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Harish-Chandra D-modules for bi-Whittaker reduction

Wen-Wei Li

Abstract

Let $D(G)$ be the algebra of algebraic differential operators on a complex reductive group $G$. Denote by $\mathbb{W}$ the bi-Whittaker quantum Hamiltonian reduction of $D(G)$, also known as the quantum Toda lattice. In this article we define the admissible $\mathbb{W}$-modules and the special case of Harish-Chandra modules, the latter being bi-Whittaker variants of the invariant holonomic systems of Hotta-Kashiwara. We then study their torsion properties up to completion relative to the Kazhdan filtration, which in general is unbounded in both directions, through the geometry of universal centralizer. In the case of regular infinitesimal characters, the corresponding $D(G)$-module is shown to be the minimal extension of an irregular connection on the open Bruhat cell. Certain ring-theoretic properties of the completion of $\mathbb{W}$ are also obtained.

Harish-Chandra D-modules for bi-Whittaker reduction

Abstract

Let be the algebra of algebraic differential operators on a complex reductive group . Denote by the bi-Whittaker quantum Hamiltonian reduction of , also known as the quantum Toda lattice. In this article we define the admissible -modules and the special case of Harish-Chandra modules, the latter being bi-Whittaker variants of the invariant holonomic systems of Hotta-Kashiwara. We then study their torsion properties up to completion relative to the Kazhdan filtration, which in general is unbounded in both directions, through the geometry of universal centralizer. In the case of regular infinitesimal characters, the corresponding -module is shown to be the minimal extension of an irregular connection on the open Bruhat cell. Certain ring-theoretic properties of the completion of are also obtained.
Paper Structure (40 sections, 68 theorems, 108 equations)

This paper contains 40 sections, 68 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results and idea of proofs
  4. Outline of this article
  5. Conventions
  6. Acknowledgements
  7. Filtered rings and modules
  8. Filtrations and completions
  9. Good filtrations
  10. Singular supports
  11. Kazhdan filtrations
  12. Ore sets
  13. Bi-Whittaker reduction
  14. Quantum Toda lattices
  15. Isogenies
  16. ...and 25 more sections

Key Result

Theorem 1.1

If $M$ is an admissible $\mathbb{W}$-module and $R$ is a subquotient of $M$ of $\mathcal{S}$-torsion, then $\widehat{R} = 0$, or equivalently $\operatorname{gr}(R) = 0$.

Theorems & Definitions (154)

  • Theorem 1.1: = Corollary \ref{['prop:torsion-cor']}
  • Proposition 1.2: = Propositions \ref{['prop:Auslander-regular']} + \ref{['prop:gldim']}
  • Proposition 1.3: = Proposition \ref{['prop:admissible-hh']} + Theorem \ref{['prop:Iwanaga']}
  • Theorem 1.4: = Theorem \ref{['prop:simplicity']}
  • Theorem 1.5: = Theorems \ref{['prop:decompletion-reg']} + \ref{['prop:minimal-extension']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 144 more