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Non-Archimedean GUE corners and Hecke modules
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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.
Forward citations
Cited by 2 Pith papers
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Hall-Littlewood-positive harmonic functionals on the algebra of symmetric functions
The cone of p2-harmonic (−t)-Hall-Littlewood-positive functionals on symmetric functions contains explicit copies of the classical cone Φ(t²) and is closed under a new twisted Kerov mixing operation.
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Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism
For products of bi-invariant random matrices over non-archimedean fields, singular numbers obey an SLLN and CLT with limits given by the corners, extending known type-A/type-C results to all split reductive groups.
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