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Hall induction for cotangent representations and wheel conditions

Danil Gubarevich

TL;DR

This work analyzes Hall-inductive structures in cotangent representations of reductive groups by extending cohomological Hall algebra techniques from quivers to cotangent stacks $T^*(V/G)$. The authors develop a framework based on vanishing cycles, dimensional reduction, and equivariant geometry to define Hall induction maps along cocharacter-induced parabolic and Levi subgroups, and prove associativity and torsion-freeness results under torus actions. A K-theoretic perspective yields wheel conditions, identifying precise divisibility constraints on the image of restriction maps in $K_G(X)$ for cotangent representations, with concrete illustrations from adjoint representations and $SL_2$-modules. These results produce structural constraints and embeddings that support a CoHA-like algebra structure in this geometric setting and connect to preprojective CoHAs via dimensional reduction. The work thus advances the understanding of Hall algebras in cotangent/invariant contexts and provides explicit conditions (wheel conditions) constraining the K-theoretic images, with potential applications to representation theory and geometric representation theory.

Abstract

In this short note we study the Hall induction of cotangent representations of reductive groups. We prove its torsion freeness in Borel-Moore homology. In K-theory we find an analog of wheel conditions verified by the image of restriction map to the fixed point and consider examples.

Hall induction for cotangent representations and wheel conditions

TL;DR

This work analyzes Hall-inductive structures in cotangent representations of reductive groups by extending cohomological Hall algebra techniques from quivers to cotangent stacks . The authors develop a framework based on vanishing cycles, dimensional reduction, and equivariant geometry to define Hall induction maps along cocharacter-induced parabolic and Levi subgroups, and prove associativity and torsion-freeness results under torus actions. A K-theoretic perspective yields wheel conditions, identifying precise divisibility constraints on the image of restriction maps in for cotangent representations, with concrete illustrations from adjoint representations and -modules. These results produce structural constraints and embeddings that support a CoHA-like algebra structure in this geometric setting and connect to preprojective CoHAs via dimensional reduction. The work thus advances the understanding of Hall algebras in cotangent/invariant contexts and provides explicit conditions (wheel conditions) constraining the K-theoretic images, with potential applications to representation theory and geometric representation theory.

Abstract

In this short note we study the Hall induction of cotangent representations of reductive groups. We prove its torsion freeness in Borel-Moore homology. In K-theory we find an analog of wheel conditions verified by the image of restriction map to the fixed point and consider examples.
Paper Structure (21 sections, 12 theorems, 136 equations)

This paper contains 21 sections, 12 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Derived constructible category
  5. Borel-Moore homology
  6. K-theory
  7. Vanishing cycles
  8. Equivariant vanishing cycles
  9. Dimensional reduction
  10. Equivariant dimensional reduction
  11. Hall induction
  12. Dynamical method
  13. KS critical Hall induction
  14. Induction for cotangent representations
  15. Associativity
  16. ...and 6 more sections

Key Result

Theorem 1.1

Under the above assumptions, the $H^\mathrm{BM}_{G\times T_s}(\mathrm{pt},\mathbb Q)$-module $H_{G\times T_s}^\mathrm{BM}(\mu_V^{-1}(0),\mathbb Q)$ is torsion free.

Theorems & Definitions (36)

  • Theorem 1.1: Theorem \ref{['tfree']}
  • Theorem 1.2: Theorem \ref{['whTheorem']}
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Theorem 4.1
  • ...and 26 more