Casimir Radial Parts via Matsuki Decomposition

TL;DR

This paper develops a rigorous mathematical framework using Matsuki's decomposition to analyze the radial parts of Casimir operators in symmetric pairs, with applications to conformal blocks and Calogero-Sutherland models.

Contribution

It introduces a novel approach to derive radial parts of Casimir operators for non-compact symmetric pairs, avoiding problematic analytical continuations.

Findings

Complete analysis of Casimir radial decomposition in Lorentzian signature

Revisiting Casimir reduction for scalar defect conformal blocks

Framework connects Casimir equations with Calogero-Sutherland models

Abstract

We use Matsuki's decomposition for symmetric pairs $(G, H)$ of (not necessarily compact) reductive Lie groups to construct the radial parts for invariant differential operators acting on matrix-spherical functions. As an application, we employ this machinery to formulate an alternative, mathematically rigorous approach to obtaining radial parts of Casimir operators that appear in the theory of conformal blocks, which avoids poorly defined analytical continuations from the compact quotient cases. To exemplify how this works, after reviewing the presentation of conformal 4-point correlation functions via matrix-spherical functions for the corresponding symmetric pair, we for the first time provide a complete analysis of the Casimir radial part decomposition in the case of Lorentzian signature. As another example, we revisit the Casimir reduction in the case of conformal blocks for two scalar defects of equal dimension. We argue that Matsuki's decomposition thus provides a proper mathematical framework for analysing the correspondence between Casimir equations and the Calogero-Sutherland-type models, first discovered by one of the authors and Schomerus.