Symmetric subgroup schemes
Jinfeng Song
TL;DR
The paper constructs a scheme-theoretic analogue of symmetric subgroups inside Chevalley group schemes using quantum symmetric pairs, producing a commutative Hopf algebra $O_A^{\\imath}$ that defines a closed subgroup scheme $\\mathbf{G}^{\\imath}$ of the Chevalley group scheme $\\mathbf{G}$ with fibers matching classical symmetric subgroups. It develops a functorial involution formalism and proves finiteness/stability properties via the $\\\imath$-canonical basis, enabling a robust quantization where $O_A^{\\imath}$ is a coisotropic quantum right subgroup of the quantized coordinate algebra. The work unifies scheme-theoretic and quantum perspectives on symmetric pairs, extends Springer’s classifications through $\\\imath$root data, and supports applications to Relative Langlands program geometry and positive characteristic settings. Overall, it provides a rigorous bridge between algebraic group schemes, quantum groups, and geometric representation theory for symmetric spaces.
Abstract
Chevalley group schemes are group schemes defined over the integers that parametrize connected reductive groups over algebraically closed fields as geometric fibers. In this paper, we construct closed subgroup schemes of Chevalley group schemes that parametrize symmetric subgroups of reductive groups as geometric fibers. Our construction relies crucially on the theory of quantum symmetric pairs and thus naturally admits a quantization. At the quantum level, this leads to the construction of coisotropic quantum right subgroups of the quantized function algebras of reductive groups.