Homology of SL2 over function fields I: parabolic subcomplexes

Abstract

The present paper studies the homology of the groups $SL_2(k[C])$ and $GL_2(k[C])$ where $C=\overline{C}\setminus\{P_1,\dots,P_s\}$ is a smooth affine curve over an algebraically closed field $k$. It is well-known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of vector bundles on the complete curve. There is a natural subcomplex of cells with non-unipotent isotropy group. The paper provides explicit formulas for the equivariant homology of this "parabolic subcomplex". These formulas also describe the homology of $SL_2(k[C])$ above degree s, with finite coefficients away from the characteristic of $k$, generalizing a result of Suslin for the case s=1.