Cartan subproduct systems

Abstract

Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $\hat{I}»$, the highest weight $G_q$-modules $V_{n\hat{I}»}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic behavior of Clebsch-Gordan coefficients we identify the corresponding Cuntz-Pimsner algebras with algebras of quantized functions on homogeneous spaces of $G$. We also show that the gauge-invariant part of the Toeplitz algebra provides a model for convergence of full matrix algebras to quantum flag manifolds, complementing and generalizing results of Landsman and Rieffel for $q=1$ and results of Vaes-Vergnioux in the rank one case for $q\ne 1$. We verify our conjecture on Clebsch-Gordan coefficients for $G=SU(n)$ and all weights that are either regular or multiples of the fundamental weight $\ddot{I}{‰}_1$. For $\hat{I}»=\ddot{I}{‰}_1$, we also provide a detailed description of the Toeplitz and Cuntz-Pimsner algebras, generalizing results of Arveson on symmetric subproduct systems.

25 pages