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Durham University

Department of Mathematical Sciences

Academic Staff

Publication details for Alexander Stasinski

Häsä, J. & Stasinski, A. (2019). Representation growth of compact linear groups. Transactions of the American Mathematical Society 372: 925-980.

Author(s) from Durham

Abstract

We study the representation growth of simple compact Lie groups
and of SLn(O), where O is a compact discrete valuation ring, as well as the
twist representation growth of GLn(O). This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions.
We determine the abscissae for a class of Mellin zeta functions which include
the Witten zeta functions. As a special case, we obtain a new proof of the
theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is
r/κ, where r is the rank and κ the number of positive roots.
We then show that the twist zeta function of GLn(O) exists and has the
same abscissa of convergence as the zeta function of SLn(O), provided n does
not divide char O. We compute the twist zeta function of GL2(O) when the
residue characteristic p of O is odd and approximate the zeta function when
p = 2 to deduce that the abscissa is 1. Finally, we construct a large part of
the representations of SL2(Fq[[t]]), q even, and deduce that its abscissa lies in
the interval [1, 5/2].