HARMONIC ANALYSIS ON THE TWISTED FINITE POINCARÉ UPPER HALF-PLANE

JORGE SOTO-ANDRADE AND JORGE VARGAS

Abstract. We prove that the induced representation from a non trivial character of the Coxeter torus of GL(2, F), for a finite field F, is multiplicity-free; we give an explicit description of the corresponding (twisted) spherical functions and a version of the Heisenberg Uncertainty Principle.

1. Introduction

Let F be a finite field, with q elements, and E be its unique quadratic extension. Put G = GL(2, F) and denote by K the Coxeter torus of G, realized as the subgroup of all matrices m_{z}(z ∈ E^{×}) of the maps w ↦ zw(w ∈ E) with respect to a fixed F - basis of E. Recall that the finite homogeneous space ℋ = G/K may be looked upon as the finite analogue of (the double cover of) the classical Poincaré Upper Half Plane (see [4]). Harmonic analysis on ℋ amounts to decompose the induced representation Ind_{K}^{G} 1 from the unit character 1 of K to G. We are interested here in the “twisted” version of this, i.e., the decomposition of the induced representation Ind_{K}^{G} Φ from a non (necessarily) trivial character Φ of K to G. The real analogue of this case has been considered in [1]. We prove that this representation is multiplicityfree, taking advantage of the fact that this is so for Ind_{K}^{G} 1 (see [4]) and reducing the computation of the multiplicities in Ind_{K}^{G} Φ to the ones in Ind_{K}^{G} 1. We also give an explicit description of the corresponding (twisted) spherical functions. Finally, we give a version of the Heisenberg Uncertainty Principle

2. The Multiplicity One Theorem for Ind_{K}^{G} Φ.

2.1. The case Φ = 1. We consider first the special case Φ = 1 in which the multiplicity one theorem follows from a geometric argument. In fact, we have

Ind_{K}^{G} 1 ≃ (L^{2}(ℋ), τ),

where L^{2}(ℋ)) stands for the space of all complex functions on ℋ endowed with the usual canonical scalar product, and τ denotes the natural representation of G in L^{2}(ℋ), defined by (τ_{g}f)(z) = f(g^{−1}.z), where z ↦ g.z is the homographic action of G on ℋ, given by

Soto-Andrade was partially supported by FONDECYT Grants 92-1041 and 1940590, DTI – U.

Chile, ICTP, ECOS-France and NSF Grant DMS-9022140 at MSRI. Vargas was partially supported by CONICET, CONICOR, SecytUNC, ICTP and TWAS.

g.z = \frac{az + b}{cz + d} forg = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G, z ∈ ℋ.

Definition 1. For all z, w ∈ ℋ, we put

D(z, w) = \frac{N(z − w)}{N(z − w̄)}

with the convention that D(z, w) = ∞ if w = z̄.

Proposition 1. D is an orbit classifying invariant function for the homographic action of G in ℋ × ℋ.

Corollary 1. The commuting algebra of (L^{2}(ℋ), τ) is commutative.

This follows from the fact that, the classifying invariant D being symmetric, the G-orbits in ℋ × ℋ are also symmetric. □

2.2. The case of general Φ. Let’s denote by ϕ the restriction of Φ to F^{×}. We will prove that every twisting of an irreducible representation π_{θ}^{d} of G (where the superscript d denotes the dimension of π and θ its character parameter) by the character (Φ + Φ^{q}) is isomorphic to a representation of the form π_{θ′}^{d′} + π_{θ′′}^{d′′}, when restricted to K. In fact we will work with the characters χ_{θ}^{d} of the irreducible representations π_{θ}^{d} of G, for which we keep the notations of [5] or [2].

Lemma 1. On K we have (Φ + Φ^{q})χ_{α,α}^{q} = χ_{Φ(α∘N)}^{q−1} + χ_{ϕα,α}^{q+1}, (Φ + Φ^{q})χ_{α,α}^{1} = χ_{ϕα,α}^{q+1} − χ_{Φ(α∘N)}^{q−1}, (Φ + Φ^{q})χ_{α,β}^{q+1} = χ_{ϕα,β}^{q+1} + χ_{α,ϕβ}^{q+1}, (Φ + Φ^{q})χ_{Λ}^{q−1} = χ_{ΦΛ}^{q−1} + χ_{Φ^{q}Λ}^{q−1}. □

Now for a character χ of G, we have χ ∘ Frob = χ on K, as it follows from the character table. Therefore ∑_{K} Φ̄(k^{q})χ(k) = ∑_{K} Φ̄(k)χ(k) because Frob is an involutive automorphism.

Hence, the multiplicity of π in Ind_{K}^{G} Φ equals \frac{1}{2} ∑_{K}(Φ̄+Φ̄^{q})(k)π(k) and so it is just the average of the multiplicities in Ind_{K}^{G} 1 of two representations of G (one of which may be virtual!)

Remark 1. Put π_{α,α}^{q+1} = π_{α}^{q} + π_{α}^{1} and π_{α∘N}^{q−1} = π_{α}^{q} − π_{α}^{1} for every α ∈ (F^{×})^{∧}. It is easy to check than in the degenerate cases α = β (for π = π_{α,β}^{q+1} ) and Λ = Λ^{q} (for π = π_{Λ}^{q−1}) we find for the multiplicities m_{1}(π)

m_{1}(π_{α,α}^{q+1}) = 1  (α ∈ (F^{×})^{∧})  (1)

and

m_{1}(π_{α∘N}^{q−1}) = −δ_{α,1}  (α ∈ (F^{×})^{∧})  (2)

Using the fact that the multiplicities of the irreducible representations of G in Ind_{K}^{G} 1 are at most one and also equations (1) and (2), we get that the multiplicities are also at most one in the more general case of Ind_{K}^{G} Φ. □

2.3. The multiplicities m_{Θ,d}(Φ) of π_{Θ}^{d} in Ind_{K}^{G} Φ for general Φ ∈ (E^{×})^{∧}. In Table 1 below, π_{Θ}^{d} denotes an irreducible representation of G, of dimension d and parameter Θ. Then d ∈ {1, q, q + 1, q − 1} and Θ is of the form {α, β}, with α, β ∈ (F^{×})^{∧} or {Λ, Λ^{q}} with Λ ∈ (E^{×})^{∧}

Table 1. The multiplicities m_{Θ,d}(Φ)

π_{Θ}^{d} m_{Θ,d}(Φ)
π_{α,α}^{1} δ_{α^{2},ϕ}
π_{α,α}^{q} δ_{α^{2},ϕ} − δ_{α∘N,Φ}
π_{α,β}^{q+1} δ_{αβ,ϕ}
π_{Λ,Λ^{q}}^{q−1} δ_{λ,ϕ} − δ_{Λ,Φ} − δ_{Λ^{q},Φ}

NOTATIONS. Here α, β ∈ (F^{×})^{∧} with α ≠ β and Φ, Λ ∈ (E^{×})^{∧} with Λ ≠ Λ^{q}, and λ (resp. ϕ ) denotes the restriction of the character Λ (resp. Φ ) to (F^{×})^{∧}.

3. The twisted spherical functions

3.1. The averaging construction. In this section G denotes an arbitrary finite group, K a subgroup of Gand Φ a one dimensional representation of K. We notice that the spherical functions for the representation Ind_{K}^{G} Φ are obtained as weighted averages of the characters of G. More precisely:

Definition 2. Let L^{1}(G) be the group algebra of G, realized as the convolution algebra of all complex functions of G and let L_{Φ}^{1}(G, K) be the convolution algebra of all complex functions f on G such that

f(kgk′) = Φ(k)f(g)Φ(k′)

for all g ∈ G, k, k′ ∈ K. For any f ∈ L^{1}(G) put

(P_{Φ}f)(g) = \frac{1}{|K|} ∑_{k∈K} Φ^{−1}(k)f(kg)

for all g ∈ G.

Notice that the operator P_{Φ} is just convolution with the idempotent function ε_{K}^{Φ} ∈ L^{1}G which coincides with |K|^{−1}Φ on K and vanishes elsewhere. Moreover L_{Φ}^{1}(G, K) may be writen as ε_{K}^{Φ} ∗L^{1}G∗ε_{K}^{Φ} and its elements f are characterized by the properties

ε_{K}^{Φ} ∗ f = f = f ∗ ε_{K}^{Φ}

Lemma 2. Let χ be the character of an irreducible representation π of G. Then P_{Φ}(χ)(e) ≠ 0 iff π appears in Ind_{K}^{G} Φ. □

Lemma 3. P_{Φ}(χ) is a non-zero function iff it doesn’t vanish for g = e. □

Proposition 2. The mapping P_{Φ} is an algebra epimorphism from the center Z(L^{1}G)) of the convolution algebra L^{1}G onto the center Z(L_{Φ}^{1}(G, K)) of the convolution algebra L_{Φ}^{1}(G, K).

Proof: We have

P_{Φ}(f_{1} ∗ f_{2}) = ε_{K}^{Φ} ∗ (f_{1} ∗ f_{2})  = (f_{1} ∗ ε_{K}^{Φ}) ∗ f_{2}
= (f_{1} ∗ ε_{K}^{Φ} ∗ ε_{K}^{Φ}) ∗ f_{2} = (ε_{K}^{Φ} ∗ f_{1}) ∗ (ε_{K}^{Φ} ∗ f_{2})
= P_{Φ}f_{1} ∗ P_{Φ}f_{2}.

since f_{1} is central and ε_{K}^{Φ} is idempotent. Moreover the dimension d of the image of Z(L^{1}G)) under P_{Φ} is the number of irreducible characters χ of G such that P_{Φ}(χ) ≠ 0; but P_{Φ}(χ) ≠ 0 iff (P_{Φ}χ)(e) ≠ 0 and, the number (P_{Φ}(χ))(e) being the multiplicity in Ind_{K}^{G} Φ of the representation π of G whose character is χ, we see that d is just the number of irreducible representations π of G appearing in Ind_{K}^{G} Φ, i. e. the dimension of the center of L_{Φ}^{1}(G, K). □

Corollary 2. The nonzero functions that satisfy the functional equation

h(x)h(y) = ∫_{K} Φ̄(k)h(xky) dk

linearly span the center of the algebra L_{Φ}^{1}(G, K).

Proof: The functions h that satisfy the above functional equation are exactly the complex multiples of the functions P_{Φ}(χ); for a proof (see [6]). Therefore the corollary follows. □

3.2. Explicit formulae for the twisted spherical functions. Define

S_{Λ}^{Φ}(a) = −(q^{2} − 1)^{−1} ∑_{(z,w)∈Γ_{a}} Φ^{−1}(z)Λ(w)

for Λ ∈ (E^{×})^{∧} and a ∈ F^{×}, where Γ_{a} denotes the set of all (z, w) ∈ E^{×} × E^{×} such that N(w) = aN(z) and T r(w) = 2(a + 1)^{−1}T r(z).

Then the spherical function ζ_{Λ}^{Φ} of G associated to the cuspidal character χ_{Λ}^{q−1} of

G is given on the representatives d(a, 1) = ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) (a ∈ F^{×}) for the K - double

HARMONIC ANALYSIS ON THE TWISTED FINITE POINCARÉ HALF-PLANE 5

cosets in G, by

ζ_{Λ}^{Φ}(d(a, 1)) = S_{Λ}^{Φ}(a) + q(q + 1)^{−1}δ_{a,1}δ_{λ,ϕ},

where λ (resp. ϕ) denotes the restriction of the character Λ (resp. Φ) of E^{×} to F^{×}.

Notice that a = 1 corresponds to the origin in ℋ and a = −1 corresponds to the antipode of the origin in ℋ. It is not difficult to check that these formulae for the spherical functions are equivalent to the ones given in [4] for the case Φ = 1.

3.3. A new form for the cuspidal spherical functions for Φ = 1 (char F ≠ 2). For a ≠ 1, one has the following new expression for the spherical functions estimated in [3]

ζ_{Λ}^{Φ}(a) = (q + 1)^{−1} ∑_{u∈U} ε(T r(u) − (a + a^{−1}))(εω)(u),

for a ≠ 1, where ε denotes the sign character of F^{×}.

4. Heisenberg Uncertainty Principle

For this section, G denotes an arbitrary finite group, K any subgroup of G and Φ any linear character of K.

Let Ĝ^{Φ} be set of allthe equivalence classes of irreducible representations of G that contain the character Φ when restricted to K. For each equivalence class we choose, once and for all, a representative (π, V_{π}). As usual, for each f in L^{1}(G), the Fourier Transform ℱ(f), valued in the class (π, V_{π}), is the linear operator ℱ in V_{π} defined by

ℱ(f)(π) := π(f) := \frac{1}{|G|} ∫_{G} f(g)π(g^{−1})dg := \frac{1}{|G|} ∑_{g∈G} f(g)π(g^{−1}).

We recall the statement of the Plancherel theorem for a function f ∈ L_{Φ}^{1}(G, K)

f(g) = \frac{1}{G} ∑_{π∈Ĝ^{Φ}} d_{π} trace(π(f)π(g));

here g ∈ G arbitrary and d_{π} := dim V_{π}.

For any complex valued function f on G, let | supp(f)| denote the number of elements of the support of f. That is, the number of points of G where f takes nonzero values.

Proposition 3 (Heisenberg Uncertainty Principle). For any nonzero function f ∈ L_{Φ}^{1}(G, K) we have

| supp(f)| ( ∑_{π∈supp(ℱ(f))} d_{π}) ≥ |G|.

Here supp(ℱ(f)) is the subset of Ĝ^{Φ} where ℱ(f) does not vanish.

Proof: For any function f on G we recall that

∥f∥_{2}^{2} = ∑_{x∈G} |f(x)|^{2}; ∥f∥_{∞} = max_{x∈G} |f(x)|; ∥f∥_{2}^{2} ≤ ∥f∥_{∞}^{2}| supp(f)|  (∗)

From now on, we fix a G−invariant inner product on V_{π}. Then T^{∗} denotes the adjoint of a linear operator T on V_{π} with respect to this inner product. Also ∥T ∥ denotes the Hilbert-Schmidt norm on End V_{π} defined by trace(T S^{∗}), for S, T ∈ End V_{π}

Since f ∈ L_{Φ}^{1}(G, K), as we pointed out before, the Plancherel Theorem says that we have that supp(f) is contained in Ĝ^{Φ} and that

f(x) = \frac{1}{G} ∑_{π∈Ĝ^{Φ}} d_{π} trace(π(f)π(x)).

The Cauchy–Schwarz inequality applied to the Hilbert-Schmidt inner product says that the first of the two following inequallities is true,

trace(π(f)π(x)) ≤ ∥π(f)∥∥π(x)∥ ≤ ∥π(f)∥,

the second inequality follows from the fact that ∥T ∥ = 1 for a unitary operator. Putting together the last two statements we get

∥f∥_{∞} ≤ \frac{1}{G} ∑_{π∈Ĝ^{Φ}} d_{π}∥ℱ(f)(π)∥

The classical Cauchy–Schwarz inequality and the fact that d_{π} = dπ^{\frac{1}{2}} dπ^{\frac{1}{2}} imply that

∥f∥_{∞}^{2} ≤ \frac{1}{|G|^{2}} ∑_{π∈Ĝ^{Φ}} d_{π}∥ℱ(f)(π)∥^{2} ∑_{π∈supp(ℱ(f))} d_{π}.

Now the L^{2}−version of Plancherel Theorem says that

∥f∥_{2}^{2} = \frac{1}{|G|} ∑_{π∈Ĝ^{Φ}} d_{π}∥ℱ(f)(π)∥^{2}.

Therefore,

∥f∥_{∞}^{2} ≤ \frac{1}{|G|}∥f∥^{2} ∑_{π∈Ĝ^{Φ}} d_{π}.

Since f is nonzero, we apply (*) to the above inequality and get the desired result. □

References

[1] Galina, E. and Vargas, J., Eigenvalues and eigenspaces for the twisted Dirac operator over SU(n, 1) and Spin(2n, 1), Trans. Amer. Math. Soc., 345 (1994), 97-113.

[2] Helversen-Pasotto, A., Représentation de Gelfand-Graev et identités de Barnes, Enseign. Math. 32 (1986), 57-77.

[3] Katz, N., Estimates for Soto-Andrade sums, J. reine angew. Math. 438 (1993), 143-161.

[4] Soto-Andrade, J., Geometrical Gel’fand Models, tensor quotients and Weil representations, Proc. Symp. Pure Maths., 47, AMS, Providence, 1987, 305-316.

[5] Soto-Andrade, J., Répresentations de certains groupes symplectiques, Mém. 55-56, Soc. Math. France, 1975.

[6] Varadarajan, Spherical functions, Springer, Berlin.

Jorge Soto-Andrade, Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile

E-mail address: sotoandr@mate.uncor.edu, sotoandr@abello.dic.uchile.cl

Jorge Vargas, FAMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba, Argentina

E-mail address: vargas@mate.uncor.edu