Non-Archimedean GUE corners and Hecke modules

TL;DR

This paper analyzes the distribution of singular numbers in $p$-adic Hermitian and alternating matrices, revealing connections to Hall-Littlewood processes and Hecke algebra modules, with implications for non-archimedean random matrix theory.

Contribution

It provides explicit joint distributions for principal corners of $p$-adic matrices and links these to Hall-Littlewood processes and Hecke modules, extending non-archimedean random matrix theory.

Findings

Distribution of singular numbers for $p$-adic matrices computed.

Alternating case yields a Hall-Littlewood process.

Hermitian case relates to a formal Hall-Littlewood process with signed transitions.

Abstract

We compute the joint distribution of singular numbers for all principal corners of a $p$-adic Hermitian (resp. alternating) matrix with additive Haar distribution, the non-archimedean analogue of the GUE (resp. aGUE) corners process. In the alternating case we find that it is a Hall-Littlewood process, explaining -- and recovering as a corollary -- results of Fulman-Kaplan. In the Hermitian case we obtain a `marginal distribution' of a formal Hall-Littlewood process with both positive and negative transition `probabilities'. The proofs relate natural random matrix operations to structural results of Hironaka and Hironaka-Sato on modules over the spherical Hecke algebra, yielding other probabilistic statements of independent interest along the way.