Author:  Kevin P. Knudson
Institution:  Northwestern University

The first cohomology with twisted coefficients of SL_n(F[t])

In an earlier work, the author showed that if F is an infinite field,
then the natural inclusion SL_n(F)-->SL_n(F[t]) induces an isomorphism
on integral homology.  In this paper, we study the extent to which
this isomorphism holds when the trivial integral coefficients are
replaced by some rational representation of SL_n(F).  The group
SL_n(F[t]) acts on such a representation via evaluation at t=0.

The main result is the following.  Denote by K the subgroup of
SL_n(F[t]) consisting of matrices which are congruent to the identity
modulo t and let C be the upper triangular subgroup of K.  Denote by
B the upper triangular subgroup of SL_n(F).  

Theorem:  The group H^1(SL_n(F[t]),V) satisfies

                                  [  H^1(SL_n(F),V)             V not Ad
             H^1(SL_n(F[t]),V)=   [
                                  [  H^1(SL_n(F),V) + Hom_B(H_1(C),V) V=Ad.


This is proved by utilizing the action of SL_n(F[t]) on a certain
Bruhat-Tits building.

This paper has been submitted for publication.

>

This paper is an updated and expanded version of the preprint "The first
cohomology with twisted coefficients of SL<sub>n</sub>(F[t])".  It now includes
a study of the abelianization of the kernel of the map SL<sub>n</sub>(F[t])
---> SL<sub>n</sub>(F) as well as a corrected proof of the main theorem
(there were minor errors in the original).


This paper will appear in the <a href="http://www.apnet.com/www/journal/ja.htm">Journal of Algebra</a>.

Note:  the file "domain.ps" is a figure.