Congruence subgroups and twisted cohomology of SL_n(F[t])
Abstract
Let F be a field of characteristic zero and let V be an irreducible representation of SL_n(F). In this paper, we compute the first cohomology of SL_n(F[t]) with coefficients in V. It agrees with H^1(SL_n(F),V) if V is not the adjoint representation, while if V = Ad, the two groups differ by an Fvector space X. We show that if n=2, X is infinite dimensional, while if n>2, dim X = 1. We also study the abelianization of the kernel of the map SL_n(F[t])>SL_n(F) given by setting t=0, where now F is any field. We conjecture that this abelianization is the adjoint representation sl_n(F) if n>2 and F is finite, and prove this in the case n=3, F=F_2, F_3.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 1998
 DOI:
 10.48550/arXiv.math/9801099
 arXiv:
 arXiv:math/9801099
 Bibcode:
 1998math......1099K
 Keywords:

 KTheory and Homology;
 Group Theory;
 20G10
 EPrint:
 24 pages, 1 figure, to appear in Journal of Algebra