The conference is preceded by a colloquium talk by Siddhartha Sahi on Friday November 10 at 4:00 p.m. which will take place in KED B005 (located at 585 King Edward Avenue). The conference will take place in Room 1006 of the FSS building.

Conference organizers: Siddhartha Sahi, Hadi Salmasian.

Acknowledgement. The workshop is sponsored by funding from the National Science Foundation and the University of Ottawa.

Schedule of Talks

Date Speaker Title and Abstract (hover your mouse pointer on titles)
Nov 10, 4:00 p.m. Siddhartha Sahi (Rutgers) Multivariate hypergeometric functions Abstract: In the 1980s I.G. Macdonald introduced a family of polynomials P_\lambda(x;q,t) indexed by partitions of length n, which form a homogeneous linear basis for the algebra of symmetric polynomials in x = (x 1,...,x n) with coefficients in the field Q(q,t). Setting q=t^\alpha and taking the limit for t converging to 1 yields polynomials with coefficients in Q(\alpha), which had been discovered earlier by H. Jack. Macdonald conjectured several properties of these polynomials, whose resolution has led to the discovery of beautiful new mathematics. Around the same time Macdonald introduced another family of functions p_F_q(x;\alpha) and p_F_q(x,y;\alpha) that he called “multivariate hypergeometric functions”. These are power series in one and two sets of variables x,y, which depend on \alpha and p + q additional parameters a = (a 1,...,a p) ; b = (b 1,...,b q), and are expressible as linear combination of normalized Jack polynomials with coefficients that are given by an explicit combinatorial formula. For \alpha=1 and \alpha=2 these functions are closely related to “hypergeometric functions of matrix argument”, which were introduced by Bochner and Herz in the 1950s for Hermitian and symmetric matrices respectively, and which have found many applications in number theory and statistics. For general \alpha the function E(x,y;\alpha ) = 0_F_0(x,y;\alpha) is a generalization of the exponential kernel e^{xy}. Thus it can be used to define a multivariate analog of the Fourier and Laplace transforms. Extrapolating from the two special values of \alpha, Macdonald made a series of conjectures about p_F_q(x,y;\alpha ) for general \alpha, which are crucial for the development of the associated harmonic analysis. We will describe recent joint work with G. Olafsson, which has succeeded in resolving many of these conjectures and made it possible to finally develop the harmonic analysis envisioned by Macdonald. The lecture will be self-contained and will not assume any special prior knowledge on part of the audience!
Nov 11, 9:00 a.m. Weiqiang Wang (Virginia) Canonical bases arising from quantum symmetric pairs Abstract: A quantum symmetric pairs (QSP) consists of (U, Ui), where U is a quantum group and Ui is a coideal subalgebra corresponding to a Lie subalgebra fixed by an involution. (The classification of QSP's of finite type corresponds to the classification of real simple Lie algebras.) We shall present an i-canonical basis theory on the modified coideal subalgebras of finite type and the tensor product U-modules. In a special case when U is of type A, the i-canonical bases admit positivity properties as well as application to super Kazhdan-Lusztig theory. We will give examples of i-divided powers, which exhibit rich q-combinatorics. This is joint work with Huanchen Bao (Maryland).
Nov 11, 1:30 p.m. Vera Seganova (Berkeley) Categories of sl(\infinity) modules and the category O for gl(m|n). Abstract: The talk concerns the idea of J. Brundan that the translation functors in the category O for gl(m|n) categorify certain module over sl(\infinity) , this idea was used later for calculating Kazhdan--Lusztig polynomials in the supercase. We will show that gl(\infinity)-modules appearing in Brundan's construction are injective in certain categories and discuss the relation of the socle filtration for these modules with representation theory of gl(m|n). Joint with C. Hoyt and I. Penkov.
Nov 12, 9:00 a.m. Nicolas Guay (Alberta) Twisted Yangians of types B-C-D and their irreducible finite dimensional representations. Abstract: I will introduce twisted Yangians associated to symmetric pairs of types B, C and D which are similar to the twisted Yangians of type A constructed by G. Olshanski around 1990 and which have since then been quite well studied. Twisted Yangians can be viewed as quantizations of certain twisted current algebras, the twist coming from an involution of the underlying finite dimensional simple Lie algebra. These twisted current algebras fit into the framework of equivariant map algebras developed by E. Neher and A. Savage, and also into the more general framework of twisted current algebras developed by M. Lau and A. Pianzola. Structural properties of twisted Yangians of types B, C and D are very similar to those of type A and an overview of those properties will be given. A lot of progress has been made towards the classification of their irreducible finite dimensional representations and a summary of the results obtained in this direction will be presented, along with some of the main ideas needed to achieve those results. This is joint work with Vidas Regelskis and Curtis Wendlandt.
Nov 12, 1:30 p.m. Alexander Alldridge (Berkeley) Gamma functions of symmetric superspaces Abstract: The Gindikin gamma functions are generalisations for Euler's gamma function which are available and understood for every symmetric cone (i.e. self-dual convex cones which are homogeneous under their group of linear automorphisms). Symmetric cones are given known to correspond to simple Jordan algebras J. The correspondence identifies generic orbits of the automorphism group of a Euclidean real form of J as symmetric cones. We introduce gamma functions for the Riemannian symmetric superspaces which appear as generic orbits of the automorphism group of a simple Jordan superalgebra and discuss some applications to representation theory.