Orthogonal Howe duality and dynamical (split) symmetric pairs

TL;DR

This paper extends the concept of dynamical algebraic structures to split symmetric pairs, establishing a duality that exchanges differential and difference operators in the context of orthogonal groups.

Contribution

It introduces dynamical fusion, K-matrix, and Weyl group concepts for split symmetric pairs, and proves a duality exchanging operators for orthogonal Lie algebras and groups.

Findings

Dynamical structures are generalized to split symmetric pairs.

A duality between $rak{so}_{2n}$ and $O_m$ is established.

Operators on both sides are shown to correspond under this duality.

Abstract

Inspired by Etingof--Varchenko's dynamical fusion, dynamical $R$-matrix, and dynamical Weyl group for Lie algebras, we introduce, for split symmetric pairs, versions of dynamical fusion, dynamical $K$-matrix, and dynamical Weyl group. We then turn to the study of $(\mathfrak{so}_{2n},O_m)$-duality and prove that the standard Knizhnik-Zamolodchikov and dynamical operators (both differential and difference) on the $\mathfrak{so}_{2n}$-side are exchanged with the symmetric pair analogs, for $O_m\subset GL_m$, on the $O_m$-side.