Coordinate rings on symmetric spaces

2024/02/13 by Huanchen Bao, Bao, Huanchen, Jinfeng Song +1 · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2402.08258

Abstract

Let G_k be a connected reductive group over an algebraically closed field k of char ≠ 2. Let θ_k be an algebraic group involution of G_k and denote the fixed point subgroup by K_k. We construct an integral model for the symmetric space K_k \backslash G_k with a natural action of the Chevalley group scheme over integers. We show the coordinate ring k[K_k \backslash G_k] admits a canonical basis, as well as a good filtration as a G_k-module. We also construct a canonical basis and an integral form for the space of K_k-biinvariant functions on k[G_k]. Our results rely on the construction of quantized coordinate algebras of symmetric spaces, using the theory of canonical bases on quantum symmetric pairs.

Cited by

• Symmetric subgroup schemes
• Based morphisms for characters of quantum symmetric pairs

Related

• Dual canonical bases and embeddings of symmetric spaces
• Study of a Z-form of the coordinate ring of a reductive group
• Symmetric subgroup schemes
• The algebra of K-invariant vector fields on a symmetric space G/K
• Canonical bases and quantum coordinate ring