Iwasawa Decomposition for Lie Superalgebras

Abstract

Let $\mathfrak{g}$ be a basic simple Lie superalgebra over an algebraically closed field of characteristic zero, and $\hat{I}¸$ an involution of $\mathfrak{g}$ preserving a nondegenerate invariant form. We prove that either $\hat{I}¸$ or $\hat{I}´\circ \hat{I}¸$ admits an Iwasawa decomposition, where $\hat{I}´$ is the canonical grading automorphism $\hat{I}´(x)=(-1{)}^{\overline{x}}x$. The proof uses the notion of generalized root systems as developed by Serganova, and follows from a more general result on centralizers of certain tori coming from semisimple automorphisms of the Lie superalgebra $\mathfrak{g}$.

23 pages; typos fixed from previous version; version appeared in Journal of Lie theory