Reflection Equation and Quantum Conjugacy Classes

Abstract

We study solutions of the Reflection Equation of the non-exceptional groups and the group G2 in connection with quantization of spherical conjugacy classes. In particular, we prove that all symmetric conjugacy classes quantized as subalgebras of endomorphisms in pseudo-parabolic Verma modules have a one-dimensional representation and admit an embedding to the function algebra on the quantum groups. We extend our studies to the Reflection Equation of basic quantum supergroups. In particular, we classify all solutions for the general linear quantum supergroup and construct invertible solutions for ortho-symplectic quantum super groups. We have generalized Letzter’s theory of quantum symmetric pairs to super-spherical pairs of basic quantum supergroups and relate them to the solutions of the Reflection Equation.