{x}N}^{P}(χ ⊗ ψ_{x}) ⊆ C^{∞}(\overline{Ω}).

Let w ∈ W and f = T (w). We need to show that f is a compactly supported function on \overline{Ω}. Let N_{D,k}, k ∈ ℤ be a family of lattices of N_{D} such that ∪_{k}N_{D,k} = N_{D} and ∩_{k}N_{D,k} = 0. Let \overline{N}_{D,k} be their dual lattices in \overline{N}_{D}. Since W is a smooth module, there exists a small integer k depending on w such that π(n)w = w for all n ∈ N_{D,k}. This implies that f is supported inside \overline{N}_{D,k}. Since W_{N} = 0 there exists an integer l depending on w such that

∫_{N_{D,l}} π(n)wdn = 0

(see 2.33 [BZ]). This implies that f is supported outside \overline{N}_{D,k}. Since \overline{Ω} is locally closed and the boundary is \{0\}, it follows that f ∈ C_{c}^{∞}(\overline{Ω}). Let ind denote smooth induction with compact support. By the Bernstein-Zelevinsky analogue of Mackey Theory (see [BZ], pages 46-47)

ind_{M_{x}N}^{P}(χ ⊗ ψ_{x}) = C_{c}^{∞}(\overline{Ω})

is irreducible P -module. Hence T (W) = C_{c}^{∞}(\overline{Ω}). Let W′ be the kernel of T. Since

dim W_{N,ψ_{x}} = dim C_{c}^{∞}(\overline{Ω})_{N,ψ_{x}}

for any x ∈ \overline{N}_{D}, it follows that W′_{N,ψ_{x}} = 0 for any x ∈ \overline{N}_{D} (2.35 [BZ]). Therefore W′ = 0 by 5.14 [BZ].

Note that the inclusion M_{x} → M induces an isomorphism M_{x}/[M_{x}, M_{x}] ≅ M/[M, M] This can be easily checked by choosing x to be in 𝔤_{D,α} ⊂ 𝔤(1), α is defined in the previous section. Hence χ is a character of M and to finish the proof we have to show that

χ(m) = | det(m)|^{\frac{s}{d}} m ∈ M.

Let GL_{2}(F) be the Levi factor of P_{2} defined above. The inclusion of GL_{2}(F) into M induces in isomorphism GL_{2}(F)/SL_{2}(F) ≅ M/[M, M]. This can be seen by looking at the action of GL_{2}(F) and M on Z. Therefore it suffices to find the restriction of χ to GL_{2}(F). In Section 8 we shall use the information on correspondences obtained in previous sections to find the character.

Let NN be the orthogonal complement of \overline{U} in N_{D}. Let J_{D}^{0} be the set of traceless elements of J_{D}. Then NN as a GL_{2}(F) × H_{D}-module is isomorphic to

F^{2} ⊗ J_{D}^{0}.

Write

ΩΩ = Ω ∩ NN.

Proposition 7.2.

(1) \overline{ΩΩ} = \{(y, z) ≠ (0, 0)| y, z ∈ J_{D}^{0} and the space F y + F z is amber \}.

(2) Let AA and BB be the subsets of \overline{ΩΩ} consisting of all pairs (y, z) such that the space F y + F z has dimension 2 and 1 respectively. Then BB is a GL_{2}(F) × H_{D}- orbit. If dim D ≥ 4 then AA is a GL_{2}(F) × H_{D}-orbit. If dim D = 2 then AA is a union of two orbits.

Proof. Recall that a two-dimensional space S ⊂ J_{D}^{0} is amber if every element is singular and

S ⊂ xJ_{D}

for all x ∈ S. As before, write

N_{D} = F + J_{D} + J_{D}^{∗} + F.

Let Q_{D} = LU_{D} ⊆ M_{D} be the stabilizer of the last summand in the above decomposition. Then [L, L] = L_{D}, U_{D} ≅ J_{D} as an L_{D}-module and Q_{D} stabilizes the partial flag

N_{D} ⊃ J_{D} + J_{D}^{∗} + F ⊃ J_{D}^{∗} + F ⊃ F.

More precisely, let u ∈ U_{D}, (a, y, z, b) ∈ N_{D} and (a′, y′, z′, b′) = u(a, y, z, b). Then

a′ = a, y′ = y + au, z′ = z + 2u × y + au × u,
b′ = b + tr(uy) + 3(u, u, z) + a det u,

where det u = (u, u, u) and u × y, the cross product, is an element of J_{D} such that

tr((u × y)x) = 3(x, y, u)

for all x ∈ J_{D} (see [Ki] page 143).

Lemma 7.3. The group Q_{D} has 4 orbits on Ω. Their representatives are

v_{1} = (1, 0, 0, 0), v_{2} = (0, x, 0, 0)
v_{3} = (0, 0, x, 0), v_{4} = (0, 0, 0, 1)

where x is any singular element in J_{D}.

Proof. Note that ℙ(Ω) = M_{D}/Q_{D}. We have to compute Q_{D}\M_{D}/Q_{D} which is the same as W_{L}\W_{M}/W_{L}, here W_{M} and W_{L} denote the Weyl groups of M_{D} and L_{D}. Since N_{D} is a miniscule representation i.e. weight vectors are all contained in one W_{M}-orbit, it follows that they are parametrized by W_{M}/W_{L}. On the other hand J_{D} is a miniscule representation of L_{D} so W_{L}\W_{M}/W_{L} has four orbits and it follows easily that v_{i}, (1 ≤ i ≤ 4) form a complete set of representatives of Q_{D}-orbits. The lemma is proved.

Let (0, y, z, 0) ∈ \overline{ΩΩ}. If y ≠ 0 then Lemma 7.3 implies that it is Q_{D} conjugated to v_{2}. Therefore y is in the L-orbit of x, hence y is singular. Since the action of GL_{2}(F) is

(y, z) ( \begin{matrix}a & b \\ c & d \end{matrix} ) = (ay + cz, by + dz)

the same argument implies that any element of F y + F z is singular.

Note that (0, y, z, 0) is Q_{D}, hence U_{D}-conjugated to (0, y, 0, 0). But this means that z = 2u × y for some u ∈ U_{D} ≅ J_{D} such that tr(uy) = 0. Since tr((u × y)v) = 3(u, y, v) for all elements v ∈ J_{D} and

6(u, y, v) = 2tr(uyv) − tr(u)tr(yv)

(use tr(y) = tr(uy) = 0), it follows that

tr((2u × y)v) = tr(u^{∗}yv)

for all v ∈ J_{D}, here u^{∗} = 2u − tr(u). Therefore z = u^{∗}y ∈ yJ_{D}. Since the same argument can be repeated for any linear combination of y and z, the first part of the proposition follows.

Let x ∈ J_{D}^{0} such that x^{2} = 0. Let Q_{1} be the parabolic subgroup of H_{D} stabilizing the line F x. Consider

B = {(ax, 0)|a ∈ F and a ≠ 0}.

Let QQ ⊂ GL_{2}(F) be the Borel subgroup stabilizing the line B. Then QQ × Q_{1} acts transitively on B and

BB = (GL_{2}(F) × H_{D}) ×_{(QQ×Q_{1})} B.

Assume now that dim D ≥ 4. Let y, z ∈ J_{D}^{0} such that the space F y + F z is amber. Let Q_{2} be the parabolic subgroup of H_{D} stabilizing the space F y + F z. Consider

A = {(ay + bz, cy + zd)|a, b, c, d ∈ F and ad − bc ≠ 0}.

Then GL_{2}(F) × Q_{2} acts transitively on A and

AA = H_{D} ×_{Q_{2}} A.

If dim D = 2 then we have two orbits of singular two-dimensional spaces. Let F y^{+} + F z^{+} and F y^{−} + F z^{−} be their representatives and Q^{+} and Q^{−} their stabilizers in SL_{3}(F). One can define A^{+} and A^{−} as above, hence

AA = SL_{3}(F) ×_{Q^{+}} A^{+} ∪ SL_{3}(F) ×_{Q^{−}} A^{−}.

The proposition is proved.

Since C_{c}^{∞}(\overline{Ω})_{U_{2}} = C_{c}^{∞}(\overline{ΩΩ}), the structure of V_{U_{2}} as GL_{2}(F) × H_{D} is given as follows.

Theorem 7.4. Assume that the residual characteristic of F is odd. V_{U_{2}} has a filtration with three quotients:

C_{c}^{∞}(AA), C_{c}^{∞}(BB), and V_{N}.

Moreover, as GL_{2}(F) × H_{D}-modules, (1) If dim D ≥ 4 then:

C_{c}^{∞}(AA) = | det |^{s} ⊗ ind_{Q_{2}}^{H_{D}}(C_{c}^{∞}(A))

(2) If dim D = 2 then:

C_{c}^{∞}(AA) = | det |^{s} ⊗ ind_{Q^{+}}^{SL_{3}}(C_{c}^{∞}(A^{+})) + | det |^{s} ⊗ ind_{Q^{−}}^{SL_{3}}(C_{c}^{∞}(A^{−}))

(3)

C_{c}^{∞}(BB) = | det |^{s} ⊗ ind_{QQ×Q_{1}}^{GL_{2}(F)×H_{D}}(C_{c}^{∞}(B))

(4)

V_{N_{D}} ≅ | det |^{t} ⊗ V_{M} + | det |^{s} ⊗ 1

where V_{M} is the minimal representation of M_{D} (center acting trivially).

In the above formulas det is the usual determinant of 2 × 2 matrices, and s and t are given by the following table:

\begin{matrix}G & s & t \\ E_{6} & 2 & 3/2 \\ E_{7} & 3 & 2 \\ E_{8} & 5 & 3 \end{matrix}

Remark: In this section we have assumed that G_{D}, and hence H_{D} are simply connected. In fact, the isogeny class of H_{D} determines the isogeny class of G_{D} and vice-versa. Note that the action of H_{D} on C_{c}^{∞}(\overline{ΩΩ}) is geometric. It comes from the (algebraic) action of H_{D} on J_{D}^{0} which extends uniquely to the group of adjoint type. Therefore we have a canonical way of extending the representation V to groups of adjoint type.

8. Θ-lifts from G_{2}(F)

In this section we compute Θ-lifts of spherical tempered representations of G_{2}(F) in all three cases. In the process we also compute the normalizing factors (i.e. coefficients s) in Theorems 7.1 and 7.4. We assume that the residual characteristic of F is odd.

We study the dual pair G_{2}(F) × F_{4}(F) in a simple group G of type E_{8} first. Let Ỹ be a spherical tempered representation of G_{2}(F). Then Ỹ = Ind_{P_{2}}^{G_{2}}(X̃) where X̃ is a spherical representation of GL_{2}(F) with the parameter (χ^{−1}| · |^{3/2}, μ^{−1}| · |^{3/2}). As before, χ and μ must be unitary characters. Let Q_{2} = L_{2}V_{2} be the maximal parabolic subgroup of F_{4}(F) stabilizing an amber projective line. In Section 6 we have shown that the action of L_{2} on the corresponding amber line gives an exact sequence

1 → SL_{3} → L_{2} → GL_{2} → 1.

One checks that ρ_{V_{2}} = | det |^{7}, here det is the usual determinant on GL_{2}. Let X be a spherical representation of GL_{2} with the parameter (χ| · |^{7/2}, μ| · |^{7/2}). Pull X back to L_{2}. Let

W = Ind_{Q_{2}}^{F_{4}} X.

Note that W is a unitarizable representation of F_{4}(F). It is quite possible that W is always irreducible but we do not know.

Theorem 8.1. Let Ỹ be a spherical tempered representation of G_{2}(F) and W the representation of F_{4}(F) defined above. Then Θ(Ỹ) ≠ {} and if W′ ∈ Θ(Ỹ) then W′ is a summand of W. Let s ∈ G_{2}(ℂ) be the Satake parameter of Ỹ. Then the Satake parameter of W is Ψ(s × ρ) where Ψ : G_{2}(ℂ) × SO_{3}(ℂ) → F_{4}(ℂ) and ρ ∈ SO_{3}(ℂ) is the Satake parameter of the trivial representation of SL_{2}(F).

Proof. Let P = MN be the maximal parabolic subalgebra of G studied in the last two sections. We described an embedding of the dual pair G_{2}(F) × F_{4}(F) in G such that G_{2}(F) ∩ P is the Heisenberg maximal parabolic subgroup P_{2}.

Yet another embedding of the dual pair G_{2}(F)×F_{4}(F) is given by the inclusion of Jordan algebras J_{F} → J (use the description of the exceptional Lie algebras given in Section 6). In this case, G_{2}(F) ⊂ M and F_{4}(F) ∩ P = Q_{4} = L_{4}V_{4}, the Heisenberg maximal parabolic subgroup of F_{4}(F). The Levi component L_{4} is isomorphic to GSp_{6}(F). Note that the inclusion GSp_{6}(F) → M induces an isomorphism GSp_{6}(F)/Sp_{6}(F) ≅ M/[M, M]. This is easily seen by considering the action of GSp_{6}(F) and M on Z, the center of both, V_{4} and N.

Let V be the minimal representation of G. As an G_{2}(F) × GSp_{6}(F)-module

V_{N} ≅ V_{M} ⊗ | det |^{3} + 1 ⊗ | det |^{5}

where det denotes the usual determinant on GSp_{6}(F) (Prop. 4.1 [S]). Let Ỹ = Ind_{P_{2}}^{G_{2}}(X̃) be a spherical tempered representation of G_{2}(F) and let Y′ be the Langlands lift of Ỹ to P GSp_{6}(F). By Theorem 5.4 Ỹ ⊗ Y′ is a quotient of V_{M}. It follows from the Frobenius reciprocity that Ỹ ⊗ W′ is a quotient of V for some W′, a subquotient of

Ind_{Q_{4}}^{F_{4}}(Y′ ⊗ | det |^{3}).

On the other hand, X̃ ⊗ W′ must be a quotient of V_{U_{2}}. By Theorem 7.4

C_{c}^{∞}(AA) = δ(det) ⊗ ind_{Q_{2}}^{H_{D}}(C_{c}^{∞}(A))

for a certain character δ which we shall now determine. Although we do not know δ, for a generic choice of χ and μ, X̃ ⊗ W′ will be a quotient of C_{c}^{∞}(AA). Since C_{c}^{∞}(A) is a regular representation of GL_{2}(F) twisted by δ, X̃ ⊗ W′ must be a quotient of

X̃ ⊗ Ind_{Q_{2}}^{F_{4}} X

where X is a representation of L_{2} pulled back from a representation of GL_{2}(F) with a parameter (χδ| · |^{−3/2}, μδ| · |^{−3/2}). We get that W′ is a subquotient of both, Ind_{Q_{4}}^{F_{4}}(Y′ ⊗ | det |^{3}) and Ind_{Q_{2}}^{F_{4}} X. This immediately implies that

δ(det) = | det |^{5}.

The knowledge of δ implies that any X̃ is a quotient of C_{c}^{∞}(AA) only and, therefore Θ-lifts of Ỹ are the summands of W.

It remains to check the statement about Satake parameters. The L-group of F_{4}(F) is F_{4}(ℂ). Let Q_{3} = L_{3}V_{3} be the maximal parabolic subgroup of F_{4} such that [L_{3}, L_{3}] = SL_{2} × SL_{3} where SL_{2} is spanned by long roots and SL_{3} by short roots. Since L_{3}(ℂ) is the L-group of L_{2}, it fits into exact sequence

1 → GL_{2} → L_{3} → P GL_{3} → 1.

Let s ∈ GL_{2}(ℂ) be the parameter (χ, μ). Obviously, the Satake parameter of W is s × ρ ∈ GL_{2} × SO(3) ⊂ L_{3}. Let G_{2} be the centralizer of SO(3) in F_{4}. Then L_{3} ∩ G_{2} = GL_{2} and s is precisely the Satake parameter of Ỹ as in Section 4.

Theorem 7.4 can be used, in a similar way, to prove converses of Theorems 4.4 and 5.4. We state results without giving details of proofs.

Theorem 8.2. Let Φ : SL_{3}(ℂ) → G_{2}(ℂ) be the inclusion of the L-groups of P GL_{3}(F) and G_{2}(F). Let Ỹ be a tempered spherical representation of G_{2}(F). The Satake parameter of Ỹ is Φ(s) for some s, a Satake parameter of a tempered spherical representation Ỹ of G_{2}(F). Note that Φ(s) = Φ(s^{∗}) where s^{∗} is a Satake parameter of Y^{∗}, the dual of Y. Then

Θ(Ỹ) = {Y, Y^{∗}}.

Theorem 8.3. Let Φ : G_{2}(ℂ) → Spin_{7}(ℂ) be the inclusion of the L-groups of G_{2}(F) and P GSp_{6}(F). Let Ỹ be a tempered spherical representation of G_{2}(F). Then

Θ(Ỹ) = {Y′}

where Y′ is the spherical tempered representation of P GSp_{6}(F) whose Satake parameter is Φ(s).

Acknowledgments

It is a pleasure to thank B. Gross for initiating this project and on many discussions during the course of this work.

A part of this work was written in the Spring of 1995. During that time the second author enjoyed hospitality of the Mathematical Sciences Research Institute and was generously supported by an NSF Postdoctoral Fellowship and a Sloan Research Fellowship.

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Department of Mathematics, Wayne State University, Detroit, MI 48202 E-mail address: kaym@math.wayne.edu

Department of Mathematics, University of Utah, Salt Lake City, UT 84112 E-mail address: savin@math.utah.edu