Toeplitz operators on Hardy spaces over $\operatorname{SL}(2,{\bf R})$: irreducibility and representations
Volume 55 / 2002
Abstract
On the non-abelian, non-compact simple rank 1 Lie group $G=\operatorname{SL}(2,{\bf R})$, we consider Hardy spaces ${\bf H}^2(G^{\bf C}_{\pm})$ defined by ${\bf L}^2$-boundary values of holomorphic functions on the complex subsemigroups $G^{\bf C}_{\pm}$ of $G^{\bf C}=\operatorname{SL}(2,{\bf C})$. These Hardy spaces are associated to the two parts of the discrete series of $G$, and give rise to equivariant projections $E_{\pm}$ and corresponding Toeplitz operators ${\rm T}_\pm (f)$, $f\in\mathcal C^0(G)$. We show that a stratification of boundary faces for $G^{\bf C}_{\pm}$ can be given, and, by a geometric construction, associate to these faces representations of the C$^*$-algebra generated by the Toeplitz operators for the respective domain, thus achieving a step $2$ composition series for this C$^*$-algebra.
