Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism

TL;DR

This paper links algebraic decompositions in split reductive groups over non-archimedean fields to Hall-Littlewood polynomials, extending prior results and establishing probabilistic limit theorems for singular numbers and corners.

Contribution

It generalizes the understanding of singular numbers and corners via the Satake isomorphism to arbitrary root systems and develops asymptotic laws for their behavior in matrix products.

Findings

Singular numbers and corners are determined by Hall-Littlewood polynomials.

Extended Van Peski's results to all root systems.

Proved strong law of large numbers and central limit theorem for these quantities.

Abstract

We establish that the singular numbers (arising from Cartan decomposition) and corners (emerging from Iwasawa decomposition) in split reductive groups over non-archimedean fields are fundamentally determined by Hall-Littlewood polynomials. Through applications of the Satake isomorphism, we extend Van Peski's results (arXiv:2011.09356, Theorem 1.3) to encompass arbitrary root systems. Leveraging this theoretical foundation, we further develop Shen's work (arXiv:2411.01104, Theorem 1.1) to demonstrate that both singular numbers and corners of such products exhibit minimal separation. This characterization enables the derivation of asymptotic properties for singular numbers in matrix products, particularly establishing the strong law of large numbers and central limit theorem for these quantities. Our results provide a unified framework connecting algebraic decomposition structures with probabilistic limit theorems in non-archimedean settings.