For tame Kazhdan-Patterson and Savin covers of GL_r, the Gelfand-Graev representation decomposes as explicit simple-type Hecke algebra modules, yielding the Whittaker dimension of discrete series as |X(λ)/S_k|.
Gelfand--Graev functor and quantum affine Schur--Weyl duality
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We explicate relations among the Gelfand--Graev modules for central covers, the Euler--Poincar\'e polynomial of the Arnold--Brieskorn manifold, and the quantum affine Schur--Weyl duality. These three objects and their relations are dictated by a permutation representation of the Weyl group. Specifically, our main result shows that for certain covers of $\mathrm{GL}(r)$ the Gelfand--Graev functor is related to quantum affine Schur--Weyl duality. Consequently, the commuting algebra of the Iwahori-fixed part of the Gelfand--Graev representation is the quotient of a quantum group.
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Gelfand--Graev representation as a Hecke algebra module of simple types of a finite central cover of $\mathrm{GL}(r)$
For tame Kazhdan-Patterson and Savin covers of GL_r, the Gelfand-Graev representation decomposes as explicit simple-type Hecke algebra modules, yielding the Whittaker dimension of discrete series as |X(λ)/S_k|.