Representation Theory of Real Reductive Groups
A conference, July 27 - July 31, 2009, at the University of Utah


Titles and Abstracts

Lecture notes and slides




Workshop Home

Conference Home

Suggestions for Wednesday Afternoon

Titles and Abstracts:

Jeffrey Adams,   Characters of nonlinear groups

Dan Barbasch,   Unitary representations of quasisplit real groups

Michel Brion,   Differential operators and differential forms on compactifications of reductive groups
The rational cohomology ring H*(G), where G is a connected Lie group, is a free exterior algebra on generators of odd degrees, by a theorem of Hopf. If G is a complex algebraic group, then the graded algebra H*(G) is equipped with a decreasing filtration (the Hodge filtration), and the associated graded is still a (bi-graded) free exterior algebra, as shown by Deligne.

The talk will present an explicit description of differential operators and differential forms on a connected complex reductive group G, which "extend" to a suitable compactification of that group. This will yield another proof of Deligne's result. Generalizations to homogeneous spaces will also be discussed.

Dan Ciubotaru,   Functors for unitary representations of real groups and affine Hecke algebras

We define exact functors from certain categories of Harish-Chandra modules of real classical groups to finite dimensional modules over an associated affine graded Hecke algebra. We show that these functors map irreducible spherical representations to irreducible spherical representations and, moreover that they preserve unitarity. This is joint work with Peter Trapa.

Patrick Delorme,   A twisted Paley-Wiener theorem for real reductive groups

(Joint work with Paul Mezo.) Let G+ (resp. G) be the group of real points of a possibly disconnected real reductive group (resp. of its identity component). We assume that G+ is generated by one of its connected components, whose set of real points G' is assumed to be non-empty. Let K be a maximal compact subgroup of the group of real points of the identity component of this algebraic group. We characterize the space of maps &pi &rarr tr(&pi(f)), where &pi is an irreducible representation of G+ and f varies over the space of smooth compactly supported functions on G' which are left and right K-finite. This work is motivated by applications to the twisted Arthur-Selberg twisted trace formula.

Michel Duflo,   Kostant-Blattner type formulas for restrictions of discrete series.

This is a joint work with Jorge Vargas. We consider cases of discrete series for a real reductive group whose restriction to a reductive subgroup is admissible, and for which the multiplicities can be computed by a formula of Kostant-Blattner type.

Roger Howe,   The (super) dual pair (O(p,q), osp(2,2)) and Maxwell's Equations

Over the years, the local theta correspondence for reductive dual pairs of subgroups of the symplectic group has repeatedly provided insight into representation theory of reductive groups and the theory of automorphic forms. The most heavily studied example of such a pair is (O(p, q), SL(2,R)), which provides the setting for the construction of modular forms, including the classical Jacobi theta functions, the η function, Hecke's construction of modular forms associated to quadratic grossencharaktere, and Maass's wave forms, as well as an analysis of Huyghens' Principle and the Bochner periodicity relations. This talk will discuss a variant of this pair, involving the indefinite orthogonal groups O(p, q) and the Lie superalgebra osp(2, 2). Among the motivations for this study is the close connection of this pair with Maxwell's equations. Many aspects of the correspondence for (O(p, q), SL(2,R)) find analogs for (O(p, q), osp(2, 2)), but in some ways this correspondence is more natural. The Casimir operator for osp(2, 2) plays a key role. This is a report on the Yale Ph.D. thesis of Dan Lu.

Toshiyuki Kobayashi,   Geometric analysis on minimal representations

Minimal representations are the smallest infinite dimensional unitary representations. The Weil representation for the metaplectic group, which plays a prominent role in number theory, is a classic example.

We may consider that minimal representations (from the viewpoint of groups) as "maximal symmetries (from the viewpoint of representation spaces)", and thus propose to use minimal representations as guiding principles to find new interactions with other fields of mathematics.

Highlighting geometric analysis on minimal representations of O(p,q), I plan to discuss conservative quantities of ultrahyperbolic equations, the generalization of the Fourier-Hankel transform on the L^2-model, and its deformation.

Shrawan Kumar,   A generalization of Cachazo-Douglas-Seiberg-Witten (CDSW) conjecture for symmetric spaces

This is a continuation of my previous work on the original CDSW conjecture. Let g be a simple Lie algebra with an involution &sigma and let k (resp. p) be the +1 (resp. -1) eigenspace of &sigma . We assume that p is an irreducible k-module. In this setting, we give a generalization of the original CDSW conjecture. The original conjecture and a brief outline of its proof will be recalled in the talk. Even though the techniques used to prove the generalization are similar to the previous work, there are added topological complications to take care of.

George Lusztig, From groups to symmetric spaces

A symmetric space is a reductive group with an involution. Any reductive group G gives rise to a symmetric space (G2,i) where i(a,b)=(b,a). Thus symmetric spaces can be viewed as generalization of reductive groups. It has happened many times that a property/construction for reductive group is a special case of a property/construction which makes sense for any symmetric space. In this lecture we will review a number of such properties/constructions.

Hisayosi Matumoto,  On existence of homomorphisms between scalar generalized Verma modules

We discuss some necessary conditions and sufficients conditions on the existence problem of the homomorphisms between scalar generalized Verma modules. In particular, for complexified minimal parabolic subalgebras of real forms, we determine such homomorphisms between those with regular infinitesimal characters except for so(m,n) with m+n odd and sp(m,m).

William McGovern,   Rational smoothness and pattern avoidance for fixed-point-free involutions

We characterize rationally smooth K-orbit closures in the flag variety for SU*(2n) via a pattern avoidance criterion on the involutions parametrizing them, showing along the way that the singular locus of any such orbit closure coincides with the rational singular locus.

Toshio Oshima,   Fractional calculus of Weyl algebra and its applications

Restrictions of zonal spherical functions and Heckman-Opdam's hypergeometric functions on one-dimensional singular lines though the origin satisfy interesting ordinary differential equations. By a unifying study of ordinary differential equations on a Riemann sphere we have a global structure of their solutions and for example we have a new proof of Gindikin-Karpelevic formula of c-functions and the Gauss summation formula of Heckman-Opdam's hypergeometric functions.

Alessandra Pantano, Complementary Series of Split Real Groups

he aim of this talk is to present some recent progress on the determination of the unitarizable minimal principal series of split real groups.

In particular, we explore the relation between genuine complementary series of the metaplectic group Mp(2n), non-spherical complementary series of the split orthogonal group SO(n+1,n) and the spherical complementary series of certain other split orthogonal groups.

This is joint work with Annegret Paul and Susana Salamanca-Riba.

Gordan Savin,   Parameterizing representations of non-linear groups.

While it is possible to write parameters of representations of linear groups in a (more or less) canonical fashion, parametrerizations for non-linear groups depend on certain choices. For example, one can parameterize representations of the two-fold cover of SL(2) using the dual pair (SL(2), O(3)) but the transfer of representations from SL(2) to O(3) depends on the choice of oscillator representation or, equivalently, on the choice of additive character. In this talk I will describe an approach to this problem for two-fold covers of simply connected Chevalley groups, and an application to dual pair correspondences arising from small representations of groups of type Bn.

Wilfried Schmid,   b-functions and canonical filtrations of Harish-Chandra modules

A minor extension of M. Saito's theory of mixed Hodge modules can be used to construct canonical filtrations on Harish-Chandra modules. They have remarkable formal properties, but are very difficult to compute explicitly. A key step is to determine a type of b-function. This is joint work with Kari Vilonen.

Michèle Vergne,   Hyperplane arrangements and Blattner formula

With Baldoni, Beck, and Cochet, we showed that an efficient way of computing vector partition functions for classical root systems is by computing residues of rational functions defined on the complement of the corresponding arrangement of hyperplanes. I will discuss applications of this method to Blattner formula for discrete series of the group U(p,q).

David Vogan,   Signatures of Hermitian forms and unitary representations

Suppose G is a real reductive Lie group. According to Gelfand's philosophy of abstract harmonic analysis, the most basic question in the representation theory of G is the classification of irreducible unitary representations. One way to approach this question is first to classify the irreducible hermitian representations: those admitting a nondegenerate but possibly indefinite invariant hermitian form. (As evidence that this is a promising path, notice that in the two classical limiting examples -- when G is compact, and when G is abelian -- a hermitian representation is automatically unitary.) Knapp and Zuckerman gave a complete classification of irreducible hermitian representations in a 1977 paper. What remains is to decide which of these representations are unitary.

I will describe joint work with Jeffrey Adams, Marc van Leeuwen, Peter Trapa, and Wai Ling Yee on formulating an algorithm to calculate the signature of the invariant Hermitian form on any irreducible Hermitian representation (and so, in particular, to determine whether the representation is unitary). The formulation of the algorithm is more or less complete; our proof that the algorithm is correct still has some missing steps. The main ingredient is the Beilinson-Bernstein proof of Jantzen's conjecture for Harish-Chandra modules; the main thing to be computed is Kazhdan-Lusztig polynomials for real groups.

Nolan Wallach,   Variations on a theme of Rino Sanchez

In his (unpublished) thesis (2003), Rino Sanchez gave a purely algebraic construction (and proof of unitarity) of an important representation of the two-fold cover of SL(3,R) originally constructed analytically by Torasso in a 1983 paper. We will give a description of Rino's method and give some extensions.

Chen-Bo Zhu,   Archimedean multiplicity one theorems

Let (G, G') be one of the following pairs of classical groups: (GL(n,R), GL(n-1,R)), (GL(n,C), GL(n-1,C)), (O(p,q), O(p,q-1) ), (O(n,C), O(n-1,C)), (U(p,q), U(p,q-1)). We consider the class of irreducible admissible smooth Frechet representations of moderate growth, for G and G' respectively. The multiplicity one theorems in the title assert that any representation of G' in this class occurs (as a quotient) with multiplicity at most one in any representation of G in the same class. The talk represents joint work with Binyong Sun. For general linear groups, this is also in dependently due to A. Aizenbud and D. Gourevitch.