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Coordinate rings on symmetric spaces

Huanchen Bao, Jinfeng Song

Abstract

Let $G_k$ be a connected reductive group over an algebraically closed field $k$ of char $\neq 2$. 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.

Coordinate rings on symmetric spaces

Abstract

Let be a connected reductive group over an algebraically closed field of char . Let be an algebraic group involution of and denote the fixed point subgroup by . We construct an integral model for the symmetric space with a natural action of the Chevalley group scheme over integers. We show the coordinate ring admits a canonical basis, as well as a good filtration as a -module. We also construct a canonical basis and an integral form for the space of -biinvariant functions on . Our results rely on the construction of quantized coordinate algebras of symmetric spaces, using the theory of canonical bases on quantum symmetric pairs.
Paper Structure (22 sections, 22 theorems, 53 equations)

This paper contains 22 sections, 22 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum groups
  4. Definitions
  5. Based modules
  6. The modified form
  7. Coordinate rings
  8. Quantum symmetric pairs
  9. $\imath$Quantum groups
  10. Based $\mathrm{U}^\imath$-modules
  11. Spherical modules
  12. Symmetric subgroups
  13. Bases for coinvariants
  14. Coinvariant spaces
  15. Coinvariant functors
  16. ...and 7 more sections

Key Result

Theorem 1

There exists a commutative ring ${}_\mathbb {Z}\xspace \mathbf{O}(K \backslash G) \subset {}_\mathbb {Z}\xspace \mathbf{O}$ over $\mathbb {Z}\xspace$, such that ${}_k \mathbf{O}(K \backslash G) = k \otimes_\mathbb {Z}\xspace {}_\mathbb {Z}\xspace \mathbf{O}(K \backslash G) \cong k[K_k \backslash G_k

Theorems & Definitions (36)

  • Theorem 1: Theorem \ref{['thm:base']} & Theorem \ref{['thm:KG']}
  • Theorem 2: Theorem \ref{['thm:fil']}
  • Theorem 3: Theorem \ref{['thm:Fbased']}
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5: BS22*Theorem 3.8
  • Remark 2.6
  • Theorem 3.1
  • ...and 26 more