Representation growth of compact linear groups

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 GL 2 (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 SL 2 (Fq[[t]]), q even, and deduce that its abscissa lies in the interval [1, 5/2]. that the abscissa of ζ SL2(O) (s) is 1 whenever char O = 0. Our computations of the abscissa ofζ GL2(O) (s) in this case, together with Proposition 3.4, give a new proof of this fact. We also show that the abscissa ofζwith q even. This does not follow from any previously known results and our computation is substantially harder than in the cases where char O = 2.The zeta function of SL 2 (F q [[t]]), q even. In Section 5, we assume that char O = 2, that is, O = F q [[t]] with q even. We give a Clifford theory construction of the representations of SL 2 (F q [[t]]/(t r )) for r even, which is completely explicit apart from the order of certain finite groups V (β, θ) (see Definition 5.7) and certain integers c ∈ {1, 2, 3} (see Lemma 4.21). We use this construction to approximate the zeta function ζ SL 2 (O) (s), and show that its abscissa lies between 1 and 5/2 (Theorem 5.9). The lower bound 1 follows from a general result of Larsen and Lubotzky [21, Proposition 6.6], but we give an independent proof of this.Section 6 is devoted to a proof of Lemma 4.21, which is crucial for our results aboutζ GL 2 (Fq[[t]]) (s) and ζ SL 2 (Fq[[t]]) (s) when q is even. The lemma gives the number of solutions, up to a factor c ∈ {1, 2, 3}, in F q [[t]]/(t i ), i ≥ 1, to the equation∆ τ ] mod (t) is a scalar plus a regular nilpotent matrix. The number of solutions depends in a delicate way on a new invariant, which we call the odd depth, of the twist orbit (i.e., orbit modulo scalars) of the matrix [ 0 1∆ τ ] (see Definition 4.19). Remark. After the present paper had been accepted for publication, Hassain M and Pooja Singla [24] announced results about the representations of SL 2 (O), p = 2, which in particular imply that the abscissa of ζ SL2(Fq[[t]]) (s) is 1. Notation.We let N stand for the set of natural numbers, not including 0.In Sections 4 and 5, we will use the Vinogradov notation f (r) ≪ g(r) for two functions f (r), g(r) of r (or of l = ⌈r/2⌉). Note that f (r) ≪ g(r) is equiv...

Representation Growth of Compact Special Linear Groups of degree two

“…Recall a ring of integers o of characteristic zero has ramification index e if 2o = p e . This also completes some of the related results mentioned in [7]. In the process of construction and enumeration of irreducible representations, we also prove the following result.…”

“…For the proof, first in Sections 4 and 5 we give a construction of a certain class of a continuous irreducible representations of SL 2 (o). We combine these results with many non-trivial results of [7] and obtain the proof of Theorem 1.1. We refer the reader to [2], [4] and [7] for more results on abscissa of convergence of compact linear groups.…”

“…We combine these results with many non-trivial results of [7] and obtain the proof of Theorem 1.1. We refer the reader to [2], [4] and [7] for more results on abscissa of convergence of compact linear groups.…”

“…For the more general case of all rings o, it is already known that α(SL 2 (o)) ∈ [1,22], where the lower bound is by the work of Larsen and Lubotzky [15,Proposition 6.6] and the upper bound is by Aizenbud-Avni [1]. Recently, Häsä-Stasinski in [7], while studying the twist representation zeta functions of GL n (o) and relating these with representation zeta function of SL n (o), improved the bounds for α(SL 2 (o)) and proved that α(SL 2 (o)) ∈ [1, 5/2] for all o and furthermore included a new proof of the fact that α(SL 2 (o)) = 1 for o such that Char(o) = 0. It is also clear from their work that the case SL 2 (o) where Char(o) = 2 is more complicated as compared to the cases of SL 2 (o) where Char(o) is zero or odd.…”

“…This work is supported in part by UGC Centre for Advanced Studies. Authors would like to immensely thank Alexander Stasinski and J. Häsä for sharing a preprint of their article [7] and providing many helpful comments on this article. The authors also greatly thank Uri Onn and Dipendra Prasad for their encouragement and very helpful comments on this work.…”