Projections in Kac-Moody Lie algebras

Abstract:

Let $\mathfrak {g}$ be a Kac-Moody Lie algebra and $\mathcal {B}$ its set of positive Borel subalgebras. If $\mathfrak {b} \in \mathcal {B}$ and $\mathfrak {p}$ is a parabolic subalgebra, let ${\operatorname {proj} _\mathfrak {p}}(\mathfrak {b}) = \mathfrak {p} \cap \mathfrak {b} + {\mathfrak {r}_n}(\mathfrak {p})$ where ${\mathfrak {r}_n}(\mathfrak {p})$ denotes the nilradical of $\mathfrak {p}$. In this paper we consider the idempotent maps ${E_{\mathfrak {p},{\mathfrak {p}^ - }}} = {\operatorname {proj} _\mathfrak {p}} \circ {\operatorname {proj} _{{\mathfrak {p}^ - }}}:\mathcal {B} \to \mathcal {B}$, where $\mathfrak {p}$ and ${\mathfrak {p}^ - }$ are opposite parabolic subalgebras with $\mathfrak {p}$ being of positive type. We consider the semigroup $M = M(\mathfrak {g})$ generated (with respect to composition) by the maps ${E_{\mathfrak {p},{\mathfrak {p}^ - }}}$. In particular we show that the maximal subgroups of $M$ are closely related to proper Levi subgroups of the Kac-Moody group associated with $\mathfrak {g}$.