Representation growth and rational singularities of the moduli space of local systems

Aizenbud, Avraham; Avni, Nir

doi:10.1007/s00222-015-0614-8

Cited by 38 publications

(127 citation statements)

References 41 publications

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“…It is easy to prove (using partitions of unity, the definable, strict

C^{1}

Sard Lemma and Fubini) that if

φ

is locally constant on

X_{0} false(F false)

then

f_{F,!} false(φ false)

is locally constant on

Y_{0} false(F false)

, see, for example, [, Proposition 3.3.1].…”

Section: Discriminants and Schwartz–bruhat Functionsmentioning

confidence: 99%

Distributions and wave front sets in the uniform non‐archimedean setting

Cluckers

Halupczok²,

Loeser

et al. 2018

Trans. London Math. Soc.

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We study some constructions on distributions in a uniform p‐adic context, and also in large positive characteristic, using model theoretic methods. We introduce a class of distributions which we call distributions of scriptC exp ‐class and which is based on the notion of scriptC exp ‐class functions from Cluckers and Halupczok [J. Ecole Polytechnique (JEP) 5 (2018) 45–78]. This class of distributions is stable under Fourier transformation and has various forms of uniform behavior across non‐archimedean local fields. We study wave front sets, pull‐backs and push‐forwards of distributions of this class. In particular, we show that the wave front set is always equal to the complement of the zero locus of a scriptC exp ‐class function. We first revise and generalize some of the results of Heifetz that he developed in the p‐adic context by analogy to results about real wave front sets by Hörmander. In the final section, we study sizes of neighborhoods of local constancy of Schwartz–Bruhat functions and their push‐forwards in relation to discriminants.

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“…It is easy to prove (using partitions of unity, the definable, strict

C^{1}

Sard Lemma and Fubini) that if

φ

is locally constant on

X_{0} false(F false)

then

f_{F,!} false(φ false)

is locally constant on

Y_{0} false(F false)

, see, for example, [, Proposition 3.3.1].…”

Section: Discriminants and Schwartz–bruhat Functionsmentioning

confidence: 99%

Distributions and wave front sets in the uniform non‐archimedean setting

Cluckers

Halupczok²,

Loeser

et al. 2018

Trans. London Math. Soc.

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“…The (FRS) property was first introduced in [AA16], where it was proved that for any semi-simple algebraic group G the commutator map [·, ·] : G × G → G is (FRS) after 21 self-convolutions. This was then used to show in [AA16] and [AA18] that if Γ is a compact p-adic group or an arithmetic group of higher rank then its representation growth is polynomial and does not depend on Γ. Explicitly, for Γ as above and every c > 40 it holds that r n (Γ) := #{irreducible n-dimensional C-representations of Γ up to equivalence} = o(n c ).…”

Section: Introductionmentioning

confidence: 99%

On singularity properties of convolutions of algebraic morphisms

Glazer

Hendel

2019

Sel. Math. New Ser.

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Let K be a field of characteristic zero, X and Y be smooth K-varieties, and let G be a algebraic K-group. Given two algebraic morphisms ϕ : X → G and ψ : Y → G, we define their convolution ϕ * ψ : X × Y → G by ϕ * ψ(x, y) = ϕ(x) · ψ(y). We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism ϕ : X → G which is dominant when restricted to each absolutely irreducible component of X, by convolving it with itself finitely many times, one can obtain a flat morphism with reduced fibers of rational singularities, generalizing the main result of our previous paper [GH]. Uniform bounds on families of morphisms are given as well. Moreover, as a key analytic step, we also prove the following result in motivic integration; if {f Qp : Q n p → C}p∈primes is a collection of functions which is motivic in the sense of Denef-Pas, and f Qp is L 1 for any p large enough, then in fact there exists ǫ > 0 such that f Qp is L 1+ǫ for any p large enough.

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“…Both the series (1) and the continued function will be denoted by ζ(P ; s). We will speak of the abscissa of convergence either of the series (1) or its meromorphic continuation (where, in the latter case, we mean the maximum of the real parts of its poles), and since one determines the other, there is no ambiguity. In the following, we will writex = (x 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…To choose a monomial P σ appearing in P , we must choose a single term from each linear factor P i of P . We start by choosing x σ (1) from as many factors as it appears in (with a non-zero coefficient). Let e 1 be the number of these factors, so that the monomial we are building contains x e1 σ (1) .…”

Section: Introductionmentioning

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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Representation growth and rational singularities of the moduli space of local systems

Cited by 38 publications

References 41 publications

Distributions and wave front sets in the uniform non‐archimedean setting

Distributions and wave front sets in the uniform non‐archimedean setting

On singularity properties of convolutions of algebraic morphisms

Representation growth of compact linear groups

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