Dual canonical bases and embeddings of symmetric spaces

TL;DR

This paper develops a quantization and integral models for affine embeddings of symmetric spaces associated with reductive groups, establishing dual canonical bases and models for canonical and wonderful compactifications.

Contribution

It constructs dual canonical bases and integral models for affine and canonical embeddings of symmetric spaces, including the wonderful compactification, in the context of reductive groups.

Findings

Coordinate rings admit dual canonical bases.

Constructed integral models for canonical embeddings.

Established models for wonderful compactifications.

Abstract

For a connected reductive group $G_k$ over an algebraically closed field $k$ of char $\neq 2$ and a fixed point subgroup $K_k$ under an algebraic group involution, we construct a quantization and an integral model of any affine embeddings of the symmetric space $G_k/K_k$. We show that the coordinate ring of any affine embedding of $G_k/K_k$ admits a dual canonical basis. We further construct an integral model for the canonical embedding (that is, an embedding which is complete, simple, and toroidal) of $G_k/K_k$. When $G_k$ is of adjoint type, we obtain an integral model for the wonderful compactification of the symmetric space.