Non-Archimedean GUE corners and Hecke modules
Abstract
We compute the joint distribution of singular numbers for all principal corners of a -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.
Keywords
Cite
@article{arxiv.2412.05999,
title = {Non-Archimedean GUE corners and Hecke modules},
author = {Jiahe Shen and Roger Van Peski},
journal= {arXiv preprint arXiv:2412.05999},
year = {2025}
}
Comments
43 pages. v2: correction to v1, which erroneously stated that the proofs were valid in positive characteristic as well as for p-adic fields. The new Appendix A explains what details are missing from the literature which would be needed to conclude this. Various other minor typos corrected