On Representations of Weyl Groups attached to Spherical Varieties

2026/07/27 by Guy Kapon, Guy Shtotland
#math.RT

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Abstract

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.

Citations

• Relative Langlands Duality
• GL(p) x GL(q)-orbit closures on the flag variety and Schubert structure constants for (p,q)-pairs

Related

• Spherical varieties, functoriality, and quantization
• Local Geometric Langlands Correspondence: the Spherical Case
• On the geometry of spherical varieties
• Orbits of spherical representations and Pyasetskii duality
• Fundamental Lemma for Rank One Spherical Varieties of Classical Types