Inner Type Isometries on Lie Groups

Let G be a connected Lie group endowed with a left invariant metric and let U = { x ∈ G : g → xgx −1 is an isometry}. In this note we study conditions on H , a compact subgroup of G , to have H = U 0 , for some left invariant metric. In the case when H is a torus and G is compact and semisimple we give a necessary and sufficient condition for this to happen. We give several examples and applications to I ( G ), the full isometry group of G, showing in particular that if G is compact and simple there exist families of metrics such that (1) U and I ( G ) are highly disconnected, (2) U = { e }, I ( G ) = G L , if g ≄ su (2).