Primitive Ideals and Hilbert Space Representations of Quantized Coordinate Algebras of Complex Semisimple Lie Groups

The primitive ideals of the coordinate algebra $ \mathcal{O} ( G ) $ of a complex semisimple Lie group $ G $ are in bijection with the points of $ G $, via the correspondence assigning to each point of $ G $ the kernel of the associated evaluation homomorphism on $ \mathcal{O} ( G ) $. This establishes a direct link between the algebraic structure of $ \mathcal{O} ( G ) $ and the geometry of $ G $.
In this talk, we investigate the quantum analogue of this classical relationship for the $ q $-deformation $ G_q $. Specifically, we establish a sharp dichotomy: primitive ideals in homogeneous Joseph strata arise as kernels of irreducible representations of $ \mathcal{O} ( G_q ) $ by bounded operators on Hilbert spaces, which provide a quantum analogue of evaluation homomorphisms at points of $ G $, whereas those in inhomogeneous Joseph strata do not. This clarifies the extent to which the primitive spectrum of $ \mathcal{O} ( G_q ) $ can be accessed through operator-theoretic methods. We also analyze the semiclassical consequences of this result in light of the fact that the primitive ideals of $ \mathcal{O} ( G_q ) $ are parametrized by the symplectic leaves of the natural Poisson structure on $ G $.
This talk is based on joint work with Christian Voigt.