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On a formula for the equivariant Euler characteristic of a $G$-sheaf

Qiangru Kuang, Francesco Sala

TL;DR

The paper extends the computation of the equivariant Euler characteristic $\chi_G(X,\mathcal{E})$ from $G$-curves to higher-dimensional $G$-varieties by leveraging a Lefschetz-Riemann-Roch isomorphism for quotient Deligne-Mumford stacks and a decomposition of equivariant K-theory via inertia data. It develops a two-tier composition framework (lower and upper) and analyzes inertia, dual cyclic subgroups, and representation-theoretic twists to express $\chi_G(X,\mathcal{E})$ as a sum of induced representations from fixed loci, weighting by fixed-point data and cyclotomic characters. The main result gives an explicit formula in terms of fixed loci $X^{\sigma}$, centralizers $C(\sigma)$, and induced characters $\mathrm{Ind}_{\sigma}^G \iota(\zeta_r^i)$, and specializes to a concrete curve formula that reproduces the established theorems of Fischbacher-Weitz and Köck when $X$ is a curve and the action is tame. Overall, the work provides a general, stack-theoretic method to compute equivariant Euler characteristics for $G$-sheaves on quotient stacks, including cases where group order is divisible by the characteristic of the base field.

Abstract

H. Fischbacher-Weitz and B. Köck computed the equivariant Euler characteristic of a $G-$sheaf on a $G$-curve $X$ over a field. Using a form of the Riemann-Roch theorem for quotient stacks proved by the second author we extend their computations to the cases where $dim(X) >1$.

On a formula for the equivariant Euler characteristic of a $G$-sheaf

TL;DR

The paper extends the computation of the equivariant Euler characteristic from -curves to higher-dimensional -varieties by leveraging a Lefschetz-Riemann-Roch isomorphism for quotient Deligne-Mumford stacks and a decomposition of equivariant K-theory via inertia data. It develops a two-tier composition framework (lower and upper) and analyzes inertia, dual cyclic subgroups, and representation-theoretic twists to express as a sum of induced representations from fixed loci, weighting by fixed-point data and cyclotomic characters. The main result gives an explicit formula in terms of fixed loci , centralizers , and induced characters , and specializes to a concrete curve formula that reproduces the established theorems of Fischbacher-Weitz and Köck when is a curve and the action is tame. Overall, the work provides a general, stack-theoretic method to compute equivariant Euler characteristics for -sheaves on quotient stacks, including cases where group order is divisible by the characteristic of the base field.

Abstract

H. Fischbacher-Weitz and B. Köck computed the equivariant Euler characteristic of a sheaf on a -curve over a field. Using a form of the Riemann-Roch theorem for quotient stacks proved by the second author we extend their computations to the cases where .
Paper Structure (9 sections, 4 theorems, 29 equations)

This paper contains 9 sections, 4 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Computation
  3. Lower composition
  4. Inclusion of geometric part of $BG$
  5. Inclusion $\tilde{R}\sigma \to R\sigma$
  6. Inertia stack of $BG$ and representations
  7. Upper composition
  8. Total composition
  9. Explicit formula for curves

Key Result

Theorem 1.1

Let $\mathcal{X}$ be a quotient stack with finite cyclotomic inertia over a base scheme $A=Spec(R)$. Then the above map gives a Lefschetz-Riemann-Roch isomorphism, which is covariant with respect to proper push-forwards of relatively tame morphisms:

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • proof