Computations in the Grothendieck Group of Stacks

Abstract: Given an algebraic group, one may consider the class of its classifying stackin the Grothendieck group of stacks. This is an invariant studied byEkedahl. For certain connected groups, called the special groups bySerre and Grothendieck, the invariant simply gives the inverse of the class ofthe group itself. It is natural to ask whether the same is true for otherconnected groups. We investigate this for the groups PGL(2) and PGL(3) under mild restrictions on the choice of base field.In the case of PGL(2), the question turns out to have a positive answer. In the case of PGL(3), we reduce the question to the computation of the invariant for thenormaliser of a maximal torus in PGL(3). The reduction involves determiningthe class of a certain gerbe over the moduli stack of elliptic curves.