Thursday, Sep. 11, 1997, 1pm-2pm, Hill 425 Siddhartha Sahi: Six parameter double affine Hecke algebras The affine Hecke algebra for a reduced and irreducible root system involves one/two parameters, and the corresponding double affine Hecke algebra, defined by Cherednik, involves two/three parameters. However, as observed by Lusztig, the affine Hecke algebra of type C admits three parameters, and we show that the double affine Hecke algebra admits six parameters. This algebra is closely connected with Koornwinder polynomials which generalize all Macdonald polynomials for classical root systems. From the structure theory of the double affine Hecke algebra we deduce a duality result which, combined with work of van Diejen, implies all the conjectures of Macdonald and Koornwinder about these polynomials.
Thursday, Sep. 18, 1997, 1pm-2pm, Hill 425 Arun Ram: Representations of affine Hecke algebras I will describe explicitly a large class of irreducible representations of affine Hecke algebras. These representations are the natural analogues of the skew shape representations of the symmetric group, for affine Hecke algebras corresponding to arbitrary root systems. In particular we are able to index a basis of these representations by a generalization of standard tableaux, for to arbitrary root systems.
Thursday, Sep. 25, 1997, 1pm-2pm, Hill 425 Alexander Dvorsky: Tensor products of singular representations. This talk reports on a joint work with Siddhartha Sahi. We consider tensor products of "small" representations of G, where G is an automorphism group of a symmetric tube domain. We establish the duality correspondence between the spectrum of such tensor product and H'-spherical representations of G', where G'/H' is some symmetric space associated with this tensor product. In particular, if G is the exceptional group E_7 (VII), among the possibilities for these symmetric spaces are the various forms of the Cayley projective plane.
Thursday, Oct. 2, 1997, 1pm-2pm, Hill 425 D. N. Verma: A small step to understanding torus invariants in the ring of the Grassmannians
Thursday, Oct. 9, 1997, 1pm-2pm, Hill 425 Julee Kim: Hecke algebras of classical groups over p-adic field and supercuspidal representations Let G be a classical group defined over p-adic field k. In order to understand the admissible representations of G, we construct a family of Hecke algebras on G. As a corollary, we also get new supercuspidal representations. When G is a symplectic group in 2n variables over k, we describe some of these algebras explicitly in terms of generators and relations. Our constructions are valid when the residual characteristic of k is large.
Thursday, Oct. 16, 1997, 1pm-2pm, Hill 425 V. C. Nanda: Arithmetic of matrices - a survey Matrices over principal ideal domains; right and left units and generalized inverses; Siegel-inverses; greates common divisor and least common multiple; congruences; arithmetical functions; Ramanujan sums; extensions to Dedekind domains - time permitting.
Thursday, Oct. 23, 1997, 1pm-2pm, Hill 425 No Seminar scheduled
Thursday, Oct. 30, 1997, 1pm-2pm, Hill 425 Eric Sommers: Computing the fundamental group of a nilpotent orbit Let G be a simple complex Lie group. In many applications in representation theory, it is necessary to compute the fundamental group of a nilpotent orbit in the Lie algebra of G. Two such applications are the Springer correspondence and the theory of Dixmier algebras. In this talk I will present a new, unified approach to computing the fundamental group. This approach is a generalization of the Bala-Carter theorem, a theorem which classifies the nilpotent orbits themselves. I will also discuss some of the applications.
Thursday, Nov. 13, 1997, 1pm-2pm, Hill 425 Andre Reznikov: Frobenius duality in automorphic forms and Sobolev norms of automorphic forms (joint work with J. Bernstein) I will discuss Frobenius duality for automorphic forms. This is a one to one correspondence between automorphic representations V of a group G with respect to a discrete subgroup L and the space of functionals on (a smooth part of) V invariant under L (i.e. the space of automorphic functionals). This is well-known due to Gelfand, Fomin, Graev and Piateski-Shapiro. I will describe an "analytic" version of this theorem. Namely I will describe Hermitian norms on V (not invariant under G!) with respect to which automorphic functionals are continuous. Some application to the theory of eigenfunctions on Riemann surfaces will be discussed.
Friday, Nov. 21, 1997, 10am-11am, Hill 425 Roman Bezrukavnikov: Asymptotic semigroups and characters of p-adic groups on elliptic elements Let G be a semisimple p-adic group. We will explain how the notion of asymptotic semigroup of a semisimple algebraic group introduced by Vinberg provides a geometric incarnation of the very old basic idea that behaviour of functions on G at infinity is controlled by induction from proper parabolic subgroups. These ideas will be used to define trace of a regular elliptic element g in G on a possibly non-admissible smooth G-module, and obtain a proof of Kazhdan's conjecture asserting that the Euler characterics
of two admissible G-modules coincides
with the L^2-scalar product of their characters as functions on the set of regular
elliptic conjugacy classes in G.
Thursday, Nov. 18, 1997 No meeting (Thanksgiving)