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<p>We establish that the singular numbers (arising from Cartan decomposition) and corners (emerging from Iwasawa decomposition) in split reductive groups over non-archimedean local fields are fundamentally determined by Hall-Littlewood polynomials. Through applications of the Satake isomorphism, we extend Van Peski’s results [<CitationRef CitationID="CR30">30</CitationRef>, Theorem 1.3] to encompass arbitrary root systems. Leveraging this theoretical foundation, we further develop Shen’s work [<CitationRef CitationID="CR28">28</CitationRef>, 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.</p>

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Gaussian universality of products over split reductive groups and the Satake isomorphism

  • 期刊论文

  • 发表日期：

    2026年7月16日

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  • Jiahe Shen

摘要

We establish that the singular numbers (arising from Cartan decomposition) and corners (emerging from Iwasawa decomposition) in split reductive groups over non-archimedean local fields are fundamentally determined by Hall-Littlewood polynomials. Through applications of the Satake isomorphism, we extend Van Peski’s results [30, Theorem 1.3] to encompass arbitrary root systems. Leveraging this theoretical foundation, we further develop Shen’s work [28, 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.