The motive of the classifying stack of the orthogonal group

Abstract

We compute the motive of the classifying stack of an orthogonal group in the Grothendieck ring of stacks over a field of characteristic different from two.

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

“…Our approach is different from that of Dhillon and Young in , and also gives an independent proof of their result. Instead of the stratification of the space of non‐degenerate quadratic forms that they use, we exploit a simpler stratification of the tautological representation of already used in .…”

“…The cases of non‐special connected groups for which has been computed include , (by Bergh in ) and when is odd (by Dhillon and Young in ). In all these cases the equality continues to hold.…”

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

“…The class of {BG} for non-special G has been computed in a few cases: for G = PGL 2 and PGL 3 by Bergh [Ber16], for SO n with odd n by Dhillon and Young [DY16] and for any n by the second author and Vistoli [TV]. In all these cases we have indeed (somewhat surprisingly) that {BG} = {G} −1 in K 0 (Stack k ).…”

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.…”