On the Motivic Class of the Classifying Stack of $G_2$ and the Spin Groups

Abstract

We compute the class of the classifying stack of the exceptional algebraic group $G_2$ and of the spin groups $\mathrm{Spin}_7$ and $\mathrm{Spin}_8$ in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks $\mathscr{B}\mathrm{Spin}_n$ can be reduced to the computation of the classes of $\mathscr{B} \Delta_n$, where $\Delta_n\subset \mathrm{Pin}_n$ is the extraspecial $2$-group, the preimage of the diagonal matrices under the projection $\mathrm{Pin}_n\to \mathrm{O}_n$ to the orthogonal group.