The motivic class of the classifying stack of the special orthogonal group

Abstract

We compute the class of the classifying stack of the special orthogonal group in the Grothendieck ring of stacks, and check that it is equal to the multiplicative inverse of the class of the 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.…”

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.…”