We define analogues of the Casimir and Dirac operators for graded affine Hecke algebras, and establish a version of Parthasarathy's Dirac operator inequality. We then prove a version of Vogan's Conjecture for Dirac cohomology. The formulation of the conjecture depends on a uniform geometric parametrization of spin representations of Weyl groups. Finally, we apply our results to the study of unitary representations.

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The main purpose of this paper is to produce a geometric realization for tempered modules of the affine Hecke algebra of type Cn with arbitrary, non-root of unity, unequal parameters, using the exotic Deligne-Langlands correspondence ([Ka09]). Our classification has several applications to the structure of the tempered Hecke algebra modules. In particular, we provide a geometric and a combinatorial classification of discrete series which contain the sgn representation of the Weyl group, equivalently, via the Iwahori-Matsumoto involution, of spherical cuspidal modules. This last combinatorial classification was expected from [HO97], and determines the L 2 -solutions for the Lieb-McGuire system. . There exists an exotic Springer correspondence ([Ka09]). Moreover, the homology groups of (classical) Springer fibers of Sp(2n) and SO(2n + 1) can be realized via the homology of a suitable exotic Springer fiber (c.f. Corollary 1.24);3. For the Hecke algebras which are known to appear in the representation theory of p-adic groups, the Deligne-Langlands-Lusztig correspondence (DLL for short) was established in [KL87, Lu89, Lu95b, Lu02a]. The connection between the eDL and DLL correspondences is non-trivial. In particular, the "lowest W -types" of a fixed irreducible module differ between the eDL/DLL correspondences;Hecke algebra has the Iwahori-Matsumoto involution which interchanges modules containing the sgn W -type and spherical modules, i.e., modules containing the triv W -type. The images of discrete series containing the sgn W -type under the Iwahori-Matsumoto involution are called spherical cuspidal in [HO97]. Therefore one may view Theorem 4.2 as a classification of spherical cuspidal modules for the affine Hecke algebra H n,m of type B n /C n with arbitrary parameter m. When the Hecke algebra comes from a p-adic group, these should be examples of Arthur representations (in the sense of [Ar89]).The organization of this paper is as follows. In §1, we fix notation and recall the basic results. Some of the material (like Corollary 1.24) is new in the sense that it was not included in [Ka09]. In §2, we present various technical lemmas needed in the sequel. In §3, we formulate and prove our main result, Theorem B. Namely, after recalling some preliminary facts in §3.1, we present our main algorithm σ → out(σ) and state Theorem B in §3.2. We analyze the weight distribution of certain special discrete series in §3.3. Then, we use the induction theorem repeatedly to prove that the module L (aσ,X out(σ) ) must be a discrete series for all σ. We also give an alternate characterization of out(σ) in §3.5. The last section §4 has various applications: we characterize those discrete series L (aσ,X out(σ) ) which contain sgn, and analyze their deformations in §4.1. We prove Theorem C in §4.2. In §4.3, we prove that for generic parameter m, the R(T )-characters of irreducible H n,m -modules are linearly independent. We explain a relation between the view-points of Lusztig and Slooten-Opdam-Solleveld in §4.4. We deduce the W n -ind...

Abstract. We define the algebraic Dirac induction map Ind D for graded affine Hecke algebras. The map Ind D is a Hecke algebra analog of the explicit realization of the Baum-Connes assembly map in the K-theory of the reduced C * -algebra of a real reductive group using Dirac operators. The definition of Ind D is uniform over the parameter space of the graded affine Hecke algebra. We show that the map Ind D defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analogue of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and Atiyah-Schmid.

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, ...