Representations of quantum symmetric pairs at roots of unity

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 $\mathrm{U}^\imath_v$.

💡 Research Summary

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The paper studies quantum symmetric pairs (U_v, U_v^ι) associated with a complex semisimple Lie algebra 𝔤 and an involution θ, focusing on the case where the quantum parameter v is an odd root of unity. Building on the De Concini‑Kac‑Procesi (DCKP) framework for quantum groups at roots of unity, the authors develop a parallel theory for the i‑quantum group U_v^ι, which is a coideal subalgebra of the Drinfeld‑Jimbo quantum group U_v.

First, they construct the De Concini‑Kac integral form U_A^ι = U^ι ∩ U_A and specialize it at v to obtain U_v^ι. They identify a coisotropic subgroup K^⊥ ⊂ G^* (the dual Poisson‑Lie group) consisting of pairs (g, θ(g)). The affine quotient X = K^⊥\G^* is a Poisson homogeneous space. By restricting the quantum Frobenius map Fr: C