Hermite polynomials and representations of the unitary group

30th International Colloquium on Group Theoretical Methods in Physics

Strasburger, Aleksander^1 ; Dziewa-Dawidczyk, D.^1

Department of Aplied Mathematics, Warsaw University of Life Sciences (SGGW), Nowoursynowska 159, Warsaw

02-787, Poland^1

关键词: Classical orthogonal polynomials;    Complex polynomials;    Gegenbauer;    Hermite polynomials;    Unitary group;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012070/pdf
DOI  :  10.1088/1742-6596/597/1/012070

来源: IOP

【 摘 要 】

Spaces of homogeneous complex polynomials in D variables form carrier spaces for representations of the unitary group U(D). These representations are well understood and their connections with certain families of classical orthogonal polynomials (Gegenbauer, Jacobi, and other) are widely studied. However, there is another realization for the action of the unitary group U(D) on polynomials, not necessarily homogeneous, in which Hermite polynomials in D variables play an important role. This action is related to the metaplectic (oscillator) representation, and was studied some time ago by one of the present authors (A. S.) and, independently, by A. Wunsche for D = 2. In this note we want to concentrate on the latter realization and describe its properties in a more comprehensible way.