Non-Archimedean GUE corners and Hecke modules

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.