On Representations of Weyl Groups attached to Spherical Varieties

To any smooth spherical $G$ variety $X$ we attach two representations of the Weyl group of $G$. The first is constructed geometrically using Borel Moore homology. The second is constructed combinatorially using the structure of the Borel orbits on $X$. Using the Fourier Sato transform, we show that both representations are isomorphic. We use this result to translate (in the cotangent case) a conjecture of Finkelberg-Ginzburg-Travkin in the relative Langlands program to a combinatorial problem. We solve this combinatorial problem for several examples.