The Smooth Representations of GL<sub>2</sub>(𝔒)

Stasinski, Alexander

doi:10.1080/00927870902829049

Cited by 17 publications

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References 17 publications

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“…In the construction of split regular representations of

{prefixGL}_{2} false(o_{r} false)

appealed to Hill's construction [, Theorem 4.6]. Since Takase has realised that this construction does not produce all the split regular representations, one should view the construction of the current paper as superseding that of , while at the same time unifying the split regular case with the cuspidal.…”

Section: Discussionmentioning

confidence: 99%

The regular representations of GLN over finite local principal ideal rings

Stasinski

Stevens

2017

Bull. London Math. Soc.

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Abstract. Let o be the ring of integers in a non-Archimedean local field with finite residue field, p its maximal ideal, and r ≥ 2 an integer. An irreducible representation of the finite group Gr = GL N (o/p r ) is called regular if its restriction to the principal congruence kernel K r−1 = 1 + p r−1 M N (o/p r ) consists of representations whose stabilisers modulo K 1 are centralisers of regular elements in M N (o/p).The regular representations form the largest class of representations of Gr which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of Gr.

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“…In the construction of split regular representations of

{prefixGL}_{2} false(o_{r} false)

Section: Discussionmentioning

confidence: 99%

The regular representations of GLN over finite local principal ideal rings

Stasinski

Stevens

2017

Bull. London Math. Soc.

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“…Let l = ⌈ r 2 ⌉ and l ′ = ⌊ r 2 ⌋, so that r = l + l ′ . To any π ∈ Irr(G r ), we associate its G l ′ -conjugacy orbit in M 2 (O l ′ ); see for example [30]. First note that if the orbit of π is scalar mod p, it means that the twist isoclass of π contains an imprimitive representation.…”

Section: 2mentioning

confidence: 99%

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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“…Green [9] calculated the characters of the irreducible representations of GL n (O 1 ). Several authors, for instance, Frobenius [6], Rohrbach [23], Kloosterman [15,16], Tanaka [30], Nobs and Wolfart [21], Nobs [20], Kutzko [17], Nagornyȋ [18], and Stasinski [27] studied the representations of the groups SL 2 (O ) and GL 2 (O ). Nagornyȋ [19] obtained partial results regarding the representations of GL 3 (O ) and Onn [22] constructed all the irreducible representations of the groups G ( 1 , 2 ) .…”

Section: Introductionmentioning

confidence: 99%

On representations of general linear groups over principal ideal local rings of length two

Singla

2010

Journal of Algebra

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We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. For general linear groups of order three and four over these rings, we construct all the irreducible representations. We show that the problem of constructing all the irreducible representations of all general linear groups over these rings is not easier than the problem of constructing all the irreducible representations of all general linear groups over principal ideal local rings of arbitrary length in the function field case.

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The Smooth Representations of GL₂(𝔒)

Cited by 17 publications

References 17 publications

The regular representations of GLN over finite local principal ideal rings

The regular representations of GLN over finite local principal ideal rings

Representation growth of compact linear groups

On representations of general linear groups over principal ideal local rings of length two

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The Smooth Representations of GL2(𝔒)

Cited by 17 publications

References 17 publications

The regular representations of GLN over finite local principal ideal rings

The regular representations of GLN over finite local principal ideal rings

Representation growth of compact linear groups

On representations of general linear groups over principal ideal local rings of length two

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The Smooth Representations of GL₂(𝔒)