Hecke algebra representations related to spherical varieties

Mars, J.; Springer, T.

doi:10.1090/S1088-4165-98-00027-2

Hecke algebra representations related to spherical varieties
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by J. G. M. Mars and T. A. Springer

Represent. Theory 2 (1998), 33-69

DOI: https://doi.org/10.1090/S1088-4165-98-00027-2

Published electronically: February 11, 1998

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Abstract:

Let $G$ be a connected reductive group over the algebraic closure of a finite field and let $Y$ be a spherical variety for $G$. We consider perverse sheaves on $G$ and on $Y$ which have a weight for the action of a Borel subgroup $B$ and are endowed with an action of Frobenius. This leads to the definition of a “generalized Hecke algebra”, attached to $G$, and of a module over that algebra, attached to $Y$. The same algebra and the same module can also be defined using constructible sheaves. Comparison of the two definitions gives, in the case of a symmetric variety $Y$ and $B$-equivariant sheaves, a geometric proof of results which Lusztig and Vogan obtained by representation theoretic means.

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