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Geometric realizations of affine Hecke algebras with unequal parameters

Jonas Antor

Abstract

We give a $K$-theoretic realization of all affine Hecke algebras with two unequal parameters including exceptional types. This extends the celebrated work of Kazhdan and Lusztig, who gave a $K$-theoretic realization of affine Hecke algebras with equal parameters, and complements results of Kato, who extended this construction to the three-parameter affine Hecke algebra of type $C$. A key idea behind our new construction is to exploit the reducibility of the adjoint representation in small characteristic. We also show that under suitable geometric conditions, our construction leads to a Deligne-Langlands style classification of simple modules. We verify these geometric conditions for $G_2$ thereby obtaining a full geometric classification of the simple modules for the affine Hecke algebra of $G_2$ with two parameters away from roots of unities.

Geometric realizations of affine Hecke algebras with unequal parameters

Abstract

We give a -theoretic realization of all affine Hecke algebras with two unequal parameters including exceptional types. This extends the celebrated work of Kazhdan and Lusztig, who gave a -theoretic realization of affine Hecke algebras with equal parameters, and complements results of Kato, who extended this construction to the three-parameter affine Hecke algebra of type . A key idea behind our new construction is to exploit the reducibility of the adjoint representation in small characteristic. We also show that under suitable geometric conditions, our construction leads to a Deligne-Langlands style classification of simple modules. We verify these geometric conditions for thereby obtaining a full geometric classification of the simple modules for the affine Hecke algebra of with two parameters away from roots of unities.
Paper Structure (16 sections, 64 theorems, 223 equations, 1 table)

This paper contains 16 sections, 64 theorems, 223 equations, 1 table.

Table of Contents

  1. Introduction
  2. A geometric realization
  3. $K$-theory of vector bundles and Steinberg varieties
  4. Affine Hecke algebras with unequal parameters
  5. Rooted representations
  6. Simple modules and perverse sheaves
  7. Centralizers and fixed points
  8. The affine Hecke algebra at a central character
  9. A Deligne-Langlands correspondence for unequal parameters
  10. The geometric conditions
  11. Positive real central characters
  12. Reduction to positive real central character
  13. The affine Hecke algebra of $G_2$
  14. Convolution, localization and the Chern character
  15. Equivariant $K$-theory
  16. ...and 1 more sections

Key Result

Theorem 1

hogeweij1982almosthiss1984adjungierten If $p$ is special for $G$, then there is a $G$-stable subspace $\mathfrak{g}_s \subset \mathfrak{g}$ whose non-zero weights are the short roots of $G$.

Theorems & Definitions (129)

  • Theorem
  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • ...and 119 more