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The coarse quotient for affine Weyl groups and pseudo-reflection groups

Tom Gannon

TL;DR

The paper introduces the coarse quotient $\mathfrak{t}^{*} \sslash W^{\text{aff}}$ for the affine Weyl group acting on the dual Cartan and develops a pointwise descent criterion that reduces global descent questions to checking fibers at field-valued points. It extends this framework to arbitrary finite group actions and, crucially, proves that for finite pseudo-reflection groups, descent to the GIT quotient is equivalent to descent for every pseudo-reflection, generalizing prior results. A robust base of results is assembled via GIT/stack-quotient machinery, the coinvariant algebra, and IndCoh/QCoh formalisms, enabling a t-structured descent theory and a groupoid-based realization of the coarse quotient for the extended affine Weyl group $\tilde{W}^{\text{aff}}$. The work connects representation-theoretic contexts (e.g., Soergel bimodules, category $\mathcal{O}$, bi-Whittaker $\mathcal{D}$-modules) with categorical representation theory, providing tools to analyze blocks, centers, and Mellin-type equivalences through the coarse quotient. Overall, the paper gives precise, pointwise criteria for descent, builds the IndCoh framework on the coarse quotient, and lays groundwork for applications in geometric representation theory and categorical structures linked to affine and extended affine Weyl groups.

Abstract

We study the coarse quotient $\mathfrak{t}^*//W^{\text{aff}}$ of the affine Weyl group $W^{\text{aff}}$ acting on a dual Cartan $\mathfrak{t}^*$ for some semisimple Lie algebra. Specifically, we classify sheaves on this space via a "pointwise" criterion for descent, which says that a $W^{\text{aff}}$-equivariant sheaf on $\mathfrak{t}^*$ descends to the coarse quotient if and only if the fiber at each field-valued point descends to the associated GIT quotient. We also prove the analogous pointwise criterion for descent for an arbitrary finite group acting on a vector space. Using this, we show that an equivariant sheaf for the action of a finite pseudo-reflection group descends to the GIT quotient if and only if it descends to the associated GIT quotient for every pseudo-reflection, generalizing a recent result of Lonergan.

The coarse quotient for affine Weyl groups and pseudo-reflection groups

TL;DR

The paper introduces the coarse quotient for the affine Weyl group acting on the dual Cartan and develops a pointwise descent criterion that reduces global descent questions to checking fibers at field-valued points. It extends this framework to arbitrary finite group actions and, crucially, proves that for finite pseudo-reflection groups, descent to the GIT quotient is equivalent to descent for every pseudo-reflection, generalizing prior results. A robust base of results is assembled via GIT/stack-quotient machinery, the coinvariant algebra, and IndCoh/QCoh formalisms, enabling a t-structured descent theory and a groupoid-based realization of the coarse quotient for the extended affine Weyl group . The work connects representation-theoretic contexts (e.g., Soergel bimodules, category , bi-Whittaker -modules) with categorical representation theory, providing tools to analyze blocks, centers, and Mellin-type equivalences through the coarse quotient. Overall, the paper gives precise, pointwise criteria for descent, builds the IndCoh framework on the coarse quotient, and lays groundwork for applications in geometric representation theory and categorical structures linked to affine and extended affine Weyl groups.

Abstract

We study the coarse quotient of the affine Weyl group acting on a dual Cartan for some semisimple Lie algebra. Specifically, we classify sheaves on this space via a "pointwise" criterion for descent, which says that a -equivariant sheaf on descends to the coarse quotient if and only if the fiber at each field-valued point descends to the associated GIT quotient. We also prove the analogous pointwise criterion for descent for an arbitrary finite group acting on a vector space. Using this, we show that an equivariant sheaf for the action of a finite pseudo-reflection group descends to the GIT quotient if and only if it descends to the associated GIT quotient for every pseudo-reflection, generalizing a recent result of Lonergan.
Paper Structure (44 sections, 56 theorems, 86 equations)

This paper contains 44 sections, 56 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. The Coarse Quotient $\mathfrak{t}^{\ast}\sslash W^{\text{aff}}$
  3. Introduction to the Coarse Quotient $\mathfrak{t}^{\ast}\sslash W^{\text{aff}}$
  4. Other Potential Definitions of $\mathfrak{t}^{\ast}\sslash W^{\text{aff}}$
  5. The Coarse Quotient $\mathfrak{t}^{\ast}\sslash W^{\text{aff}}$ Via Groupoids
  6. Field-Valued Points
  7. Descent to the Coarse Quotient for Finite Groups
  8. Generalities on Descent to the Coarse Quotient
  9. Descent to the Coarse Quotient for Pseudo-Reflection Groups
  10. Local Descent to the Coarse Quotient
  11. Motivation
  12. Representation Theoretic Motivation
  13. Connections to Categorical Representation Theory
  14. Conventions
  15. Acknowledgements
  16. ...and 29 more sections

Key Result

Theorem 1.1

(CoherentFullyFaithful, Various Conditions for Wext Equivariant Sheaf to Satisfy Coxteter Descent) The pullback map induced by $\mathfrak{t}^{\ast}/W^{\text{aff}} \to \mathfrak{t}^{\ast}\sslash W^{\text{aff}}$ is fully faithful. Moreover, a $W^{\text{aff}}$-equivariant sheaf $\mathcal{F}$ on $\mathf

Theorems & Definitions (118)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Remark 1.9
  • Proposition 1.10
  • ...and 108 more