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Lie Groups Quantum Mathematics Seminar

Metaplectic representations of affine Hecke algebras and Weyl groups

Vidya Venkateswaran, Center for Communications Research at Princeton

Location:  Hill 705
Date & time: Friday, 26 October 2018 at 12:00PM - 1:00PM

  • Abstract:   In recent work, Chinta and Gunnells discovered a surprising new action of the Weyl group W associated to an irreducible reduced root system on the space of rational functions; it is a key ingredient in their construction of Weyl group multiple Dirichlet series. Their action depends on a metaplectic parameter n, and at n=1, one recovers the standard Weyl group action. However, their formulas are complicated and showing that they actually define a W-action is non-trivial. Their proof relies on lengthy case-by-case computations with rational functions.In this talk, I'll discuss some recent work with Siddhartha Sahi and Jasper Stokman. We construct a representation of the associated affine Hecke algebra in a natural way (as a quotient of a certain induced module), and provide formulas for the action of the generators in terms of metaplectic divided-difference operators. We then show that the Chinta-Gunnells W-action can be obtained from this via a Baxterization procedure. This gives an independent and uniform proof that it is indeed an action, and allows us to generalize with extra parameters. I'll discuss applications of our work to metaplectic Whittaker functions, as well as some ongoing work on constructing metaplectic analogues of Macdonald polynomials.

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