Frobenius–Schur indicators of unipotent characters and the twisted involution module

Geck, Meinolf; Malle, Gunter

doi:10.1090/s1088-4165-2013-00430-2

Cited by 5 publications

(6 citation statements)

References 19 publications

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“…We also mention that when W is finite, the irreducible decomposition of M q 2 has a surprising interpretation in terms of the "Fourier transform" of a set of "unipotent characters" attached to (W, S). This phenomenon, which is studied in various cases in the articles [3,7,15,20,23], gives one more indication that M q 2 deserves consideration not only in the crystallographic case.…”

Section: "Twisted Kazhdan-lusztig Theory"mentioning

confidence: 98%

Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the universal case

Marberg

2014

Represent. Theory

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Abstract. Let (W, S) be a Coxeter system and let w → w * be an involution of W which preserves the set of simple generators S. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements w ∈ W with w −1 = w * ) naturally generates a module of the Hecke algebra of (W, S) with two distinguished bases. The transition matrix between these bases defines a family of polynomials P σ y,w which one can view as "twisted" analogues of the much-studied Kazhdan-Lusztig polynomials of (W, S). The polynomials P σ y,w can have negative coefficients, but display several conjectural positivity properties of interest. This paper reviews Lusztig's construction and then proves three such positivity properties for Coxeter systems which are universal (i.e., having no braids relations), generalizing previous work of Dyer. Our methods are entirely combinatorial and elementary, in contrast to the geometric arguments employed by Lusztig and Vogan to prove similar positivity conjectures for crystallographic Coxeter systems.

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Section: "Twisted Kazhdan-lusztig Theory"mentioning

confidence: 98%

Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the universal case

Marberg

2014

Represent. Theory

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“…The fact that ρ C indeed is equal to the character originally constructed in [Ko] is shown in [GeMa,Rem. 2.2].…”

Section: Conjecture 57 (Kottwitz [Ko §1]) Let C Be a Union Of Conjmentioning

confidence: 74%

“…Let Φ be the parabolic subsystem defined by W . Then, for any w ∈ C W (σ), we have ε σ (w) = (−1) k , where k is the number of positive roots in Φ which are sent to negative roots by w (see also [GeMa,Rem. 2.2]).…”

Section: Definition 64 ([mentioning

confidence: 99%

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Conjugacy classes of involutions and Kazhdan–Lusztig cells

Bonnafé¹,

Geck

2014

Represent. Theory

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Abstract. According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group S n are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz's conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types which, combined with further recent work, settles this conjecture in general.

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“…(More precisely, Kottwitz computed the irreducible constituents of certain representations induced from the centralizers of involutions in W . The sum of these induced representations is isomorphic to ̺ W , although this is not an obvious fact; see [22,Remark 2.2] for a detailed explanation.) Kottwitz proved this formula in the classical cases, while Casselman [14] carried out the calculations necessary to check it in the exceptional ones.…”

Section: Introductionmentioning

confidence: 99%

How to compute the Frobenius–Schur indicator of a unipotent character of a finite Coxeter system

Marberg¹

2013

Advances in Mathematics

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For each finite, irreducible Coxeter system (W, S), Lusztig has associated a set of "unipotent characters" Uch(W ). There is also a notion of a "Fourier transform" on the space of functions Uch(W ) → R, due to Lusztig for Weyl groups and to Broué, Lusztig, and Malle in the remaining cases. This paper concerns a certain W -representation ̺ W in the vector space generated by the involutions of W . Our main result is to show that the irreducible multiplicities of ̺ W are given by the Fourier transform of a unique function ǫ : Uch(W ) → {−1, 0, 1}, which for various reasons serves naturally as a heuristic definition of the Frobenius-Schur indicator on Uch(W ). The formula we obtain for ǫ extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which W is a Weyl group. We include in addition a succinct description of the irreducible decomposition of ̺ W derived by Kottwitz when (W, S) is classical, and prove that ̺ W defines a Gelfand model if and only if (W, S) has type A n , H 3 , or I 2 (m) with m odd. We show finally that a conjecture of Kottwitz connecting the decomposition of ̺ W to the left cells of W holds in all non-crystallographic types, and observe that a weaker form of Kottwitz's conjecture holds in general. In giving these results, we carefully survey the construction and notable properties of the set Uch(W ) and its attached Fourier transform.

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Frobenius–Schur indicators of unipotent characters and the twisted involution module

Cited by 5 publications

References 19 publications

Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the universal case

Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the universal case

Conjugacy classes of involutions and Kazhdan–Lusztig cells

How to compute the Frobenius–Schur indicator of a unipotent character of a finite Coxeter system

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