On the phenomena of constant curvature in the diffusion-orthogonal polynomials

TL;DR

This paper investigates the structure of diffusion-orthogonal polynomials with maximal boundary degree, showing they originate from reflection groups in dimensions 2 and 3, but not in higher dimensions.

Contribution

It proves that such polynomial systems are derived from reflection groups in low dimensions, highlighting a boundary phenomenon and leaving open questions in higher dimensions.

Findings

Systems with maximal boundary degree come from reflection groups in dimensions 2 and 3.

The proof techniques are algebraic and complex analytic, involving double coverings of complex space.

The phenomenon does not extend straightforwardly to dimensions 4 and higher.

Abstract

We consider the systems of diffusion-orthogonal polynomials, defined in the work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why these systems with boundary of maximal possible degree should always come from the group, generated by reflections. Our proof works for the dimensions $2$ (on which this phenomena was discovered) and $3$, and fails in the dimensions $4$ and higher, leaving the possibility of existence of diffusion-orthogonal systems related to the Einstein metrics. The methods of our proof are algebraic / complex analytic in nature and based mainly on the consideration of the double covering of $\mathbb{C}^d$, branched in the boundary divisor. Author wants to thank Stepan Orevkov, Misha Verbitsky and Dmitry Korb for useful discussions.