arXiv subjects

Maarten van Pruijssen

Publications and source records attributed to Maarten van Pruijssen.

14 recordsLinked to original sources

arXiv2026

An elementary approach to non-symmetric shift operators and their q-analogs

We give an algebraic construction of shift operators for the non-symmetric Heckman-Opdam polynomials and the non-symmetric Macdonald-Koornwinder polynomials. To each linear character of the finite Weyl group, we associate forward and backward shift operators, which are differential-reflection and difference-reflection operators that satisfy certain transmutation relations with the (Dunkl-)Cherednik operators. In the Heckman-Opdam case, the construction recovers the non-symmetric shift operators of Opdam and Toledano Laredo for the sign character. Furthermore, in rank one, we recover the rank-one non-symmetric shift operators previously obtained by the authors and Schlösser.

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arXiv2025

Multiplicity free induction for the pairs $(\mathrm{GL}_{2}\times\mathrm{GL}_{2},\mathrm{diag}(\mathrm{GL}_{2}))$ and $(\mathrm{SL}_{3},\mathrm{GL}_{2})$ over finite fields

We classify the irredible representations of $\mathrm{GL}_{2}(q)$ for which the induction to the product group $\mathrm{GL}_{2}(q)\times\mathrm{GL}_{2}(q)$, under the diagonal embedding, decomposes multiplicity free. It turns out that only the irreducible representations of dimensions $1$ and $q-1$ have this property. We show that for $\mathrm{GL}_{2}(q)$ embedded into $\mathrm{SL}_{3}(q)$ via $g\mapsto\mathrm{diag}(g,\det g^{-1})$ none of the irreducible representations of $\mathrm{GL}_{2}(q)$ induce multiplicity free. In contrast, over the complex numbers, the holomorphic representation theory of these pairs is multiplicity free and the corresponding matrix coefficients are encoded by vector-valued Jacobi polynomials. We show that similar results cannot be expected in the context of finite fields for these examples.

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arXiv2024

Non-symmetric Jacobi polynomials of type $BC_{1}$ as vector-valued polynomials Part 1: spherical functions

We study non-symmetric Jacobi polynomials of type $BC_{1}$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials yields a new expression of the non-symmetric Jacobi polynomials of type $BC_1$ in terms of the symmetric Jacobi polynomials of type $BC_{1}$. In this interpretation, the Cherednik operator, that has the non-symmetric Jacobi polynomials as eigenfunctions, corresponds to two shift operators for the symmetric Jacobi polynomials of type $BC_{1}$. We show that the non-symmetric Jacobi polynomials of type $BC_{1}$ with so-called geometric root multiplicities, interpreted as vector-valued polynomials, can be identified with spherical functions on the sphere $S^{2m+1}=\mathrm{Spin}(2m+2)/\mathrm{Spin}(2m+1)$ associated with the fundamental spin-representation of $\mathrm{Spin}(2m+1)$. The Cherednik operator corresponds to the Dirac operator for the spinors on $S^{2m+1}$ in this interpretation.

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arXiv2024

Non-symmetric Jacobi polynomials of type $BC_1$ as vector-valued polynomials Part 2: Shift operators

We study non-symmetric Jacobi polynomials of type $BC_1$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials allows us to introduce shift operators for the non-symmetric Jacobi polynomials. The shift operators are differential-reflection operators and we present four of these operators that are fundamental in the sense that they generate all shift operators. Moreover, the symmetrizations of these fundamental shift operators are the fundamental shift operators for the symmetric Jacobi polynomials of type $BC_1$. For the realization of non-symmetric Jacobi polynomials of type $BC_1$ as invariant $\mathbb{C}^2$-valued Laurent polynomials, we introduce a homomorphism that is analogous to the Harish-Chandra homomorphism for the symmetric Jacobi polynomials of type $BC_1$. For geometric root multiplicities, the non-symmetric Jacobi polynomials of type $BC_1$ can be interpreted as spherical functions and we show that our Harish-Chandra homomorphism in this context is related to the Lepowsky homomorphism via the radial part map.

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arXiv2023

Vector-valued Heckman-Opdam polynomials: a Steinberg variation

We develop a theory of Jacobi polynomials for parabolic subgroups of finite reflection groups that specializes to the cases studied by Heckman and Opdam in which the whole group and the trivial group are considered. For the intermediate cases we combine results of Steinberg and Heckman and Opdam to obtain new examples of families of vector-valued orthogonal polynomials with properties similar to those of the usual Jacobi polynomials. Most notably we show that these polynomials, when suitably interpreted as vector-valued polynomials, are determined up to scaling as simultaneous eigenfunctions of a commutative algebra of differential operators. We establish an example in which the vector-valued Jacobi polynomials can be identified with spherical functions for a higher $K$-type on a compact symmetric pair with restricted root system of Dynkin type $A_{2}$. We also describe how to obtain new examples of matrix-valued orthogonal polynomials in several variables.

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arXiv2021

On the extended weight monoid and its applications to orthogonal polynomials

Given a connected simply connected semisimple group G and a connected spherical subgroup K we determine the generators of the extended weight monoid of G/K, based on the homogeneous spherical datum of G/K. Let H be a reductive subgroup of G and let P be a parabolic subgroup of H for which G/P is spherical. A triple (G,H,P) with this property is called multiplicity free system and we determine the generators of the extended weight monoid of G/P explicitly in the cases where (G,H) is strictly indecomposable. The extended weight monoid of G/P describes the induction from H to G of an irreducible H-representation V whose lowest weight is a character of P. The space of regular End(V)-valued functions on G that satisfy F(hgk)=hF(g)k for all h,k in H and all g in G, is a module over the algebra of H-biinvariant regular functions on G. We show that under a mild assumption this module is freely and finitely generated. As a consequence the spherical functions of such a type V can be described as a family of matrix-valued orthogonal polynomials.

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arXiv2019

An application of spherical geometry to hyperkähler slices

This work is concerned with Bielawski's hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice to the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy slice $S\subseteq\mathfrak{g}:=\mathrm{Lie}(G)$, defining it to be the hyperkähler quotient of $T^*(G/H)\times (G\times S)$ by a maximal compact subgroup of $G$. This hyperkähler slice is empty in some of the most elementary cases (e.g. when $S$ is regular and $(G,H)=(\operatorname{SL}_{n+1},\operatorname{GL}_{n})$, $n\geq 3$), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$-regularity of $(G,H)$. This $\mathfrak{a}$-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$. We also provide a classification of the $\mathfrak{a}$-regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop's results on moment map images and Losev's algorithm for computing Cartan spaces.

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arXiv2017

Matrix elements of irreducible representations of $\mathrm{SU}(n+1)\times\mathrm{SU}(n+1)$ and multivariable matrix-valued orthogonal polynomials

In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case $\mathrm{SU}(n+1)$ meets all the conditions that we impose in Part 1. For any $k\in\mathbb{N}_{0}$ we obtain families of orthogonal polynomials in $n$ variables with values in the $N\times N$-matrices, where $N=\binom{n+k}{k}$. The case $k=0$ leads to the classical Heckman-Opdam polynomials of type $A_{n}$ with geometric parameter. For $k=1$ we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever $n\ge2$. We also give explicit expressions of the spherical functions that determine the matrix weight for $k=1$. These expressions are used to calculate the spherical functions that determine the matrix weight for general $k$ up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case $n=1$. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for $(n,k)$ equal to $(2,1)$ and $(3,1)$.

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arXiv2016

Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs

In this paper we present a method to obtain deformations of families of matrix-valued orthogonal polynomials that are associated to the representation theory of compact Gelfand pairs. These polynomials have the Sturm-Liouville property in the sense that they are simultaneous eigenfunctions of a symmetric second order differential operator and we deform this operator accordingly so that the deformed families also have the Sturm-Liouville property. Our strategy is to deform the system of spherical functions that is related to the matrix-valued orthogonal polynomials and then check that the polynomial structure is respected by the deformation. Crucial in these considerations is the full spherical function $Ψ_{0}$, which relates the spherical functions to the polynomials. We prove an explicit formula for $Ψ_{0}$ in terms of Krawtchouk polynomials for the Gelfand pair $(\mathrm{SU}(2)\times\mathrm{SU}(2),\mathrm{diag}(\mathrm{SU}(2)))$. For the matrix-valued orthogonal polynomials associated to this pair, a deformation was already available by different methods and we show that our method gives same results using explicit knowledge of $Ψ_{0}$. Furthermore we apply our method to some of the examples of size $2\times2$ for more general Gelfand pairs. We prove that the families related to the groups $\mathrm{SU}(n)$ are deformations of one another. On the other hand, the families associated to the symplectic groups $\mathrm{Sp}(n)$ give rise to a new family with an extra free parameter.

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arXiv2014

Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One

We present a method to obtain infinitely many examples of pairs $(W,D)$ consisting of a matrix weight $W$ in one variable and a symmetric second-order differential operator $D$. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs $(G,K)$ of rank one and a suitable irreducible $K$-representation. The heart of the construction is the existence of a suitable base change $Ψ_{0}$. We analyze the base change and derive several properties. The most important one is that $Ψ_{0}$ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group $G$ as soon as we have an explicit expression for $Ψ_{0}$. The weight $W$ is also determined by $Ψ_{0}$. We provide an algorithm to calculate $Ψ_{0}$ explicitly. For the pair $(\mathrm{USp}(2n),\mathrm{USp}(2n-2)\times\mathrm{USp}(2))$ we have implemented the algorithm in GAP so that individual pairs $(W,D)$ can be calculated explicitly. Finally we classify the Gelfand pairs $(G,K)$ and the $K$-representations that yield pairs $(W,D)$ of size $2\times2$ and we provide explicit expressions for most of these cases.

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arXiv2014

Multiplicity free induced representations and orthogonal polynomials

Let $(G,H)$ be a reductive spherical pair and $P\subset H$ a parabolic subgroup such that $(G,P)$ is spherical. The triples $(G,H,P)$ with this property are called multiplicity free systems and they are classified in this paper. Denote by $π^{H}_μ=\mathrm{ind}_{P}^{H}μ$ the Borel-Weil realization of the irreducible $H$-representation of highest weight $μ\in P^{+}_{H}$ and consider the induced representation $\mathrm{ind}_{P}^{G}χ_μ=\mathrm{ind}_{H}^{G}π^{H}_μ$, a multiplicity free induced representation. Some properties of the spectrum of the multiplicity free induced representations are discussed. For three multiplicity free systems the spectra are calculated explicitly. The spectra give rise to families of multi-variable orthogonal polynomials which generalize families of Jacobi polynomials: they are simultaneous eigenfunctions of a commutative algebra of differential operators, they satisfy recurrence relations and they are orthogonal with respect to integrating against a matrix weight on a compact subset. We discuss some difficulties in describing the theory for these families of polynomials in the generality of the classification.

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arXiv2013

Matrix Valued Orthogonal Polynomials for Gelfand Pairs of Rank One

In this paper we study matrix valued orthogonal polynomials of one variable associated with a compact connected Gelfand pair (G,K) of rank one, as a generalization of earlier work by Koornwinder and subsequently by Koelink, van Pruijssen and Roman for the pair (SU(2) x SU(2),SU(2)), and by Grünbaum, Pacharoni and Tirao for the pair (SU(3),U(2)). Our method is based on representation theory using an explicit determination of the relevant branching rules. Our matrix valued orthogonal polynomials have the Sturm--Liouville property of being eigenfunctions of a second order matrix valued linear differential operator coming from the Casimir operator, and in fact are eigenfunctions of a commutative algebra of matrix valued linear differential operators coming from $U(g_c)^K$.

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arXiv2012

Matrix-valued orthogonal polynomials related to (SU(2)\times SU(2),diag), II

In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the P_n's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials P_n. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2)\times SU(2).

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arXiv2011

Matrix Valued Orthogonal Polynomials related to (SU(2) x SU(2),diag)

The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are studied. By restriction to the subgroup A the matrix-valued spherical functions are diagonal. For suitable set of representations we take these diagonals into a matrix-valued function, which are the full spherical functions. Their orthogonality is a consequence of the Schur orthogonality relations. From the full spherical functions we obtain matrix-valued orthogonal polynomials of arbitrary size, and they satisfy a three-term recurrence relation which follows by considering tensor product decompositions. An explicit expression for the weight and the complete block-diagonalization of the matrix-valued orthogonal polynomials is obtained. From the explicit expression we obtain right-hand sided differential operators of first and second order for which the matrix-valued orthogonal polynomials are eigenfunctions. We study the low-dimensional cases explicitly, and for these cases additional results, such as the Rodrigues formula and being eigenfunctions to first order differential-difference and second order differential operators, are obtained.

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