# CHAPTER 6

# Group-Theoretic Facts about *G*_{geom} and *G*_{arith}

**Theorem 6.1.** *Suppose N in P*_{arith} is geometrically semisimple. Then G_{geom,N} is a normal subgroup of G_{arith,N}.

*Proof.* Because *N* is geometrically semisimple, the group *G*_{geom,N} is reductive, so it is the fixer of its invariants in all finite dimensional representations of the ambient *G*_{arith,N}. By noetherianity, there is a finite list of representations of *G*_{arith,N} such that *G*_{geom,N} is the fixer of its invariants in these representations. Taking the direct sum of these representations, we get a single representation of *G*_{arith,N} such that *G*_{geom,N} is the fixer of its invariants in that single representation. This representation corresponds to an object *M* in <*N>*_{arith}, and a *G*_{geom,N-} invariant in ...