Representation growth of compact linear groups

2017/10/25 by Häsä, Jokke, Stasinski, Alexander
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1710.09112

Abstract

We study the representation growth of simple compact Lie groups and of SL_n(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GL_n(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 GL_n(O) exists and has the same abscissa of convergence as the zeta function of SL_n(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(\mathbbF_q[[t]]), q even, and deduce that its abscissa lies in the interval [1, 5/2].

Related

• Representation Growth of Linear Groups
• Representation growth and representation zeta functions of groups
• Representation Growth of Compact Special Linear Groups of degree two
• On representation zeta functions for special linear groups
• On representation zeta functions of groups and a conjecture of Larsen and Lubotzky