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Representation growth of compact linear groups

Häsä, J.; Stasinski, A.

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Authors

J. Häsä



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].

Journal Article Type Article
Acceptance Date May 18, 2018
Online Publication Date Apr 18, 2019
Publication Date Apr 18, 2019
Deposit Date Jun 14, 2018
Publicly Available Date Apr 18, 2019
Journal Transactions of the American Mathematical Society
Print ISSN 0002-9947
Electronic ISSN 1088-6850
Publisher American Mathematical Society
Peer Reviewed Peer Reviewed
Volume 372
Issue 2
Pages 925-980
DOI https://doi.org/10.1090/tran/7618
Public URL https://durham-repository.worktribe.com/output/1328940

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