Nonexistence of invariant semigroups in affine symmetric spaces

Abstract

Let \(\left( G,L,\tau \right) \) be an affine symmetric space with \(G\) a simple Lie group, \(\tau \) an involutive automorphism of \(G\) and \(L\) an open subgroup of the \(\tau \)-fixed point group \(G^{\tau }\). It is proved here that the existence of a proper semigroup \(S\subset G\) with \({\rm int}S\neq \emptyset \) and \(L\subset S\) implies that \(\left( G,L,\tau \right) \) is of Hermitian type, as conjectured by Hilgert and Neeb [4]. When \(S\) exists, it turns out that it leaves invariant an open \(L\)-orbit in a minimal flag manifold of \(G\). A byproduct of our approach is an alternate proof of the maximality of the compression semigroup of an open orbit (see Hilgert and Neeb [31]).