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Research Report MRR98-066
Étale Slices for Representation Varieties in Characteristic p
Benjamin M. S. Martin
Abstract:
Let R(F,G) be the variety of representations of a finitely
generated group F into a connected reductive algebraic group
G,
and let C(F,G) be the variety of closed conjugacy classes
of representations. We examine the question of whether an étale
slice for the conjugation action of G exists through a
representation \rho \in R(F,G) when the ground field
k has
characteristic p>0. We show that an étale slice through
\rho may exist for the action of an enlarged group \widehat G,
even when there is no étale slice for the G-action. As an
application, we generalise a result known to hold in characteristic
zero, which expresses the tangent space to C(F,G) at the
conjugacy class of a suitable representation \rho as a subspace
of the 1-cohomology H1(F,g(\rho)) of an
F-module
g(\rho). A similar result holds in characteristic p, but
with H1(F,g(\rho)) replaced by a quotient of
H1(F,g(\rho)).
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