Dan Barbasch

• Cornell University

Let G be a real reductive group in Harish-Chandra’s class. We derive some consequences of the theory of coherent continuation representations to the counting of irreducible representations of G with a given infinitesimal character and a given bound of the complex associated variety. When G is a real classical group (including the real metaplectic g...

In the case of complex symplectic and orthogonal groups, we find \((\mathfrak {g}, K)-\)modules with the property that their \(K-\)structure matches the structure of regular functions on the closures of nilpotent orbits. This establishes a version of the Orbit Method of Kirillov-Kostant-Souriau as proposed by Vogan. In the process, we give another...

We determine all genuine special unipotent representations of real spin groups and quaternionic spin groups and show in particular that all of them are unitarizable. We also show that there are no genuine special unipotent representations of complex spin groups.

Let G be a special linear group over the real, the complex or the quaternion, or a special unitary group. In this note, we determine all special unipotent representations of G in the sense of Arthur and Barbasch–Vogan, and show in particular that all of them are unitarizable.

In analogy with the Barbasch–Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this d...

In this paper, we classify all unitary representations with non-zero Dirac cohomology for complex Lie group of Type E8. This completes the classification of Dirac series for all complex simple Lie groups.

Let G be a special linear group over the real, the complex or the quaternion, or a special unitary group. In this note, we determine all special unipotent representations of G in the sense of Arthur and Barbasch-Vogan, and show in particular that all of them are unitarizable.

We determine all genuine special unipotent representations of real spin groups and quaternionic spin groups. In particular, we show that all of them are unitarizable.

This paper computes the Dirac cohomology HD(π) of irreducible unitary Harish-Chandra modules π of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are shown to be unitarily induced from unipotent representations. When nonzero, there is a unique, multiplicity free K-type...

In this paper, we compute the character formula of the Brylinski model for all classical nilpotent varieties $\overline{\mathcal{O}}$. As a consequence, one can compute the multiplicities of all $K-$types of the ring of regular functions $R(\overline{\mathcal{O}})$ for all classical nilpotent varieties.

Let $G$ be a complex classical Lie group. Up to equivalence, this paper classifies all the irreducible unitary Harish-Chandra modules with non-zero Dirac cohomology for $G$. Moreover, we prove that any such module $\pi$ has a unique spin-lowest $K$-type which occurs with multiplicity one. This confirms a couple of conjectures raised by Pandzic and...

We present recent joint work with Peter Trapa on the notion of twisted Dirac index and its applications to (twisted) characters and to extensions of modules in a short and informal way. We also announce some further generalizations with applications to Lefschetz numbers and automorphic forms.

The Peter–Weyl idempotent $e_{\mathscr{P}}$ of a parahoric subgroup $\mathscr{P}$ is the sum of the idempotents of irreducible representations of $\mathscr{P}$ that have a nonzero Iwahori fixed vector. The convolution algebra associated with $e_{\mathscr{P}}$ is called a Peter–Weyl Iwahori algebra . We show that any Peter–Weyl Iwahori algebra is Mo...

The Peter-Weyl idempotent $e_{\mathcal{P}}$ of a parahoric subgroup ${\mathcal{P}}$ is the sum of the idempotents of irreducible representations of $\mathcal{P}$ which have a nonzero Iwahori fixed vector. The convolution algebra associated to $e_{\mathcal{P}}$ is called a Peter-Weyl Iwahori algebra. We show any Peter-Weyl Iwahori algebra is Morita...

In the case of complex classical groups, we find $(\mathfrak{g},K)$-modules with the property that their $K$-structure matches the structure of regular functions on the closures of nilpotent orbits.

This chapter describes properties of unipotent representations in relation to the Θ-correspondence and relations of the K-structure to rational function of coadjoint orbits in the Lie algebra. The main focus is on the complex case.

This paper provides a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type $D$. Precisely, let $(\tilde G, \tilde K)$ be the complexification of the groups $(\tilde G_0,\tilde K_0)=(Spin(2n,\mathbb C), Spin(2n))$. We compute $\tilde K$-spectra of the regular...

The results in this paper provide a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits. Precisely, let $\tilde{G}=\tilde{Spin}(a,b)$ with $a+b=2n$, the nonlinear double cover of $Spin(a,b)$, and let $\tilde{K}=Spin(a)\times Spin(b)$ be the maximal compact subgroup of $\tilde{G}$. We...

Work of Bezrukavnikov–Kazhdan–Varshavsky uses an equivariant system of trivial idempotents of Moy–Prasad groups to obtain an Euler–Poincaré formula for the r–depth Bernstein projector. We establish an Euler–Poincaré formula for natural sums of depth zero Bernstein projectors (which is often the projector of a single Bernstein component) in terms of...

Work of Bezrukavnikov--Kazhdan--Varshavsky uses an equivariant system of trivial idempotents of Moy--Prasad groups to obtain an Euler--Poincar\'e formula for the r--depth Bernstein projector. We establish an Euler--Poincar\'e formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter--Weyl i...

This paper provides a construction of the unipotent representations for classical complex groups in terms of the Theta correspondence as introduced and studied by R. Howe. The K-type structure of unipotent representations is obtained as a consequence of the character formulas for unipotent representations of D. Vogan and the author. This provides a...

Let G be a real reductive Lie group with maximal compact sub- group K. We generalize the usual notion of Dirac index to a twisted version, which is nontrivial even in case G and K do not have equal rank. We compute ordinary and twisted indices of standard modules. As applications, we study extensions of Harish-Chandra modules and twisted characters...

Let G be a real reductive Lie group with maximal compact sub- group K. We generalize the usual notion of Dirac index to a twisted version, which is nontrivial even in case G and K do not have equal rank. We compute ordinary and twisted indices of standard modules. As applications, we study extensions of Harish-Chandra modules and twisted characters...

In this paper, we recover certain known results about the ladder representations of \(\mathrm{GL}(n, \mathbb{Q}_{p})\) defined and studied by Lapid, Mínguez, and Tadić. We work in the equivalent setting of graded Hecke algebra modules. Using the Arakawa–Suzuki functor from category O to graded Hecke algebra modules, we show that the determinantal f...

In this paper, we consider the star operations for (graded) affine Hecke algebras which preserve certain natural filtrations. We show that, up to inner conjugation, there are only two such star operations for the graded Hecke algebra: the first, denoted $\star$, corresponds to the usual star operation from reductive $p$-adic groups, and the second,...

In this paper, we recover certain known results about the ladder representations of GL(n, Q_p) defined and studied by Lapid, Minguez, and Tadic. We work in the equivalent setting of graded Hecke algebra modules. Using the Arakawa-Suzuki functor from category O to graded Hecke algebra modules, we show that the determinantal formula proved by Lapid-M...

We study star operations for Iwahori-Hecke algebras and invariant hermitian forms for finite dimensional modules over (graded) affine Hecke algebras.

In this paper we study the problem of computing the Dirac cohomology of the special unipotent representations of the real groups Sp(2n,R) and U(p,q).

In this paper, we generalize the results of Barbasch-Moy to affine Hecke algebras of arbitrary isogeny class with geometric unequal parameters, and extended by groups of automorphisms of the root datum. When the Bushnell-Kutzko theory of types gives a Hecke algebra of the form considered in this paper, our results establish a transfer of unitarity...

In this paper, we review the construction of the Dirac operator for graded affine Hecke algebras and calculate the Dirac cohomology of irreducible unitary modules for the graded Hecke algebra of gl(n).

Using Lusztig's geometric classification, we find the reducibility points of a standard module for the affine Hecke algebra, in the case when the inducing data is generic. This recovers the known result of Muić and Shahidi for representations of split p-adic groups with Iwahori-spherical Whittaker vectors. We also give a necessary (but insufficient...

This paper studies unitary representations with Dirac cohomology for complex groups, in particular relations to unipotent representations

We define an analogue of the Casimir element for a graded affine Hecke algebra \( \mathbb{H} \), and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology H D (X) of an \( \mathbb{H} \)-module X, and show that H D (X) carries a representation of a canonical double cover of the Weyl group \( \wi...

We establish a transfer of unitarity for a Bernstein component of the category of smooth representations of a reductive p-adic group to the associated Hecke algebra, in the framework of the theory of types, whenever the Hecke algebra is an affine Hecke algebra with geometric parameters, in the sense of Lusztig (possibly extended by a group of autom...

Using Lusztig’s geometric classification, we find the reducibility points of a standard module for the affine Hecke algebra, in the case when the inducing data is generic. This recovers the known result of [MS] for representations of split p-adic groups with Iwahori-spherical Whittaker vectors. We also give a necessary (insufficient) condition for...

The aim of this paper is to give an exposition of recent progress on the determination of the unitarizable Langlands quotients of minimal principal series for reductive groups over the real or p-adic fields in characteristic 0.

This paper gives the classification of the Whittaker unitary dual for affine graded Hecke algebras of type E. By the Iwahori-Matsumoto involution, this is also equivalent to the classification of the spherical unitary dual for type E. Together with some results of Barbasch and Moy (D. Barbasch and A. Moy, Unitary spherical spectrum for p-adic class...

This paper gives a complete classification of the unitary irreducible spherical representations of split real and p-adic groups. The results were obtained around 2000, the changes to the new version are expository.

In this paper, we prove two results about the unitary dual of graded a-ne Iwahori-Hecke algebras. The flrst one, theorem 2.4, is in the case of the Hecke algebra with equal parameters which arises from split p-adic groups. It says that multiplicities of W-types in irreducible spherical modules are constant over the faces of root hyperplane arrangem...

Let k be a nonarchimedean local field of an arbitrary characteristic and G=G(k) the k-rational points of a connected reductive algebraic group G defined over k. If L is a compact open subgroup of G, let us define C c (G/L) as the space of compactly supported functions on G which are right L-invariant. The traditional formulation of the group versio...

Suppose G is a simple reductive p-adic group with Weyl group W . We give a classification of the irreducible representations of W which can be extended to real hermitian representations of the associated graded Hecke algebra H. Such representations correspond to unitary representations of G which have a small spectrum when restricted to an Iwahori...

In this paper the authors develop a method to compute the local character expansion of a depth zero representation of a p-adic group. The main idea is to use the generalized Gelfand-Graev characters for finite groups as test functions to plug into the character formula. This is possible due to results of Waldspurger on the validity of the local cha...

Let G be a split reductive p-adic group. Then the determination of the unitary representations with nontrivial Iwahori fixed vectors can be reduced to the determination of the unitary dual of the corresponding Iwahori-Hecke algebra. In this paper we study the unitary dual of the Iwahori-Hecke algebras corresponding to the classical groups. We deter...

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