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

TL;DR

This paper establishes a relative Kazhdan Lusztig isomorphism for the symmetric pair (GL_{2n}, Sp_{2n}), linking Iwahori-invariant functions on the quotient to modules over the affine Hecke algebra, with applications to distinguished representations.

Contribution

It introduces and proves a relative version of the Kazhdan Lusztig isomorphism for (GL_{2n}, Sp_{2n}), extending the classical theory to symmetric pairs.

Findings

Proves the relative Kazhdan Lusztig isomorphism for (GL_{2n}, Sp_{2n})

Provides a new proof for conditions on distinguished representations

Connects equivariant K-theory with representation theory of symmetric pairs

Abstract

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.