Multivariate hypergeometric functions
Friday, 10 November 2017 - 4:00 pm to 4:30 pm
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Speaker: Siddhartha Sahi (Rutgers University)

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!