Multivariate $q$-Hypergeometric polynomials as zonal spherical functions over a local field (Various Issues relating to Representation Theory and Non-commutative Harmonic Analysis)

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  • Kawamura, Koei
    Department of Mathematics, Faculty of Science, Kyoto University

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Other Title
  • 局所体上の帯球関数として現れる多変数$q$-超幾何多項式 (表現論と非可換調和解析をめぐる諸問題)
  • 局所体上の帯球関数として現れる多変数q-超幾何多項式
  • キョクショタイ ジョウ ノ タイキュウ カンスウ ト シテ アラワレル タヘンスウ q-チョウキカ タコウシキ

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Abstract

Krawtchouk polynomials are orthogonal polynomials which are defined by the use of the hypergeometric function. They have a group theoretic intepretation as zonal spherical functions on wreath products of symmetric groups (Dunkl, 1976). As a generalization, multivariate Krawtchouk polynomials have an intepretation as zonal spherical functions on complex reflection groups (Mizukawa, 2004). And affine q-Krawtchouk polynomials, one of q-analogues of Krawtchouk polynomials, are also zonal spherical functoins on matrices over a finite field (Delsarte, 1978). In this paper we define new generalizations of Krawtchouk polynomials, that is, ∞-variate Krawtchouk polynomials, multivariate affine q-Krawtchouk polynomials, and ∞-variate affine q-Krawtchouk polynomials. And we show they have also interpretations as zonal spherical functoins on groups concerning finite or non-Archimedean local field.

Journal

  • RIMS Kokyuroku

    RIMS Kokyuroku 2031 15-32, 2017-05

    京都大学数理解析研究所

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