ABSTRACT
We discuss certain questions dealing with Weil representations of the symplectic group Sp(2n,K), where K is a finite or a nonarchimedean local field. We also consider the case where K is a finite ring. In the case that K is a local field of characteristic 0, we consider the problem of realizing the Weil representation over a number field. References for the local field case are [MVW], [K], [RR], [W]. For the case of a finite fields, see [I], [G]. For the case of a finite ring see [CMS1], [CMS2]. We first define the symplectic and Heisenberg groups, which can be de-
fined for any field K. For our purposes in studying the Weil representation, K will be one of: the real numbers R, a finite field of characteristic not 2, or a local field. In the case that K is a local field, it is the field of fractions of a complete discrete valuation ring O having maximal ideal P and finite residue class field k = GF (q) where q is a power of a prime p 6= 2. Let V be a vector space over a field K, of even dimension 2n. We suppose
that 〈 , 〉 is a nondegenerate alternating bilinear form on V . The symplectic group is by definition
Sp = Sp(V ) = {g ∈ GL(V ) : 〈vg, wg〉 = 〈v, w〉 , for all v, w ∈ V }. Let V have K-basis {x1, x2, . . . , xn, y1, y2, . . . , yn}. We define our alter-
nating form by
〈xi, xj〉 = 〈yi, yj〉 = 0, 1 ≤ i, j ≤ n, 〈xi, yj〉 = δij , 1 ≤ i < j ≤ n. We define X and Y to be the K-span of the sets {x1, x2, . . . , xn} and {y1, y2, . . . , yn} respectively, so V = X⊕Y , and 〈x, y〉 = 0 for x ∈ X, y ∈ Y . For some of the results mentioned below, K is taken to be a commutative
ring; then we assume that V is a free K-module of even rank 2n. To discuss the Weil representation, one starts with the Heisenberg group
H = H(V ) which as a set is given by
H = K × V 81
with multiplication given by
(c1, v1)(c2, v2) = (c1 + c2 + 〈v1, v2〉 , v1 + v2), c1, c2 ∈ V, v1, v2 ∈ K. It is not difficult to see that H is a nilpotent group of class 2, whose center is
Z(H) = {(c, 0) : c ∈ K}, so Z(H) is isomorphic to the additive group of K, denoted K+. The symplectic group Sp acts on the Heisenberg group H by
(c, v)g = (c, vg), g ∈ Sp, (c, v) ∈ H. Note that under this action, each element of the center Z(H) is fixed by Sp. Fix a continuous character λ ∈ Hom(K+,C∗). If K is a finite field or a
local field or the real numbers, there is an irreducible representation, the Schro¨dinger representation, S of H with the property that S(c, 0) is scalar multiplication by λ(c); one says that S has central character λ. We describe how S can be constructed. One can also make this construction in the case that K is a finite ring. In this case we assume that K is Z/p`Z for some odd prime p, or more generally, K = O/P ` where O is a discrete valuation ring having 2 as a unit, where P is the maximal ideal of O. One takes V = K2n as above. We assume that the character λ is primitive, in the sense that its kernel contains no nontrivial ideal of K. Since 〈x, y〉 = 0 for x, y ∈ X, then the subset A of H given by A =
K ×X = {(c, x) : x ∈ X, c ∈ K} is an abelian normal subgroup of H. We extend the character λ to a character λ′ ∈ Hom(A,C∗) by defining
λ′(c, x) = λ(c), x ∈ X. In the case that K is a finite field, or more generally O/P `, we define
indHA λ ′ to be the set of all functions φ from H to C∗ such that
φ(ah) = λ′(a)φ(h), for all a ∈ A, h ∈ H. Given h ∈ H, we have the operator S(h) on indHA λ′ defined by (1) (S(h)φ)(h′) = φ(h′h), h, h′ ∈ H. We can find functions in indHA λ
′ by taking coset representatives T of A in H, taking any function φ defined on T , and extending φ to H by φ(at) = λ′(a)φ(t). A natural choice of coset representatives is to take T to be the subset of H given by {(0, y) : y ∈ Y }. Then indHA λ′ is identified with the set S(Y ) of all complex functions on Y . With this identification, the action of the operator S(h), given in equation (1) on φ ∈ indHA (λ), is now given, if h = (c, x+ y), by
(2) S(c, x+ y)φ(y′) = λ(c+ 〈y, x〉+ 2 〈y′, x〉)φ(y + y′), y′ ∈ Y, φ ∈ S(Y ). Then S(Y ) gives us an irreducible representation ofH with central character λ, and is the unique irreducible representation, up to equivalence, with this property.
