We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.

Durham Research Online
You are in:

Representation growth of compact linear groups.

Häsä, J. and Stasinski, A. (2019) 'Representation growth of compact linear groups.', Transactions of the American Mathematical Society., 372 (2). pp. 925-980.


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].

Item Type:Article
Full text:Publisher-imposed embargo
(AM) Accepted Manuscript
File format - PDF
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
Download PDF (Revised version)
Publisher Web site:
Publisher statement:Accepted manuscript available under a CC-BY-NC-ND licence.
Date accepted:18 May 2018
Date deposited:15 June 2018
Date of first online publication:18 April 2019
Date first made open access:18 April 2019

Save or Share this output

Look up in GoogleScholar