Peter E. Trapa

Unitary shimura correspondences for split real groups

We find a relationship between certain complementary series representations for nonlinear coverings of split simple groups, and spherical complementary series for (different) linear groups. The main technique is Barbaschs method of calculating some intertwining operators purely in terms of the Weyl group

Characters of Springer representations on elliptic conjugacy classes

For a Weyl group W , we investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of W in the homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the W -character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of W and the Dirac operator for graded affine Hecke algebras play key roles.

Annihilators and Associated Varieties of Aq(λ) Modules for U(p, q)

Vogan has conjectured that the cohomologically induced modules Aq(λ) in the weakly fair range exhaust all unitary representations of U(p, q) with certain kinds of real integral infinitesimal character. To prove a statement like this, it is essential to identify these modules among the set of all irreducible Harish-Chandra modules. Barbasch and Vogan have parametrized this latter set in terms of their annihilators and asymptotic supports (or, equivalently, associated varieties). In this paper, we identify the weakly fair Aq(λ) in this parametrization by combining known results about their asymptotic supports together with an explicit computation of their annihilators. In particular, this determines all vanishing and coincidences among the Aq(λ) in the weakly fair range, and gives the Langlands parameters of these modules.

Algebraic and analytic Dirac induction for graded affine Hecke algebras

We define the algebraic Dirac induction map IndD for graded affine Hecke algebras. The map IndD 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 IndD is uniform over the parameter space of the graded affine Hecke algebra. We show that the map IndD 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 analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

Leading-term cycles of Harish-Chandra modules and partial orders on components of the Springer fiber

We use the geometry of characteristic cycles of Harish-Chandra modules for a real semisimple Lie group GR to prove an upper triangularity relationship between two bases of each special representation of a classical Weyl group. One basis consists of Goldie rank polynomials attached to primitive ideals in the enveloping algebra of the complexified Lie algebra g; the other consists of polynomials that measure the Euler characteristic of the restriction of an equivariant line bundle on the flag variety for g to an irreducible component of the Springer fiber. While these two bases are defined only using the structure of the complex Lie algebra g, the relationship between them is closely tied to the real group GR. More precisely, the order leading to the upper triangularity result is a suborder of closure order for the orbits of the complexification of a maximal compact subgroup of GR on the flag variety for g.

The exotic Robinson–Schensted correspondence☆

Abstract We study the action of the symplectic group on pairs of a vector and a flag. Considering the irreducible components of the conormal variety, we obtain an exotic analogue of the Robinson–Schensted correspondence. Conjecturally, the resulting cells are related to exotic character sheaves.

The adjoint representation in rings of functions

Let G be a connected, simple Lie group of rank n defined over the complex numbers. To a parabolic subgroup P in G of semisimple rank r, one can associate n−r positive integers coming from the theory of hyperplane arrangements (see P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167-189; Coxeter arrangements, in Proc. of Symposia in Pure Math., Vol. 40 (1983) Part 2, 269-291). In the case r=0, these numbers are just the usual exponents of the Weyl group W of G. These n−r numbers are called coexponents. Spaltenstein and Lehrer-Shoji have proven the observation of Spaltenstein that the degrees in which the reflection representation ofW occurs in a Springer representation associated to P are exactly (twice) the coexponents (see N. Spaltenstein, On the reflection representation in Springer’s theory, Comment. Math. Helv. 66 (1991), 618-636 and G. I. Lehrer and T. Shoji, On flag varieties, hyperplane complements and Springer representations of Weyl groups, J. Austral. Math. Soc. (Series A) 49 (1990), 449-485). On the other hand, Kostant has shown that the degrees in which the adjoint representation of G occurs in the regular functions on the variety of regular nilpotents in g := Lie(G) are the usual exponents (see B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404). In this paper, we extend Kostant’s result to Richardson orbits (or orbit covers) and we get a statement which is dual to Spaltenstein’s. We will show that the degrees in which the adjoint representation of G occurs in the regular functions on an orbit cover of a Richardson orbit associated to P are also the coexponents.

Duality for nonlinear simply laced groups

We establish a character multiplicity duality for a certain natural class of nonlinear (nonalgebraic) groups arising as two-fold covers of simply laced real reductive algebraic groups. This allows us to extend part of the formalism of the local Langlands conjecture to such groups.

Kazhdan-Lusztig algorithms for nonlinear groups and applications to Kazhdan-Patterson lifting

We establish an algorithm to compute characters of irreducible Harish-Chandra modules for a large class of nonalgebraic Lie groups. (Roughly speaking the class of groups consists of those obtained as nonlinear double covers of linear groups in Harish-Chandras class.) We then apply this theory to study a particular group (the universal cover of the real general linear group), and discover a symmetry of the character computations encoded in a character multiplicity duality. Using this duality theory, we reinterpret a kind of representation-theoretic Shimura correspondence for the general linear group geometrically, and find that it is dual to an analogous lifting for indefinite unitary groups. It seems likely that this example is illustrative of a general framework for studying similar correspondences.

Derived functor modules arising as large irreducible constituents of degenerate principal series

For the groups G = Sp(p, q), SO∗(2n), and U(m,n), we consider degenerate principal series whose infinitesimal character coincides with a finite-dimensional representation of G. We prove that each irreducible constituent of maximal Gelfand-Kirillov dimension is a derived functor module. We also show that at an appropriate “most singular” parameter, each irreducible constituent is weakly unipotent and unitarizable. Conversely we show that any weakly unipotent representation associated to a real form of the corresponding Richardson orbit is unique up to isomorphism and can be embedded into a degenerate principal series at the most singular integral parameter (apart from a handful of very even cases in type D). We also discuss edge-of-wedgetype embeddings of derived functor modules into degenerate principal series. 1