Abstract 

Let G be a group, and for a natural number n, let r(n, G) denote the number of n-dimensional complex irreducible representations of G up to isomorphism. If G is topological, we consider the continuous representations. The study of the asymptotic behavior of the sequence r(n, G) has interesting applications, for instance in studying singularities of the moduli space of local systems. A main tool for this analysis is the representation zeta function, which is a Dirichlet generating function that encodes this sequence. 

In this talk, we will focus on the representation zeta function of the rank-2 symplectic group over a compact discrete valuation ring with finite residue field of odd characteristic. Using Clifford theory, the irreducible representations of the finite quotients of this group can be organised in terms of the adjoint orbits of the symplectic group acting on the underlying abelian group of its Lie algebra over the finite residue field. 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.