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Orthogonal Howe duality and dynamical (split) symmetric pairs

Elijah Bodish, Artem Kalmykov

TL;DR

The paper develops a comprehensive boundary/dynamical framework for split symmetric pairs, focusing on the orthogonal Howe duality (\mathfrak{so}_{2n}, O_m). It constructs boundary analogues of fusion, dynamical Weyl groups, and Casimir-type connections, and proves a quantum bispectral duality that equates dynamical boundary operators with R-/twisted-Yangian structures across the two sides. It also establishes classical (KZ/Casimir) bispectral dualities and conjectures about monodromy relations to quantum boundary Weyl groups, supported by explicit rank-1 and rank-2 analyses. Together these results extend Tarasov–Varchenko-type dualities to symmetric-pair boundaries, providing a coherent algebraic-analytic bridge between so_{2n}-spinor/exterior-algebra representations and O_m actions, with potential applications to cyclotomic Gaudin systems and boundary-integrable models.

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.

Orthogonal Howe duality and dynamical (split) symmetric pairs

TL;DR

The paper develops a comprehensive boundary/dynamical framework for split symmetric pairs, focusing on the orthogonal Howe duality (\mathfrak{so}_{2n}, O_m). It constructs boundary analogues of fusion, dynamical Weyl groups, and Casimir-type connections, and proves a quantum bispectral duality that equates dynamical boundary operators with R-/twisted-Yangian structures across the two sides. It also establishes classical (KZ/Casimir) bispectral dualities and conjectures about monodromy relations to quantum boundary Weyl groups, supported by explicit rank-1 and rank-2 analyses. Together these results extend Tarasov–Varchenko-type dualities to symmetric-pair boundaries, providing a coherent algebraic-analytic bridge between so_{2n}-spinor/exterior-algebra representations and O_m actions, with potential applications to cyclotomic Gaudin systems and boundary-integrable models.

Abstract

Inspired by Etingof--Varchenko's dynamical fusion, dynamical -matrix, and dynamical Weyl group for Lie algebras, we introduce, for split symmetric pairs, versions of dynamical fusion, dynamical -matrix, and dynamical Weyl group. We then turn to the study of -duality and prove that the standard Knizhnik-Zamolodchikov and dynamical operators (both differential and difference) on the -side are exchanged with the symmetric pair analogs, for , on the -side.
Paper Structure (71 sections, 59 theorems, 371 equations)

This paper contains 71 sections, 59 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. Boundary Weyl group
  3. Boundary dynamical Weyl group
  4. Simplified series expressions
  5. Quantum bispectral duality
  6. Classical bispectral duality
  7. Structure of the paper
  8. Lie theory notation
  9. Cartan involution
  10. Roots and generators
  11. Iwasawa decomposition
  12. Chevalley involution and generators
  13. Integrable representations
  14. Weights and representations
  15. Boundary Weyl groups
  16. ...and 56 more sections

Key Result

Theorem 1.3

The operators $A^{\mathfrak{k},s_i}_X(\lambda)$ satisfy the dynamical braid relation where the number of factors on each side of the equation is the order of $(s_i s_j)$ in $W$, the Weyl group of $\mathfrak{g}$.

Theorems & Definitions (164)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['thm:boundary_braid_relations']}
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: Propositions \ref{['prop:so_r_operator_equal_bweyl']} and \ref{['prop:twisted_rmatrix_equal_dyn_operator']} and Theorems \ref{['thm:so_dynamical_bkz_bispectrality']} and \ref{['thm:orthogonal_duality_qkz_bdyn']}
  • Theorem 1.8: Definition \ref{['D:boundary-Casimir']} and Theorem \ref{['thm::bcasimir_flat']}
  • Remark 1.9
  • Theorem 1.10: Theorem \ref{['prop::so_bkz_cas_duality']} and Theorem \ref{['thm:orthogonal_duality_bcas_kz']}
  • ...and 154 more