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Quantum wreath products and $p$-adic general linear group

Valentin Buciumas, Chun-Ju Lai

Abstract

We study the pro-$p$ Iwahori-Hecke algebra and its Gelfand-Graev modules for the $p$-adic general linear group and its metaplectic covers. We develop the theory of quantum wreath products of skew polynomial type and use it to provide transparent descriptions of these Hecke algebras and their modules that were previously inaccessible through standard $p$-adic methods. We introduce the notion of (anti)spherical and Kashiwara-Miwa-Stern modules for these quantum wreath products for the first time and interpret the $p$-adic Gelfand-Graev modules in terms of these new modules. As an application, we study the structure theory for the corresponding $p$-adic pro-$p$ Schur algebras and obtain an explicit basis and multiplication rules. Moreover, we give algebraic (re)proofs of several results of $p$-adic interest including the existence of PBW basis for the pro-$p$ metaplectic and Iwahori-Hecke algebras, identification of the Iwahori-Schur algebra with the quantum affine Schur algebra, and failure of the local Shimura correspondence at the pro-$p$ level.

Quantum wreath products and $p$-adic general linear group

Abstract

We study the pro- Iwahori-Hecke algebra and its Gelfand-Graev modules for the -adic general linear group and its metaplectic covers. We develop the theory of quantum wreath products of skew polynomial type and use it to provide transparent descriptions of these Hecke algebras and their modules that were previously inaccessible through standard -adic methods. We introduce the notion of (anti)spherical and Kashiwara-Miwa-Stern modules for these quantum wreath products for the first time and interpret the -adic Gelfand-Graev modules in terms of these new modules. As an application, we study the structure theory for the corresponding -adic pro- Schur algebras and obtain an explicit basis and multiplication rules. Moreover, we give algebraic (re)proofs of several results of -adic interest including the existence of PBW basis for the pro- metaplectic and Iwahori-Hecke algebras, identification of the Iwahori-Schur algebra with the quantum affine Schur algebra, and failure of the local Shimura correspondence at the pro- level.
Paper Structure (44 sections, 33 theorems, 143 equations)

This paper contains 44 sections, 33 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. On $p$-adic groups, Hecke algebras and Whittaker modules
  3. Hecke algebras
  4. Whittaker modules
  5. $p$-adic Schur algebras
  6. Metaplectic groups
  7. On quantum wreath products and wreath modules
  8. Quantum wreath Products
  9. Skew polynomial quantum wreath products
  10. Descent to the Iwahori level
  11. Wreath modules
  12. Wreath modules for skew polynomial quantum wreath products
  13. Pro-$p$ metaplectic Schur algebras
  14. Polynomial Quantum Wreath Products
  15. Polynomial quantum wreath products
  16. ...and 29 more sections

Key Result

Theorem 1

We have an isomorphism of algebras $\widetilde{\cH}^{(1)} \cong \bbA \wr \cH(d).$

Theorems & Definitions (68)

  • Theorem 1: Theorem \ref{['thm:PQWP2']}
  • Corollary 2
  • Theorem 3: Theorem \ref{['thm:VGG2']}
  • Theorem 4
  • Proposition 2.1.1
  • proof
  • Definition 2.1.2
  • Example 2.1.3
  • Example 2.1.4
  • Remark 2.1.5
  • ...and 58 more