Representation zeta functions of split extensions of $SL_2^m(O)$

TL;DR

This paper studies the representation zeta functions of split extensions of $SL_2^m(O)$, showing they factor into known components and computing explicit examples for certain families and subgroups.

Contribution

It proves the factorization of representation zeta functions for split extensions of $SL_2^m(O)$ and computes these functions explicitly for specific infinite families.

Findings

Representation zeta functions factor as a product of component functions.

Explicit formulas for zeta functions of two infinite families of extensions.

Computed zeta functions for a broad class of subgroups of $SL_2^m(O)$.

Abstract

We consider the representation growth of split extensions of $SL_2^m(O)$. We prove that the corresponding representation zeta functions factor as a product of the representation zeta function of $SL_2^m(O)$ and the relative representation zeta function associated to the extension. We make use of our result by computing the zeta functions for two infinite families of split extensions of $SL_2^m(O)$ explicitly. Along the way, we compute the representation zeta functions of a large class of subgroups of $SL_2^m(O)$.