Representation Growth of Compact Special Linear Groups of degree two

Hassain, M; Singla, Pooja

doi:10.48550/arxiv.1807.06570

Cited by 1 publication

(5 citation statements)

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“…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…”

Section: Introductionmentioning

confidence: 86%

“…In any case, our results show that the only remaining thing needed in order to compute the exact abscissa of convergence for SL 2 (F q [[t]]), q even, is the function u r (β, θ). Very recently, M and Singla [24] have obtained strong bounds on u r (β, θ) which imply that the abscissa is 1.…”

Section: This Upper Bound Converges Formentioning

confidence: 99%

“…, y w , the equation may therefore not have a solution, but if it does, there are two possibilities for x w . ix) Equations (24) and (26) for m > w. For each m > w, we have the equations . The first one corresponds to type (24) and the second to type (26).…”

Section: Moreovermentioning

confidence: 99%

“…ix) Equations (24) and (26) for m > w. For each m > w, we have the equations . The first one corresponds to type (24) and the second to type (26). Since m > w, we have 2m − w > m and 2m − w − 1 > m − 1, so we can solve each equation from the left hand side.…”

Section: Moreovermentioning

confidence: 99%

“…Suppose first that x 0 = 0. For each m > w, we can solve from the equation of type (24) y 2m−w = h 2m−w (y 0 , . .…”

Section: Moreovermentioning

confidence: 99%

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Representation growth of compact linear groups

Häsä

Stasinski

2019

Trans. Amer. Math. Soc.

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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...

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Section: Introductionmentioning

confidence: 86%

Section: This Upper Bound Converges Formentioning

confidence: 99%