Representations of quantum symmetric pairs at roots of unity

Abstract

Let $\theta$ be an involution of a complex semisimple Lie algebra $\mathfrak{g}$ and $(\mathrm{U}_v,\mathrm{U}^\imath_v)$ be the associated quantum symmetric pair at an odd root of unity $v$. In this paper, generalizing the approach of De Concini-Kac-Procesi for quantum groups, we study the structures and irreducible representations of the iquantum group $\mathrm{U}^\imath_v$. We establish a Frobenius center of $\mathrm{U}^\imath_v$ as a coideal subalgebra of the Frobenius center of the quantum group $\mathrm{U}_v$. Via a quantum Frobenius map, we show that the Frobenius center of $\mathrm{U}^\imath_v$ is isomorphic to the coordinate algebra of a Poisson homogeneous space $\mathcal{X}$ of the dual Poisson-Lie group $G^*$. We define a filtration on $\mathrm{U}^\imath_v$ such that the associated graded algebra is $q$-commutative. Using this filtration, we show that the full center of $\mathrm{U}^\imath_v$ is generated by the Frobenius center and the Kolb-Letzter center, and we determine the degree of $\mathrm{U}^\imath_v$. We show that irreducible representations of $\mathrm{U}^\imath_v$ are parametrized by $\theta$-twisted conjugacy classes. We determine the maximal dimension of those irreducible representations, and show that the dimension of an irreducible representation is maximal if the corresponding twisted conjugacy class has maximal dimension. We also study the branching problem for irreducible $\mathrm{U}_v$-modules when restricting to $\mathrm{U}^\imath_v$.