Weil Representations of Symplectic Groups over Non-Principal Rings

Abstract

Let W : Sp (2n, R) → GL(X) be a Weil representation of the symplectic group of rank 2n over a finite commutative ring R of odd characteristic. This is a complex representation of degree |R|n defined in terms of the action of Sp (2n, R) on a two-step nilpotent group called Heisenberg group. We address the problem of decomposing the Sp (2n, R)-module X into irreducible constituents. The problem can easily be reduced to the case when R is local and quasi-Frobenius. Further, the case when R is a principal ring has already been solved. This was achieved by means of the following recursive property of the Weil representation: precisely two irreducible constituents of X do not admit trivial action by any congruence subgroup of Sp (2n, R); the remaining irreducible constituents lie inside an Sp (2n, R)submodule Y of X that affords a Weil representation for a quotient symplectic group Sp (2n, T ). We show here that this recursive property of Y holds only when R is principal, failing in all other cases. This failure opens the following Pandora box: given any finite commutative quasi-Frobenius local ring R0 of odd characteristic, we can choose R so that R0 is quotient of R and every complex irreducible character of Sp (2n,R0) enters Y when inflated to Sp (2n, R). Thus, the problem of decomposing the Weil module X into irreducible constituents is, in general, as difficult as the problem of finding all complex irreducible characters of all symplectic groups Sp (2n, R0). In spite of this, we manage to identify submodules of X that do admit either a Weil representation or the tensor product of various Weil representations for a quotient symplectic group.