Quantized Algebras of Functions on Homogeneous Spaces with Poisson Stabilizers

Abstract

Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0 < q < 1. We study a quantization C(G q /K q ) of the algebra of continuous functions on G/K. Using results of Soibelman and Dijkhuizen-Stokman we classify the irreducible representations of C(G q /K q ) and obtain a composition series for C(G q /K q ). We describe closures of the symplectic leaves of G/K refining the well-known description in the case of flag manifolds in terms of the Bruhat order. We then show that the same rules describe the topology on the spectrum of C(G q /K q ). Next we show that the family of C*-algebras C(G q /K q ), 0 < q ≤ 1, has a canonical structure of a continuous field of C*-algebras and provides a strict deformation quantization of the Poisson algebra . Finally, extending a result of Nagy, we show that C(G q /K q ) is canonically KK-equivalent to C(G/K).