Toeplitz operators on Hardy spaces over $\operatorname{SL}(2,{\bf R})$: irreducibility and representations

Alexander Alldridge, Harald Upmeier
2002 Banach Center Publications  
On the non-abelian, non-compact simple rank 1 Lie group G = SL(2, R), we consider Hardy spaces H 2 (G C ± ) defined by L 2 -boundary values of holomorphic functions on the complex subsemigroups G C ± of G C = SL(2, C). These Hardy spaces are associated to the two parts of the discrete series of G, and give rise to equivariant projections E ± and corresponding Toeplitz operators T ± (f ), f ∈ C 0 (G). We show that a stratification of boundary faces for G C ± can be given, and, by a geometric
more » ... 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. 2000 Mathematics Subject Classification: Primary 47B35, 22D25; Secondary 22E46, 32A25.
doi:10.4064/bc55-0-9 fatcat:ah4l3e5oxffelo3rahgrzbnaju