arXiv subjects

Guy Shtotland

Publications and source records attributed to Guy Shtotland.

4 recordsLinked to original sources

arXiv2026

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.

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arXiv2026

Distinction of the Steinberg representation and the dual group of a symmetric space

We study the distinction of the Steinberg representation of a split reductive group $G$ with respect to a split symmetric subgroup $H \subset G$. We relate this distinction problem to a problem about the existence of a non-zero harmonic function on a certain hyper-graph related to $X = G/H$. We verify the relative local Langlands conjecture for the Steinberg representation by showing that over a $p$-adic field the Steinberg representation is $H$-distinguished if and only if its Langlands parameter factors through the dual group of $X$.

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arXiv2026

Iwahori Matsumoto presentation for modules of Iwahori fixed functions on symmetric spaces

We study the space $S(X)^I$ of smooth functions on a symmetric space $X=G/H$ invariant to the action of an Iwahori subgroup $I$, as a module over $\mathcal{H}(G,I)$, the Iwahori Hecke algebra of a p-adic group $G$. We present a description of this module that generalizes the description given to $\mathcal{H}(G,I)$ by Iwahori and Matsumoto.

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arXiv2026

Relative Kazhdan Lusztig isomorphism for $GL_{2n}/Sp_{2n}$

The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a $p$-adic group to the equivariant $K$ theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne Langlands conjectures concerning the classification of smooth irreducible representations with an Iwahori fixed vector. In this work we state and prove a relative version of the Kazhdan Lusztig isomorphism for the symmetric pair $(GL_{2n},Sp_{2n})$. The relative isomorphism is an isomorphism between the module of compactly supported Iwahori invariant functions on $X=GL_{2n}/Sp_{2n}$ and another module over the affine Hecke algebra constructed using equivariant $K$ theory and the relative Langlands duality. We use this isomorphism to give a new proof of a condition on $X$ distinguished representations.

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