On the pre-lambda-ring structure on the Grothendieck ring of stacks and the power structures over it

Abstract

We describe a pre-lambda-structure on the Grothendieck ring of stacks (originally studied by Torsten Ekedahl) and the corresponding power structures over it, discuss some of their properties and give some explicit formulae for the Kapranov zeta-function for some stacks. In particular, we show that the nth symmetric power of the class of the classifying stack BGL(1) of the group GL(1) coincides, up to a power of the class L of the affine line, with the class of the classifying stack BGL(n).