Motivic classes of classifying stacks of some semi-direct products

TL;DR

This paper investigates the motivic classes of classifying stacks associated with certain semi-direct products, demonstrating triviality under specific conditions and extending results to complex reflection groups.

Contribution

It introduces conditions under which the motivic class of classifying stacks of semi-direct products is trivial, advancing understanding in algebraic geometry and group actions.

Findings

Motivic class of classifying stacks of T[m] and G is trivial under certain assumptions.

Triviality of motivic class of BW for many complex reflection groups W.

Provides new tools for studying classifying stacks in algebraic geometry.

Abstract

Let k be a field, let G be a finite group and let T be a split k-torus on which G acts multiplicatively, and for every m greater than 1 denote by T[m] the m-torsion subgroup of T. Under a suitable assumption on m, we show that the motivic class of the classifying stack of the semi-direct product of T[m] and G in K_0(Stacks_k) is trivial. As a consequence, we prove that the motivic class of BW is trivial for a large class of complex reflection groups W.