Congruence subgroups and twisted cohomology of SL_n(F[t])

Details Congruence subgroups and twisted cohomology of SL_n(F[t]) Congruence subgroups and twisted cohomology of SL_n(F[t])

1998-01-21

arxiv.org/abs/math/9801099v1

J. Algebra <b>207</b> (1998), 695-721

Mathematics

K-Theory and Homology

24 pages, 1 figure, to appear in Journal of Algebra

Scientific paper

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 F-vector 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.

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