Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proof of the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}(n)$

Kristaps John Balodis

TL;DR

This work proves the $p$-adic Kazhdan-Lusztig hypothesis for $GL_n(F)$ by fusing Zelinski–Langlands multisegment theory with Chriss–Ginzburg’s geometric framework on affine Hecke algebras. It establishes that standard representations $S_Q(a)$ are the Langlands-standard models, and that multiplicities $[S_Z(a):Z(b)]$ and $[S_Q(a):Q(b)]$ coincide via Aubert duality, allowing a reduction to simple inertial support and a geometric interpretation through Vogan varieties. The paper then aligns Zel’s geometric setup with CG’s, transferring results from the semi-simple simply connected setting to $GL_n(F)$ (via a PGL$_n$ bridge) and proving the proposition for simple support before assembling the general case. As a result, it provides a formal, comprehensive proof of the $p$-adic KL hypothesis for the general linear group over a $p$-adic field, clarifying gaps and offering a robust framework for potential generalizations using Solleveld’s graded Hecke algebras. The work also records several technical lemmas and proofs that experts had taken for granted, contributing to a rigorous, publishable foundation for these connections.

Abstract

In this article, we prove the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}_n(F)$. While the approach via graded affine Hecke algebras due to recent work of Solleveld leads to more general results, this article serves to completes and clarifies the approach via affine Hecke algebras of Chriss and Ginzburg. In particular, this article serves as an opportunity to articulate several results which are undoubtedly known to experts, but have not been formally recorded in the literature.

Proof of the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}(n)$

TL;DR

This work proves the -adic Kazhdan-Lusztig hypothesis for by fusing Zelinski–Langlands multisegment theory with Chriss–Ginzburg’s geometric framework on affine Hecke algebras. It establishes that standard representations are the Langlands-standard models, and that multiplicities and coincide via Aubert duality, allowing a reduction to simple inertial support and a geometric interpretation through Vogan varieties. The paper then aligns Zel’s geometric setup with CG’s, transferring results from the semi-simple simply connected setting to (via a PGL bridge) and proving the proposition for simple support before assembling the general case. As a result, it provides a formal, comprehensive proof of the -adic KL hypothesis for the general linear group over a -adic field, clarifying gaps and offering a robust framework for potential generalizations using Solleveld’s graded Hecke algebras. The work also records several technical lemmas and proofs that experts had taken for granted, contributing to a rigorous, publishable foundation for these connections.

Abstract

In this article, we prove the -adic Kazhdan-Lusztig hypothesis for . While the approach via graded affine Hecke algebras due to recent work of Solleveld leads to more general results, this article serves to completes and clarifies the approach via affine Hecke algebras of Chriss and Ginzburg. In particular, this article serves as an opportunity to articulate several results which are undoubtedly known to experts, but have not been formally recorded in the literature.

Paper Structure

This paper contains 14 sections, 19 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Representation Theory of $\mathrm{GL}_n(F)$
  3. Standard representations
  4. Between the Langlands and Zelevinsky classification theorems
  5. Relating $S_Z(a)$ and $S_Q(a)$
  6. Reduction to multisegments of simple inertial support
  7. Reduction to simple inertial support
  8. Standard representations as modules over affine Hecke algebras
  9. The Geometry of Vogan Varieties
  10. The work of Chriss and Ginzburg
  11. Relations to the representation theory of $\mathrm{PGL}_n(F)$
  12. The Langlands correspondence and Vogan varieties
  13. The case of simple support
  14. The general case

Key Result

Theorem 2.1

For any irreducible admissible representation $(V, \pi)$ of $G(F)$, there exists a standard parabolic $P=MU$ and an irreducible tempered representation $\tau$ of $M(F)$, and $\lambda\in\mathfrak{a}_P^{\ast, +}$ such that $\pi$ is the unique irreducible quotient of $I_P^G(e^\lambda\otimes\tau)$, wher

Theorems & Definitions (51)

  • Theorem 2.1: Kon, Theorem 3.5
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5: Does Not Proceed
  • Example 2.2
  • Definition 6
  • Lemma 2.3
  • proof
  • ...and 41 more