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Cohomological Mackey formula for quotient stacks

Lucien Hennecart

TL;DR

This work extends cohomological Donaldson--Thomas theory to the setting of equivariant Landau--Ginzburg models by formulating a Mackey-type formula for the critical cohomology $\mathrm{H}^*_{\mathrm{crit}}(V/G,f)$. Central to the construction are the induction and restriction morphisms organized by a Coxeter complex built from weights of $V$ and $\mathfrak{g}$, together with localized (Euler-class) twists that enable precise functorial compatibility. The main result proves a cohomological Mackey formula that governs the interaction between induction and restriction across cells and flats, with torus-equivariant and $2d$-dimensional reductions providing alternative viewpoints and consistency checks. These structures yield a localized induction-restriction system on critical cohomology, and the framework connects to broader themes in CoHA, Langlands-type functors, and hyperplane-arrangement categorifications, suggesting extensions to global stacks and BPS-type invariants. The paper also provides explicit computations in low-rank examples, highlighting concrete shuffle-like descriptions when the potential vanishes, and clarifying equivariance under the Weyl group and the role of Euler-class localization in the Mackey formalism.

Abstract

In this paper, we construct a restriction morphism on the critical cohomology of an equivariant Landau-Ginzburg model associated to a representation of a reductive group equipped with an invariant function. We show a compatibility formula between the restriction and induction maps as a Mackey-type formula, thereby giving the critical cohomology the structure of a localized induction-restriction system.

Cohomological Mackey formula for quotient stacks

TL;DR

This work extends cohomological Donaldson--Thomas theory to the setting of equivariant Landau--Ginzburg models by formulating a Mackey-type formula for the critical cohomology . Central to the construction are the induction and restriction morphisms organized by a Coxeter complex built from weights of and , together with localized (Euler-class) twists that enable precise functorial compatibility. The main result proves a cohomological Mackey formula that governs the interaction between induction and restriction across cells and flats, with torus-equivariant and -dimensional reductions providing alternative viewpoints and consistency checks. These structures yield a localized induction-restriction system on critical cohomology, and the framework connects to broader themes in CoHA, Langlands-type functors, and hyperplane-arrangement categorifications, suggesting extensions to global stacks and BPS-type invariants. The paper also provides explicit computations in low-rank examples, highlighting concrete shuffle-like descriptions when the potential vanishes, and clarifying equivariance under the Weyl group and the role of Euler-class localization in the Mackey formalism.

Abstract

In this paper, we construct a restriction morphism on the critical cohomology of an equivariant Landau-Ginzburg model associated to a representation of a reductive group equipped with an invariant function. We show a compatibility formula between the restriction and induction maps as a Mackey-type formula, thereby giving the critical cohomology the structure of a localized induction-restriction system.
Paper Structure (94 sections, 54 theorems, 160 equations)

This paper contains 94 sections, 54 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Donaldson--Thomas theory of equivariant Landau--Ginzburg models
  3. Induction and restriction morphisms and functors
  4. Mackey formula for representations of finite groups
  5. Restriction morphism and cohomological integrality
  6. Localized coproduct on cohomological Hall algebras
  7. Eisenstein series and constant term functors
  8. Springer theory
  9. Perverse sheaves on hyperplane arrangements
  10. Main results
  11. Coxeter complex
  12. The critical cohomological system
  13. Critical induction morphisms
  14. $2d$ induction morphisms
  15. Restriction morphism
  16. ...and 79 more sections

Key Result

Proposition 1.1

For any cells $C,C'\in\mathfrak{C}$ and flat $F\in\mathfrak{F}$ such that $C\preceq C'\preceq F$, we have

Theorems & Definitions (107)

  • Proposition 1.1: =Proposition \ref{['proposition:associativity3d']}
  • Theorem 1.2: =§\ref{['subsection:parabolicinduction2d']}+Proposition \ref{['proposition:associativity2d']}
  • Theorem 1.3: Cohomological dimensional reduction and induction = Proposition \ref{['proposition:comparison2d3dmultiplications']}
  • Proposition 1.4: =Proposition \ref{['proposition:coassociativityrestriction']}
  • Definition 1.5: =Definition \ref{['definition:braidingoperators']}
  • Theorem 1.6: Mackey formula=Theorem \ref{['theorem:mackeyformula']} and Corollary \ref{['corollary:Mackey3cells']}
  • Proposition 2.1
  • Proposition 2.2
  • proof : Sketch of proof
  • Definition 2.3
  • ...and 97 more