Representations of quantum symmetric pairs at roots of unity
Jinfeng Song, Weinan Zhang
TL;DR
This work extends the De Concini–Kac framework to quantum symmetric pairs at odd roots of unity, constructing a Frobenius center $Z_0^ ext{imath}$ inside the iquantum group $ ext{U}^ ext{imath}_v$ and identifying it with the coordinate algebra of a Poisson homogeneous space $ ext{X}$ via a quantum Frobenius map. A filtration yields a graded, $q$-commutative algebra whose center controls representation-theoretic data, enabling a complete description of the full center as generated by $Z_0^ ext{imath}$ and the Kolb–Letzter center $Z_1^ ext{imath}$, and fixing the degree of $ ext{U}^ ext{imath}_v$. Irreducible modules are parametrized by $ heta$-twisted conjugacy classes, with dimensions constrained by the dimension of the corresponding twisted class; maximal dimensions occur for regular classes. The paper also develops small quantum symmetric pairs, branching laws under restriction from $ ext{U}_v$, relative braid group symmetries, and a compatibility theory between Frobenius maps and these symmetries, providing a cohesive algebraic and Poisson-theoretic picture linking representation theory to twisted conjugacy and symplectic leaves. Together, these results generalize De Concini–Kac–Procesi’s program to a broad class of iquantum groups and offer tools for further geometric and categorical explorations in quantum symmetric pairs at roots of unity.
Abstract
Let $θ$ 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 $θ$-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 $\mathrm{U}^\imath_v$.