Journal of Lie Theory Volume 14 (2004) 1–9 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Weil Representations of SL*(2,A) for a Locally Profinite Ring A with Involution

Roberto Johnson and José Pantoja ^{∗}

Communicated by D. Poguntke

Abstract. We construct, via a complex G−bundle space, a Weil representation for the group G = SL_{∗}(2, A), where (A, ∗) is a locally profinite ring with involution. The construction is obtained using maximal isotropic lattices and Heisenberg groups.

1. Preliminaries. Let (A, ∗) be a locally profinite ring with involution , i.e. a unitary locally

compact and totally disconnected ring with an involutive anti-automorphism a ⟶ a^{∗}, a ∈ A. Let Z_{s}(A) be the subring of central symmetric elements of A.

We define the group GL_{∗}(2, A) of matrices g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) with a, b, c, d

∈ A, such that:

1. ab^{∗} = ba^{∗}, cd^{∗} = dc^{∗}

2. a^{∗}c = c^{∗}a, b^{∗}d = d^{∗}b

3. ad^{∗} − bc^{∗} = a^{∗}d − c^{∗}b is an invertible central symmetric element of A, i.e. an element of Z_{s}(A)^{×}.

We set det_{∗}(g) = ad^{∗} − bc^{∗} = a^{∗}d − c^{∗}b; then

g^{−1} = [det_{∗}(g)]^{−1} ( \begin{matrix}d^{∗} & −b^{∗} \\ −c^{∗} & a^{∗} \end{matrix} )

We observe that the function det_{∗} : GL_{∗}(2, A) ⟶ Z_{s}(A)^{×} is an epimorphism so that G = SL_{∗}(2, A) = Ker det_{∗} is a normal subgroup of GL_{∗}(2, A).

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

∗ Both authors have been partially supported by FONDECYT grant 1990029, PICS (CNRS-CONICYT) and Universidad Católica de Valparaíso

In what follows we will assume that Z_{s}(A) = F is a p-adic field. We denote by O_{F} the ring of integers of F, P_{F} is the maximal ideal of O_{F}, ϖ is a generator of P_{F} and k_{F} is the residual field of F which has q elements.

Some such rings are: A = M_{n}(F), F a p-adic field, with ∗ the transposition; A = K a separable quadratic extension of F, F as above with ∗ the non trivial Galois element; A =^{⋀0}V ⊕^{⋀1}V ⊕^{⋀2}V where V is a two dimensional vector space over a p-adic field F with basis (e_{1}, e_{2}) and ∗ is given by the basis transposition (e_{1}, e_{2}) to (e_{2}, e_{1}).

2. General Setting Let H be a locally profinite group and Γ a subgroup of Aut(H). Let

(π, V) be an irreducible smooth (complex) representation of H such that π^{γ} ≃ π (π^{γ} = π ∘ γ) for every γ in Γ.

If γ ∈ Γ then there exists T_{γ} ∈ Aut_{C}(V) such that T_{γ}π(x) = πγ(x)T_{γ} for every x ∈ H.

Set G be the semidirect product of Γ and H. For (γ, h) in G we define π̃(γ, h) in Aut_{C}(V) by

π̃(γ, h) = T_{γ}π(h).

Proposition 2.1. The endomorphism π̃, defined above, is a projective extension of π to G. Proof. We want to prove that T_{γδ}^{−1}T_{γ}T_{δ} is a scalar.

Since T_{γ}T_{δ}π(x) = T_{γ}π(δ(x))T_{δ} = π(γδ(x))T_{γ}T_{δ} and T_{γδ}π(x) = π(γδ(x))T_{γδ} then

T_{γδ}^{−1}T_{γ}T_{δ}π(x) = π(x)T_{γδ}^{−1}T_{γ}T_{δ}

It follows, by Schur’s Lemma, that T_{γδ}^{−1}T_{γ}T_{δ} = σ(γ, δ)id_{V} , for a cocycle σ.

We compute now π̃(γ, h)π̃(δ, k). We have

π̃(γ, h)π̃(δ, k) = σ(γ, δ)T_{γδ}π(δ^{−1}(h))π(k).
Since π̃((γ, h)(δ, k)) = π̃(γδ, δ^{−1}(h)k) = T_{γδ}π(δ^{−1}(h)k) we get
π̃(γ, h)π̃(δ, k) = σ(γ, δ)π̃((γ, h)(δ, k)).

Therefore π̃ is a projective representation of G with cocycle σ.

We recall now the definition of compact induction, c-Ind, as we will use it: Let L be a an open subgroup of H, compact modulo the centre of H, and let (ρ, W) be a smooth representation of L. Let V denote the space of compactly supported modulo the centre of H functions f : H → W with the property f(lh) = ρ(l)f(h), l ∈ L, h ∈ H. The group acts on this space by right translation of functions; the implied representation is smooth. We will assume now that

(π, V) = c − Ind_{L}^{H} ρ, where L is an open, compact modulo the centre, subgroup of H and ρ is a one dimensional representation of L.

We assume also that ρ^{γ} = ρ on L^{γ} ∩ L, where L^{γ} = γ(L) and ρ^{γ}(y) = ρ(γ^{−1}(y)) with y ∈ L^{γ}. We can define, similarly,

(π_{γ}, V_{γ}) = c − Ind_{L^{γ}}^{H} ρ^{γ}.

Let S_{γ} be a non zero intertwining operator from (π, V) to (π_{γ^{−1}}, V_{γ^{−1}}). So S_{γ} is an isomorphism between π and π_{γ^{−1}} when π (and then π_{γ^{−1}} ) is irreducible. Then S_{γ}π(x) = π_{γ^{−1}}(x)S_{γ}.

We define now I_{γ} : V_{γ^{−1}} ⟶ V by (I_{γ}(f)) (x) = f(γ^{−1}(x)). The operator I_{γ} is well defined and intertwining, in fact, I_{γ}(f(lx)) = ρ(l)f(x) and I_{γ}π_{γ^{−1}}(x) = π(γ(x))I_{γ}. On the other hand, we have that I_{γ}S_{γ} : V ⟶ V is an intertwining operator since I_{γ}S_{γ}π(x) = π(γ(x))I_{γ}S_{γ}. Let us define T_{γ} = I_{γ}S_{γ}. We want to compute the cocycle σ. In order to do this we look first at I_{γ} on V_{δ}, Since γ^{−1}(h) ∈ δ(L) implies that h ∈ γδ(L) , we have (I_{γ}f)(hx) = f(γ^{−1}(h)γ^{−1}(x)). We can define I_{γ,δ} : V_{γ^{−1}δ} ⟶ V_{δ} by (I_{γ,δ}f)(x) = f(γ^{−1}x), and S_{δ,γ} : V_{γ^{−1}} ⟶ V_{γ^{−1}δ^{−1}} by S_{δ,γ} = I_{γ,δ^{−1}}^{−1}S_{δ}I_{γ,1} a computation shows that S_{δ,γ} is an intertwining map.

Since the operators S_{δ,γ} ∘ S_{γ} : V ⟶ V_{γ^{−1}δ^{−1}} and S_{δγ} : V ⟶ V_{γ^{−1}δ^{−1}} are both intertwining, the irreductibility of V implies that they differ on a scalar i.e. S_{δ,γ} ∘ S_{γ} = kS_{δγ}.

Lemma 2.2. The intertwining operators defined above satisfy the equation I_{δ} ∘ I_{γ,δ^{−1}} = I_{δγ}. Proof. Straightforward.

We finally show that k = σ(δ, γ): Since S_{δ,γ} ∘ S_{γ} = kS_{δγ} we have I_{γ,δ^{−1}}^{−1}S_{δ}I_{γ}S_{γ} = kS_{δγ}. So S_{δ}I_{γ}S_{γ} = kI_{γ,δ^{−1}}S_{δγ} and then I_{δ}S_{δ}I_{γ}S_{γ} = kI_{δ}I_{γδ^{−1}}S_{δγ}. Using Lemma 2.2 we get I_{δ}S_{δ}I_{γ}S_{γ} = kI_{δγ}S_{δγ} i.e. T_{δ}T_{γ} = kT_{δ}_{γ}.

3. Heisenberg Construction Given a F−vector space W we can define H = F⊕W which has a structure

of group with respect to

(a, w) · (a′, w′) = (a + a′ + B(w, w′), w + w′)

where B : W × W ⟶ F is a non-degenerate alternating form. If M is any subgroup of W we write M̃ = F⊕M, which is a subgroup of H. Definition 3.1. Let M be an any subset of W. We define M^{∗} = {w ∈ W |

B(m, w) ∈ O_{F} ∀m ∈ M} and M^{⊥} = {w ∈ W | B(m, w) = 0 ∀m ∈ M}.

Observation 3.2.

a) If M is a F−subspace of W, then M^{∗} = M^{⊥}. In fact, the inclusion M^{⊥} ⊂ M^{∗} is obvious. On the other hand, since αB(m, w) = B(αm, w) we have that w ∈ M^{∗} implies that αB(m, w) ∈ O_{F} ∀m ∈ M ∀α ∈ F, so B(m, w) = 0.

b) Another fact that we will use later, is the following

[(a, w), (a′, w′)] = (2B(w, w′), 0).

Let 𝔏 be a maximal isotropic lattice i.e. 𝔏 is compact and open and 𝔏^{∗} = 𝔏. Set

𝔏̃ = F⊕𝔏 and let ψ be a character of F of conductor O_{F}. Define ψ_{𝔏} on 𝔏̃ by ψ_{𝔏}(a, l) = ψ(a) for a ∈ F.

Proposition 3.3. With the above notation and assuming that 2 ∈ O_{F}^{×} we have:

a) ψ_{𝔏} is a character of 𝔏̃.

b) If we define Int_{H}(ψ_{𝔏}) = {h ∈ H | Hom_{𝔏̃∩𝔏̃^{h}} (ψ_{𝔏}, ψ_{𝔏}^{h}) ≠ 0}, where 𝔏̃^{h} = h𝔏̃h^{−1} and ψ_{𝔏}^{h}(x) = ψ_{𝔏}(h^{−1}xh) for any x ∈ 𝔏̃^{h}, then Int_{H}(ψ_{𝔏}) is equal to 𝔏̃. Proof. a) ψ_{𝔏}((a, w)(a′, w′)) = ψ_{𝔏}(a + a′ + B(w, w′), w + w′), since 𝔏 is a maximal isotropic lattice, B(w, w′) ∈ O_{F}. Then ψ_{𝔏}((a, w)(a′, w′)) = ψ(a)ψ(a′) = ψ_{𝔏}(a, w)ψ_{𝔏}(a′, w′).

b) If (a, w) ∈ H Since (−a, −w)(α, y)(a, w) = (α + 2B(y, w), y) and 𝔏̃ ◁ H, we have ψ_{𝔏}^{(a,w)} = ψ_{𝔏} on 𝔏̃ ∩ (−a, −w)𝔏̃(a, w) = 𝔏̃ if and only if 2B(y, w) ∈ O_{F} ∀y ∈ 𝔏 if and only if B(y, w) ∈ O_{F} ∀y ∈ 𝔏 (given that 2 ∈ O_{F}^{×}) and this is the case if and only if w ∈ 𝔏.

Now let Π_{𝔏} = c − IndH ; 𝔏̃ ψ_{𝔏} be the compact induction of the character ψ_{𝔏} from 𝔏̃ to H as defined in Section 2.

Proposition 3.4. The representation Π_{𝔏} defined above is an irreducible admissible supercuspidal representation of H. Proof. The representation Π_{𝔏} is the Heisenberg representation realized in the lattice model (see [5], Chapter 2). Stone-von Neumann theorem implies that Π_{𝔏} is a smooth irreducible (thus admissible) representation. Then, using theorem 1 of [2], we get that it is supercuspidal.

Now let Γ be the subgroup of Aut(H) of all automorphism γ : H ⟶ H such that γ_{|F} = id_{F} and γ_{|W} is a symplectic linear automorphism. The subgroup Γ acts transitively over the set Θ of all maximal isotropic lattices in W, by 𝔏^{γ} = γ(𝔏) (γ ∈ Γ and 𝔏 ∈ Θ). Furthermore ψ_{𝔏}^{γ} = ψ_{𝔏} on 𝔏^{γ} ∩ 𝔏 where ψ_{𝔏}^{γ}(y) = ψ_{𝔏}(γ^{−1}(y)), ∀y ∈ 𝔏^{γ}.

On the other hand, by Proposition 3.4 , (Π_{𝔏}, V_{𝔏}) = c − IndH ; 𝔏̃ ψ_{𝔏} is an irreducible admissible supercuspidal representation of H, where V_{𝔏} = {f : H → C |f(lx) = ψ_{𝔏}(l)f(x), ∀l ∈ 𝔏, ∀x ∈ H, f compactly supported modulo the centre of H}. So, we can define (Π_{𝔏^{γ}}, V_{𝔏^{γ}}) = c − IndH ; 𝔏̃^{γ} ψ_{𝔏}^{γ}, where V_{𝔏^{γ}} = {f : H → C |f(lx) = ψ_{𝔏}^{γ}(l)f(x), ∀l ∈ 𝔏^{γ}} and now the general set-up of Section 2 applies.

Define the function τ_{γ} : H → C by

τ_{γ}(xy) = \{ \begin{matrix}ψ_{𝔏}^{γ}(x)ψ_{𝔏}(y) & if & x ∈ 𝔏̃^{γ}, y ∈ 𝔏̃ \\ 0 & & otherwise \end{matrix}

Note that τ_{γ} is well defined since ψ_{𝔏}^{γ} = ψ_{𝔏} on 𝔏̃^{γ} ∩ 𝔏̃. For any f in the space of

Π_{𝔏} we can define Υ_{γ}(f) : H ⟶ C by

Υ_{γ}(f)(x) = ∫_{H/F} τ_{γ}(y)f(y^{−1}x)dy

for an appropriate Haar measure on W = H/F. We can observe that Υ_{γ} : V_{𝔏} ⟶ V_{𝔏^{γ^{−1}}} is a non zero intertwining operator and since Π_{𝔏} is irreducible (and also Π_{𝔏^{γ^{−1}}} ), we have that Υ_{γ} is an isomorphism.

We define now I_{γ} : V_{𝔏^{γ^{−1}}} ⟶ V_{𝔏} by (I_{γ}f)(x) = f(γ^{−1}(x)) and so we have, as in section 2, that T_{γ} = I_{γ}Υ_{γ} is an intertwining of V_{𝔏} which verify

T_{δ} ∘ T_{γ} = σ(δ, γ)T_{δγ}.

4. Lagrangians Let S be a left A- module whose F-dimension is n. We note that S is a

right A- module with sa = a^{∗}s, a ∈ A, s ∈ S.

Let b : S × S ⟶ F be a non degenerate bilinear symmetric form such that

b(x_{1}a, x_{2}) = b(x_{1}, ax_{2}) (a ∈ A; x_{1}, x_{2} ∈ S).

We set now W = S⊕S and define B : W × W ⟶ F by B(x, y) = b(x_{1,}y_{2}) − b(y_{1}, x_{2}) for x = (x_{1}, x_{2}) and y = (y_{1}, y_{2}) in W. Observe that B is a non degenerate alternating form and we can define M^{⊥} = {w ∈ W | B(w, m) = 0, ∀m ∈ M} for any O_{F}−submodule M of W. The following properties are straightforward.

If either M is an F−subspace or if M is a compact open O_{F}−submodule of W (an O_{F}−lattice in W) then (M^{⊥})^{⊥} = M.

If M, N are any O_{F}−submodules which satisfy (M^{⊥})^{⊥} = M and (N^{⊥})^{⊥} = N then (M ∩ N)^{⊥} = M^{⊥} + N^{⊥}.

We call an O_{F}−submodule M of W isotropic if M is an O_{F}−submodule of M^{⊥}. We say that M is maximal isotropic if M = M^{⊥}

Fixing an additive (continuous) character ψ of F of conductor O_{F}, we can define the function χ : W × W ⟶ T, where T is the group of complex numbers of module one, by

χ(x, y) = (ψ ∘ B)(x, y) ((x, y) ∈ W × W).

which is a symplectic bicharacter.

Definition 4.1. Let M be a subset of W. The orthogonal component M^{∗} of M is the set of y ∈ W such that χ(x, y) = 1, for every x ∈ M.

Observation 4.2. In the case where M is a F−subspace of W we have that M^{∗} is also a F−subspace of W and M^{⊥} = M^{∗}.

Definition 4.3. Let L be a F−subspace of W such that L^{⊥} = L. L is called a Lagrangian subspace of W.

Observation 4.4. If M is an F−subspace of W then M is maximal isotropic if and only if M is Lagrangian.

Lemma 4.5. Let W and χ be as above. Let L and L′ be two Lagrangian subspaces of W. Then there exists a symplectic basis {w_{1}, w_{2}, …, w_{n}, w′_{1}, w′_{2}, …, w′_{n}} of W, i.e.

1. χ(w_{j}, w′_{j}) ≠ 1, j = 1, …, n

2. χ(w_{i}, w_{j}) = χ(w′_{i}, w′_{j}) = 1 for every i, j.

3. χ(w_{i}, w′_{j}) = 1 for every i ≠ j such that:

L = Fw_{1}⊕Fw_{2}⊕ ⋯ ⊕Fw_{k}⊕Fw_{k+1}⊕ ⋯ ⊕Fw_{n}
L′ = Fw′_{1}⊕Fw′_{2}⊕ ⋯ ⊕Fw′_{k}⊕Fw_{k+1}⊕ ⋯ ⊕Fw_{n}

Proof. See Lemma 1.4.6. in [4].

Corollary 4.6. Given a Lagrangian L, there exists a Lagrangian L′ such that W = L⊕L′. Proof. If L = ⟨w_{1}, w_{2}, …, w_{n}⟩ , then L is a proper subspace of ⟨w_{2}, …, w_{n}⟩^{⊥}. We consider an element v_{1} ∈ ⟨w_{2}, …, w_{n}⟩^{⊥} − L. Then χ(w_{1}, v_{1}) ≠ 1. Now we can pick an element v_{2} ∈ ⟨w_{1}, w_{3}, w_{4,} …, w_{n}, v_{1}⟩^{⊥} − ⟨w_{1}, w_{2}, w_{3}, …, w_{n}, v_{1}⟩^{⊥} , and so χ(w_{2}, v_{2}) ≠ 1. By induction we have {w_{1}, v_{1}}, {w_{2}, v_{2}}, …, {w_{n}, v_{n}} such that χ(w_{i}, v_{i}) ≠ 1, i = 1, …, n; χ(w_{i}, w_{j}) = χ(v_{i}, v_{j}) = 1 for every i, j and χ(w_{i}, v_{j}) = 1 for every i ≠ j. Hence L′ = ⟨v_{1}, v_{2}, …, v_{n}⟩ is such that W = L⊕L′.

Corollary 4.7. There exists a maximal isotropic O_{F}−lattice 𝔏 in W.

Let L be a Lagrangian in W and define ψ_{L} on L̃ = F ⊕ L as above. Let Π_{L} = c − IndH ; L̃ ψ_{L} and consider the group H = F ⊕ W. Let 𝔏̃ = F ⊕ 𝔏, 𝔏 a maximal isotropic O_{F}−lattice in W.

Now we can define the function ρ : H ⟶ C by

ρ(z) = \{ \begin{matrix}ψ_{L}(x)ψ_{𝔏}(y) & if & z = x · y, x ∈ L̃, y ∈ 𝔏̃ \\ 0 & if & z ∉ L̃𝔏̃ = F⊕(L + 𝔏) \end{matrix}

Note that ρ is well defined since ψ_{L} = ψ_{𝔏} on L̃ ∩ 𝔏̃ and L̃ ∩ 𝔏̃ = F⊕(L ∩ 𝔏).

For any f in the space of Π_{𝔏} we can define S(f) : H ⟶ C by

S(f)(x) = ∫_{H/F} ρ(y)f(y^{−1}x)dy.

Given an O_{F}−lattice 𝔐 submodule of 𝔏, we define the function

ρ_{𝔐}(z) = \{ \begin{matrix}ψ_{L}(x)ψ_{𝔏}(y) & if & z = x · y, x ∈ L̃, y ∈ 𝔐̃ \\ 0 & if & z ∉ L̃𝔐̃ = F⊕(L + 𝔐). \end{matrix}

Proposition 4.8. The map S defined above is an H−isomorphism from Π_{𝔏} to Π_{L} Proof. Let f_{0} be the function, in the space of Π_{𝔏}, defined by

f_{0}(z) = \{ \begin{matrix}ψ_{𝔏}(z) & if & z ∈ 𝔏̃ \\ 0 & & otherwise \end{matrix}

and

f_{𝔐}(z) = \{ \begin{matrix}ψ_{𝔐}(z) & if & z ∈ 𝔐̃ \\ 0 & & otherwise \end{matrix}

A computation shows S(f_{0}) = ρ and S(f_{𝔐}) = ρ_{𝔐}.

Since S is different from 0 and Π_{𝔏} is irreducible, S is injective.

We will prove now that S is onto. To this end we prove that the space of Π_{L} is equal to ⟨{ρ_{𝔐} | 𝔐 ⊂ 𝔏}⟩ . First, S(f_{𝔐}) = ρ_{𝔐} so ⟨{ρ_{𝔐} | 𝔐 ⊂ 𝔏}⟩ ⊂ Π_{L}. On the other hand any f in Π_{L} has support compact modulo 𝔏̃ and it is locally constant. From this, it can be seen that any function f is a linear combination of ρ_{𝔐} for different lattices 𝔐 ⊂ 𝔏. Hence we can conclude that S is an isomorphism.

Define now T : Π_{L} ⟶ Π_{𝔏} by

T (f)(x) = ∫_{H/F} θ(y)f(y^{−1}x)dy

where θ is given by

θ(z) = \{ \begin{matrix}ψ_{𝔏}(x)ψ_{L}(y) & if & z = x · y, x ∈ 𝔏, y ∈ L \\ 0 & if & z ∉ 𝔏L \end{matrix}

We have that T ≠ 0 and by Schur´ s Lemma [1] [3], T S = cI, so T S(f_{0}) = cf_{0} which implies c = 1, and finally

T S = I_{Π_{𝔏}}

5. Connections over SL_{∗}(2, A).

The group G = SL_{∗}(2, A) acts naturally by matrix multiplication on W by fixing the bicharacter χ,

χ(gx, gy) = χ(x, y) (x, y ∈ W)

We define a complex G−bundle space 𝔉 = (𝔈, p, Γ, τ) by:

1. Γ = {L | L a Lagrangian of W }

2. Fix a Haar measure dw on W and dw_{L} on a Lagrangian L such that d\overline{w_{L}} is the unique Haar measure on W/L which verify that dw = d\overline{w_{L}}dw_{L}.

For each Lagrangian L we consider the set 𝔈_{L} of all functions f : W ⟶ C which are locally constant, compactly supported modulo L, and such that f(w + l) = χ(w, l)f(w) for every w ∈ W and l ∈ L.

We set

𝔈 = ⋃_{L∈𝔡}^{·}𝔈_{L}

and we define an inner product on each 𝔈_{L} by

⟨f, h⟩ = ∫_{W/L} f(w)\overline{h(w)}dw_{L} (f, h ∈ 𝔈_{L})

3. Let p : 𝔈 ⟶ Γ be the canonical projection which sends each f of 𝔈_{L} to L.

4. The group G acts on 𝔈 and Γ by

[τ_{g}(f)](w) = f(g^{−1}w) (f ∈ 𝔈, g ∈ G, w ∈ W)

and by

τ_{g}(L) = gL (L ∈ 𝔡, g ∈ G)

respectively.

Lemma 5.1. Let L be a Lagrangian subspace of W. Let M be an O_{F}−lattice of W. We set

g_{M}(w) = \{ \begin{matrix}\overline{χ(x, c)} & if & w = x + c ∈ L + M \\ 0 & & otherwise. \end{matrix}

Then, the set {g_{M} | M be an O_{F}−lattice of L} span 𝔈_{L} as a C-vector space. Proof. For each f in 𝔈_{L} we can pick an O_{F}−lattice M such that Supp(f) = L + M. We use that f is locally constant and M is compact, to write f as linear combination of g_{M′} as above.

Let L and L′ be Lagrangians included in a fixed maximal O_{F}−lattice 𝔏 in W. As we have seen, there are two isomorphisms, namely S_{L} : Π_{𝔏} ⟶ Π_{L} and S_{L′} : Π_{𝔏} ⟶ Π_{L′} with T_{L} : Π_{L} ⟶ Π_{𝔏} and T_{L′} : Π_{L′} ⟶ Π_{𝔏} as the respective inverses.

We now define isomorphisms γ̃_{L′,L} : Π_{L} ⟶ Π_{L′}, by
γ̃_{L′,L} = S_{L′} ∘ T_{L}

Let Λ^{L} : Π_{L} ⟶ 𝔈_{L} be defined by Λ^{L}(f)(w) = f(0, w), for f ∈ Π_{L} and w ∈ W, and let, Ω^{L} : 𝔈_{L} ⟶ Π_{L} be defined by Ω^{L}(f)(a, w) = ψ(a)f(w), for f ∈ 𝔈_{L} and (a, w) ∈ L̃. A computation shows that Λ^{L} and Ω^{L} are inverse to each other and both are intertwining operators.

We can define now isomorphisms (which we will call connections) γ_{L,L′} :

𝔈_{L} → 𝔈_{L′} by γ_{L,L′} = Λ^{L′} ∘ γ̃_{L,L′} ∘ Ω^{L}.

Then the diagram

Π_{L} \underrightarrow{γ̃_{L,L′}}  Π_{L′}
Ω^{L} ↑  ↓ Λ^{L′}
𝔈_{L} ⟶^{γ_{L,L′}} 𝔈_{L′}

is commutative.

We obtain

Theorem 5.2. The set Γ = {γ_{L′,L} | L′, L ∈ 𝔡} is a family of G-equivariant connections over the fiber bundle 𝔉 which verifies, for L, L′, L′′ ∈ 𝔡; f, f′ ∈ 𝔈_{L}; h ∈ 𝔈_{L′} g ∈ G the following properties:

1. γ_{L,L′} ∘ γ_{L′,L} = γ_{L,L} = id_{𝔈_{L}} 1. ⟨γ_{L′,L}(f), h⟩ = ⟨f, γ_{L,L′}(h)⟩

2. ⟨γ_{L′,L}(f), γ_{L′,L}(f′)⟩ = ⟨f, f′⟩

3. γ_{L,L′′} ∘ γ_{L′′,L′} ∘ γ_{L′,L} = S_{W} (L; L′, L′′)id_{𝔈_{L}}

4. where S_{W} (L; L′, L′′) is a constant.

5. τ_{g} ∘ γ_{L′,L} = γ_{gL′,gL} ∘ τ_{g}

Note that S_{W} (L; L′, L′′) is the analogous of the Maslov index in [4] and this theorem is comparable with theorem 1.4 in [6].

References

[1] Bernstein, J., and A. Zelevinsky, Representations of the group GL(n, F)where F is a non-Arquimidian local field, Russian Math. Surveys 31 (1976), 1– 68.

[2] Bushnell, C. J., Induced representations of locally profinite groups, J. Alg. 134, (1990), 104–114.

[3] Cartier, P., Representations of p-adic groups, Aut. Forms, Representations, and L-functions, “Proc. of Sym. in Pure Math.” vol 33, Amer. Math. Soc, 1997.

[4] Lion, G., et M. Vergne, The Weil Representation, Maslov index and Theta Series, Prog. Math., 6, Birkhäuser-Verlag, (1980).

[5] Moeglin, C., M.-F. Vignéras, and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Math. 1291, Springer-Verlag, Berlin, (1987).

[6] Perrin, P., Représentations de Schrödinger, Indice de Maslov et groupe métaplectique, “Non Conm. Harmonic Analysis and Lie Groups,” Springer-Verlag, Berlin, (1981).

Roberto Johnson Universidad Católica de Valparaíso Chile rjohnson@ucv.cl José Pantoja Universidad Católica de Valparaíso Chile jpantoja@ucv.cl

Received January 25, 2001 and in final form July 16, 2003