Motives of classifying stacks of orthogonal groups

Abstract

(Based on joint work with Ajneet Dhillon.) The Grothendieck ring of stacks is a dimensional completion of the usual Grothendieck ring of varieties. If $G$ is a linear algebraic group, then its classifying stack has a motivic class $[BG]$ in this ring. The understanding of $[BG]$ is an important problem because of its relevance to the computation of invariants of moduli problems in algebraic geometry. If $G$ is a special algebraic group, such as $GL_n$, $SL_n$ or $Sp_{2n}$, then $[BG]=[G]^{-1}$. For non-special groups, the motive $[BG]$ is much more difficult to understand. In this talk I will describe a solution to this problem for the orthogonal groups. I will also discuss an application of this result to Donaldson-Thomas theory with orientifolds.