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Lie Group Quantum Mathematics Seminar

Quasi-polynomials, partial symmetry, and metaplectic Whittaker functions

Vidya Venkateswaran, Center for Communications Research at Princeton

Location:  Hill 705
Date & time: Friday, 22 September 2023 at 12:10PM - 1:10PM

In joint work with Siddhartha Sahi and Jasper Stokman, we introduced quasi-polynomial generalizations of (nonsymmetric and symmetric) Macdonald polynomials for arbitrary root systems. In this talk, we discuss some recent results on (anti-) symmetric quasi-polynomials. For GL_r, we show that there is a bijection between (anti-)symmetric quasi-polynomials and partially (anti-)symmetric polynomials. Moreover, under this bijection, the q \rightarrow \infty limit of quasi-polynomial generalizations of (anti-)symmetric Macdonald polynomials map to Macdonald polynomials with prescribed symmetry (as introduced in 2000 by Baker-Dunkl-Forrester), in the same limit. Translating our results to the metaplectic context, we provide a precise statement and proof of a conjecture of Brubaker-Buciumas-Bump-Gustafsson on a ``duality" between metaplectic spherical Whittaker functions and non-metaplectic parahoric Whittaker functions. These results extend to other root systems as well if we restrict to a certain subspace of the quasi-polynomials.

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