Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds
Abstract
We introduce the radial part of the Laplace-Beltrami operator on partial flag manifolds and study its eigenfunctions, the elementary zonal spherical functions. These eigenfunctions can be related to orthogonal polynomials on the simplex of Hermitian matrices invariant under simultaneous conjugation by unitary matrices with respect to the complex matrix-variate Dirichlet distribution. Using a matrix-variate version of Koornwinder's method for constructing orthogonal polynomials in multiple variables from orthogonal polynomials in one variable, we construct a family of pairwise orthogonal polynomials in terms of Hermitian Jacobi polynomials and conjecture that some of these are elementary zonal spherical functions. Furthermore, we recall the definition of multivariate Schur polynomials and show how they can be used to obtain information about elementary zonal spherical functions. In particular, we give some partial results in the direction that the radial part of the Laplace-Beltrami operator on partial flag manifolds is strongly triangular with respect to the multivariate Schur polynomials. The elementary zonal spherical functions can then be constructed by applying the Gram-Schmidt orthogonalisation process to the multivariate Schur functions.
Related
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BibTeX
@misc{kuijper2026jacobi,
title = {{Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds}},
author = {Teije Kuijper},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.25760},
eprint = {2607.25760},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.25760},
}
RIS
TY - UNPB
TI - Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds
AU - Teije Kuijper
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2607.25760
UR - https://arxiv.org/abs/2607.25760
AB - We introduce the radial part of the Laplace-Beltrami operator on partial flag manifolds and study its eigenfunctions, the elementary zonal spherical functions. These eigenfunctions can be related to orthogonal polynomials on the simplex of Hermitian matrices invariant under simultaneous conjugation by unitary matrices with respect to the complex matrix-variate Dirichlet distribution. Using a matrix-variate version of Koornwinder's method for constructing orthogonal polynomials in multiple variables from orthogonal polynomials in one variable, we construct a family of pairwise orthogonal polynomials in terms of Hermitian Jacobi polynomials and conjecture that some of these are elementary zonal spherical functions. Furthermore, we recall the definition of multivariate Schur polynomials and show how they can be used to obtain information about elementary zonal spherical functions. In particular, we give some partial results in the direction that the radial part of the Laplace-Beltrami operator on partial flag manifolds is strongly triangular with respect to the multivariate Schur polynomials. The elementary zonal spherical functions can then be constructed by applying the Gram-Schmidt orthogonalisation process to the multivariate Schur functions.
ER -