Lattices and automorphisms of Lie groups (Q1110663)

Let G be a connected Lie group and L a lattice in G. The automorphisms of G are in general not determined by their values on L. This paper is about the possible ambiguities: the group F(L) of automorphisms of G fixing L pointwise. It is shown that the identity component \(F(L)_ 0\) of F(L) is a real algebraic group consisting of inner automorphisms and that it is the product of a vector subgroup and a connected compact subgroup. The same results are first proved for F(L) itself, when G is simply connected. In this case \(F(L)_ 0\) has finite index in F(L).