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)

Abstract

超幾何型・選点系の直交多項式であるKrawtchouk多項式は, 対称群の輪状積(wreath product)の帯球関数という群論的解釈を持つ(Dunkl, 1976). その一般化として, 多変数Krawtchouk多項式は複素鏡映群上の帯球関数という解釈を持つ(Mizukawa, 2004). またq-analogueの一つであるaffine q-Krawtchouk多項式についても, 有限体上の行列群の帯球関数という解釈がある(Delsarte, 1978). そこで本稿では, まずKrawtchouk多項式の新たな一般化として, ∞変数Krawtchouk多項式, 多変数affine q-Krawtchouk多項式, ∞変数affine q-Krawtchouk多項式を定義する. そして, それらが全て一貫した形で, 有限体および非アルキメデス的局所体に関連する群の帯球関数として捉えられることを述べる.

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.