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 Spin 7 and 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 BSpin n can be reduced to the computation of the classes of B∆n, where ∆n ⊂ Pinn is the "extraspecial 2-group", the preimage of the diagonal matrices under the projection Pinn → On to the orthogonal group.

On the Mixed Tate property and the motivic class of the classifying stack of a finite group

“…The equality {B k G}{G} = 1 in K 0 (Stacks k ) holds for many connected linear algebraic groups G. For example, it holds if G is special (i.e. H 1 (K, G) = 0 for every field extension K/k), G = PGL 2 and PGL 3 by [2], G = SO n by [11] and [31], and G a split group of type G 2 , Spin 7 and Spin 8 by [27]. In all these examples B k G is stably rational.…”

The Transition Function of G2 over S6

“…We identify G 2 with Aut O ⊂ SO (7), the automorphism group of the Cayley octonions. It is a classical fact that there is a fibration p : G 2 → S 6 , which makes G 2 a locally trivial SU(3)bundle over S 6 .…”

On the motivic class of an algebraic group

“…Computations for non-special G have been carried out for PGL 2 or PGL 3 in [2], for SO n and n odd in [4], for SO n and n even or O n for any n in [18], and for Spin 7 , Spin 8 and G 2 in [15]. In each of these cases, (1.3) was found to be true.…”

“…(cf. [18, §1] and [15,Remark 4.1]) Is it true that, for a connected linear algebraic group G, the following two conditions are equivalent?…”