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

Guy Shtotland

TL;DR

The paper develops a relative Kazhdan–Lusztig isomorphism for the symmetric pair $(GL_{2n},Sp_{2n})$, linking the affine Hecke algebra action on $S(X)^I$ with a relative dual K-theoretic/Langlands-dual framework via the variety $ ilde ext{L}$ and the relative Steinberg geometry. It establishes a suite of isomorphisms among $H_q$-modules, $K$-theory modules, and Borel–Moore homology, including $H^{BM}_{top}( ilde ext{L}) ensor sgn o b C[Backslash X]$ and $IM(K^{G^ ext{∨} imes b C^ imes}( ilde ext{L})) o M_q$, thereby enabling a geometric criterion for $X$-distinguished representations generated by $I$-fixed vectors. The work yields a concrete description of $I$-orbits on $X$, a cellular-fibration mechanism to reduce to Levi factors, and a fixed-point analysis that imposes necessary conditions on Deligne–Langlands parameters; in particular it recovers a parity condition on Zelevinsky parameters (even-length segments) for distinguished representations, via an alternative, geometry-driven route. It also discusses conjectural extensions to the Lusztig–Vogan framework in the relative/ hyperspherical setting and provides a sheaf-theoretic analysis of the distinguished representations, with explicit quiver-representation examples illustrating the approach.

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.

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

TL;DR

The paper develops a relative Kazhdan–Lusztig isomorphism for the symmetric pair , linking the affine Hecke algebra action on with a relative dual K-theoretic/Langlands-dual framework via the variety and the relative Steinberg geometry. It establishes a suite of isomorphisms among -modules, -theory modules, and Borel–Moore homology, including and , thereby enabling a geometric criterion for -distinguished representations generated by -fixed vectors. The work yields a concrete description of -orbits on , a cellular-fibration mechanism to reduce to Levi factors, and a fixed-point analysis that imposes necessary conditions on Deligne–Langlands parameters; in particular it recovers a parity condition on Zelevinsky parameters (even-length segments) for distinguished representations, via an alternative, geometry-driven route. It also discusses conjectural extensions to the Lusztig–Vogan framework in the relative/ hyperspherical setting and provides a sheaf-theoretic analysis of the distinguished representations, with explicit quiver-representation examples illustrating the approach.

Abstract

The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a -adic group to the equivariant 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 . The relative isomorphism is an isomorphism between the module of compactly supported Iwahori invariant functions on and another module over the affine Hecke algebra constructed using equivariant theory and the relative Langlands duality. We use this isomorphism to give a new proof of a condition on distinguished representations.
Paper Structure (20 sections, 54 theorems, 36 equations)

This paper contains 20 sections, 54 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. General conjectures
  3. Methods of proof
  4. Structure of the paper
  5. Acknowledgment
  6. Preliminaries
  7. The convolution construction
  8. Cellular Fibration Lemma
  9. Borel orbits and irreducible components
  10. Borel Moore homology and the $W$ action
  11. A version of Conjecture 1.2 of finkelberg2023lagrangiansubvarietieshypersphericalvarieties
  12. Iwahori orbits and the module $S(X)^I$
  13. Cellular fibration argument
  14. The module $K^{G^\vee\times \mathbb{C}^\times}(\Lambda)$
  15. Fixed points and central characters
  16. ...and 5 more sections

Key Result

Theorem 1.1

Kazhdan1987ProofOT,reeder The irreducible smooth representations of $G$ with an $I$ fixed vector are parametrized by conjugacy classes of triples $(t,n,\chi)$ with $t\in G^\vee$ semi simple, $n\in \mathfrak{g}^{\vee}$ such that $Ad_t(n)=q_rn$. The element $\chi$ is an irreducible representation of t

Theorems & Definitions (127)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 117 more