Dirac cohomology for graded affine Hecke algebras

Barbasch, Dan; Ciubotaru, Dan; Trapa, Peter E.

doi:10.1007/s11511-012-0085-3

Cited by 44 publications

(87 citation statements)

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“…Hence, H D (X) is equipped with a W -representation structure. A crucial result in the Dirac cohomology theory, so-called Vogan's conjecture [BCT,Theorem 4.4], states that if X is irreducible and H D (X) = 0, then the W -isotypic component in H D (X) determines the central character of X. A goal of this paper is to compute the Dirac cohomology of standard modules.…”

Section: Main Results For Any Complex Reductive Group H Let Hmentioning

confidence: 99%

“…In order to apply the result, we need the fact that for the Iwahori-Matsumoto dual of a tempered module is 'tempered' in the sense of [BCT]. By suitably rescaling on V , we first reduce our statement to the case that E = E he,e,1,ζ for some θ-quasidistingusihed e. Now using the argument in the proof of Lemma 4.7, we have that IM (E he,e,1,ζ ) is unitary and so the hypothesis of [BCT,Proposition 5.7] is satisfied. We now consider (1).…”

Section: Vanishing Theoremmentioning

confidence: 99%

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A vanishing theorem for Dirac cohomology of standard modules

Chan

2018

Advances in Mathematics

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Abstract. This paper studies the Dirac cohomology of standard modules in the setting of graded Hecke algebras with geometric parameters. We prove that the Dirac cohomology of a standard module vanishes if and only if the module is not twistedelliptic tempered. The proof makes use of two deep results. One is some structural information from the generalized Springer correspondence obtained by S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of tempered modules by Barbasch-Ciubotaru-Trapa and Ciubotaru.We apply our result to compute the Dirac cohomology of ladder representations for type An. For each of such representations with non-zero Dirac cohomology, we associate to a canonical Weyl group representation. We use the Dirac cohomology to conclude that such representations appear with multiplicity one.

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Section: Main Results For Any Complex Reductive Group H Let Hmentioning

confidence: 99%

Section: Vanishing Theoremmentioning

confidence: 99%

A vanishing theorem for Dirac cohomology of standard modules

Chan

2018

Advances in Mathematics

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“…In particular, 1 / ∈ W (a) and κ 1 = 0. Theorem 2.7 specializes to the formula for D 2 from [BCT,Theorem 3.5]:…”

Section: Motivated By the Analogy Between The Theory Of Graded Affinementioning

confidence: 99%

“…We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld [Dr]. We generalize in this way, the Dirac cohomology theory for Lusztig's graded affine Hecke algebras defined in [BCT] and further developed in [BCT,COT,Ci1,CH,Cha2]. We apply these constructions to the case of the symplectic reflection algebras defined by Etingof-Ginzburg [EG], particularly to rational Cherednik algebras for real or complex reflection groups.…”

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confidence: 99%

Dirac cohomology for symplectic reflection algebras

Ciubotaru

2015

Sel. Math. New Ser.

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Abstract. We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld [Dr]. We generalize in this way, the Dirac cohomology theory for Lusztig's graded affine Hecke algebras defined in [BCT] and further developed in [BCT,COT,Ci1,CH,Cha2]. We apply these constructions to the case of the symplectic reflection algebras defined by Etingof-Ginzburg [EG], particularly to rational Cherednik algebras for real or complex reflection groups. As applications, we give criteria for unitarity of modules in category O and we show that the 0-fiber of the Calogero-Moser space admits a description in terms of a certain "Dirac morphism" originally defined by Vogan for representations of real reductive groups. Contents1. Introduction 1 2. The Dirac operator for Drinfeld's Hecke algebras 3 3. Vogan's Dirac morphism 9 4. Symplectic reflection algebras 14 5. Applications: unitarity, the Calogero-Moser space 21 References 271. Introduction 1.1. The Dirac operator has played an important role in the representation theory of real reductive groups, see for example [AS], [Ko], [Pa], and the monograph [HP2]. The notion of Dirac cohomology for admissible (g, K)-modules of real reductive groups was introduced by Vogan [Vo] around 1997. The Dirac cohomology of a (g, K)-module is a certain finite dimensional representation of (a pin cover of) the maximal compact subgroup K. One of the main ideas, "Vogan's conjecture", proved by Huang and Padžić [HP1] says that, if nonzero, the Dirac cohomology of an irreducible module X uniquely determines the infinitesimal character of X.It is a pleasure to thank B. Krötz and E. Opdam for the invitation to give a series of lectures on the theory of the Dirac operator for Hecke algebras at the Spring School "Representation theory and geometry of reductive groups", Heiligkreuztal 2014, where some of these ideas crystallized. I also thank J.S. Huang, K.D. Wong, and the referee for corrections, helpful comments, and references. 1.2. Motivated by the analogy between the theory of graded affine Hecke algebras H of reductive p-adic groups, as defined by Lusztig [Lu1], and certain elements of the representation theory of real reductive groups, in joint work with Barbasch and Trapa [BCT], we defined a Dirac operator and the notion of Dirac cohomology for H-modules. This theory was subsequently developed in several papers, including [COT,CT1,CH,Cha2] and it led to interesting results, such as a geometric realization in the kernel of global Dirac operators of the irreducible discrete series H-modules [COT], a partial analogue of the realization of discrete series representations for real semisimple groups by Atiyah-Schmid [AS] and Parthasarathy [Pa]. An important element that occurs in these constructions for H is a certain pin cover W of finite Coxeter groups W whose representations turned out to have surprising relations with the geometry of the nilpotent cone, see [Ci1,CH,CT1], also [Cha1] for noncrystallographic Coxeter groups. In particular, ...

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“…In [12], we introduced a Dirac operator and the notion of Dirac cohomology in the setting of the graded Hecke algebras defined by Drinfeld, extending in this way the construction from [2] for Lusztig's graded affine (see [19]) to be related to the one into two-sided cells (or families) of representations, in the sense of Lusztig [20] for real reflection groups, and [24] for complex reflection groups. At least when W is a real reflection group, the conjecture is that the two partitions coincide.…”

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confidence: 99%

One-W-type Modules for Rational Cherednik Algebra and Cuspidal Two-sided Cells

Ciubotaru¹

2018

Bulletin of the Inst. of Math. Academia Sinica(N.S.)

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We classify the simple modules for the rational Cherednik algebra H0,c that are irreducible when restricted to W , in the case when W is a finite Weyl group. The classification turns out to be closely related to the cuspidal two-sided cells in the sense of Lusztig. We compute the Dirac cohomology of these modules and use the tools of Dirac theory to find nontrivial relations between the cuspidal Calogero-Moser cells, in the sense of Bellamy, and the cuspidal two-sided cells.

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Dirac cohomology for graded affine Hecke algebras

Cited by 44 publications

References 21 publications

A vanishing theorem for Dirac cohomology of standard modules

A vanishing theorem for Dirac cohomology of standard modules

Dirac cohomology for symplectic reflection algebras

One-W-type Modules for Rational Cherednik Algebra and Cuspidal Two-sided Cells

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