On representation zeta functions of Sp_4 over compact discrete valuation rings

Abstract

Let G be a group, and let r(n,G) denote the number of n-dimensional complex irreducible representations of G up to isomorphism. When G is a topological group, we consider its continuous representations. We assume that G has polynomial representation growth, that is, the number of complex irreducible representations of G of dimension at most n is bounded by a polynomial in n. The representation zeta function of G is a generating function that encodes the sequence r(n,G) and provides a means to study its asymptotic behaviour.

We study the representation zeta function of the rank 2 symplectic group over a compact discrete valuation ring with finite residue field of odd characteristic. Every finite dimensional continuous complex representation of this group factors through a finite quotient. Thus, our focus shifts to the representation zeta functions of these finite quotients, namely, the symplectic groups of rank 2 over local principal ideal rings of length k with finite residue field of odd characteristic.

Using Clifford theory, the irreducible representations of these finite groups can be organised in terms of the adjoint orbits for the action of the symplectic group over the finite residue field on the underlying abelian group of its Lie algebra. We begin by classifying these adjoint orbits into regular, decomposable, 2-primary and nilpotent types. We study the part of the representation zeta function encoding the representations corresponding to each type of orbit separately. The difficulty in the computation varies significantly with the type of the orbit. Show more