The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials {P_n}_n, with respect to a weight W, that are eigenfunctions of a second order differential operator D. The weight W and the differential operator D were found in [12], using some aspects of the theory of the spherical functions associated to the complex projective spaces. We also find other second order differential operator E symmetric with respect to W and we describe the algebra generated by D and E.
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 dif…
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 ex…
The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. We generalize the Kudla-Millson relation between intersection numbers of cycles and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the generating series of the cycles give rise to vector-valued Siegel modular forms. The underlying correspondence between the highest weights of the orthogonal and the symplectic group coincides with the one obtained by Adams for which we provide a geometric interpretation.
We present a friendly introduction to the very detailed results in [9,10,11] and as an illustration we discuss here the issue of {\em linearization of products}. We find some interesting new phenomena.
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 matr…
We write spectral decomposition of the hypergeometric differential operator on the contour $Re z=1/2$ (multiplicity of spectrum is 2). As a result, we obtain an integral transform that differs from the Jacobi (or Olevsky) transform. We also try to do a step towards vector-valued special functions and construct a 3F2 orthogonal basis in the space of functions having values in 2-dimensional space. This basis is lying in an analytic continuation of continuous dual Hahn polynomials with respect to number $n$ of a polynomial. In Addendum, we discuss a vector-value analog of the Meixner--Pollachek o…
The main purpose of this paper is to compute all irreducible spherical functions on $G={SL}(2,{\mathbb C})$ of arbitrary type $δ\in \hat K$, where $K={SU}(2)$. This is accomplished by associating to a spherical function $Φ$ on $G$ a matrix valued function $H$ on the three dimensional hyperbolic space ${\mathbb H}=G/K$. The entries of $H$ are solutions of two coupled systems of ordinary differential equations. By an appropriate twisting involving Hahn polynomials we uncouple one of the systems and express the entries of $H$ in terms of Gauss' functions ${}_2F_1$. Just as in the compact instance…
For each irreducible module of the symmetric group on $N$ objects there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to certain Hermitian forms. These polynomials were studied by the author and J.-G. Luque using a Yang-Baxter graph technique. This paper constructs a matrix-valued measure on the $N$-torus for which the polynomials are mutually orthogonal. The construction uses Fourier analysis techniques. Recursion relations for t…
This paper presents theoretical analysis and software implementation for real harmonics analysis on the special orthogonal group. Noncommutative harmonic analysis for complex-valued functions on the special orthogonal group has been studied extensively. However, it is customary to treat real harmonic analysis as a special case of complex harmonic analysis, and there have been limited results developed specifically for real-valued functions. Here, we develop a set of explicit formulas for real-valued irreducible unitary representations on the special orthogonal group, and provide several operat…
The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in \cite{T2}. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in \cite{T2} to identify the corresponding matrix hypergeometric functions ${}_nF_m$. We prove that, if n=m+1, this functions are analytic …
Algebraic interpretation of discrete families of matrix valued orthogonal polynomials
Centre de recherches mathématiques, Université de Montréal（蒙特利尔大学数学研究中心）；IVADO and Centre de recherches mathématiques, Université de Montréal（IVADO与蒙特利尔大学数学研究中心）
The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of arbitrary type $δ\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by associating to a spherical function $Φ$ on $G$ a matrix valued function $H$ on the complex projective plane $P_2(\mathbb{C})=G/K$. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair $(G,K)$. But the relation of spherical functions…
It is well-known that polynomials decompose into spherical harmonics. This result is called separation of variables or the Fischer decomposition. In the paper we prove the Fischer decomposition for spinor valued polynomials in $k$ vector variables of ${\mathbb R}^m$ under the stable range condition $m\geq 2k$. Here the role of spherical harmonics is played by monogenic polynomials, that is, polynomial solutions of the Dirac equation in $k$ vector variables.
The spaces of invariants and the zonal spherical functions associated with quantum super 2-shpheres defined by $\Bbb{C}_{q}(osp(1,2))$ are discussed. Connection between the zonal spherical functions and orthogonal $q$-polynomials from the Askey-Wilson scheme is investigated.
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 o…
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 u…
The aim of this paper is to determine all irreducible spherical functions of the pair (G,K)=(SU(n+1),U(n)), where the highest weight of their K-types are of the form (m+l,...,m+l,m,...,m). Instead of looking at a spherical function Φof type πwe look at a matrix-valued function H defined on a section of the K-orbits in an affine subvariety of P_n(C). The function H diagonalizes, hence it can be identified with a column vector-valued function. The irreducible spherical functions of type πturn out to be parameterized by S={(w,r)\in Z x Z : 0\leq w, 0 \leq r \leq l, 0\leq m+w+r}. A key result to c…
In this work we start by determining all irreducible spherical functions $Φ$ of any $K $-type associated to the pair $(G,K)=(\SO(4),\SO(3))$. The functions $P=P(u)$ corresponding to the irreducible spherical functions of a fixed $K$-type $π_\ell$ are appropriately packaged into a sequence of matrix valued polynomials $(P_w)_{w\ge0}$ of size $(\ell+1)\times(\ell+1)$. Finally we prove that $\widetilde P_w={P_0}^{-1}P_w$ is a sequence of matrix orthogonal polynomials with respect to a weight matrix $W$. Moreover, we show that $W$ admits a second order symmetric hypergeometric operator $\widetilde…
We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explicitly.Symmetry breaking operators at exceptional discrete parameters are thoroughly studied. We obtain closed formulae for the functional equations which the composition of the the symmetry breaking operators with the Knapp-Stein intertwining operators of $G$ and G' satisfy, and u…
We prove the existence of $\mathrm{GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of $\mathrm{GSO}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb{H}}$, arising from forms of $\mathrm{GSO}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (…
In this paper, we determine all irreducible spherical functions Φof any K -type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by associating to Φa vector valued function H=H(u) of a real variable u, which is analytic at u=0 and whose components are solutions of two coupled systems of ordinary differential equations. By an appropriate conjugation involving Hahn polynomials we uncouple one of the systems. Then this is taken to an uncoupled system of hypergeometric equations, leading to a vector valued solution P=P(u) whose entries are Gegenbauer's polynomials. Afterward, we …
Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as sl(2,C)-modules. As finite-dimensional irreducible sl(2,C)-modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Guerlebeck. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.
There is a space of vector-valued nonsymmetric Jack polynomials associated with any irreducible representation of a symmetric group. Singular polynomials for the smallest singular values are constructed in terms of the Jack polynomials. The smallest singular values bound the region of positivity of the bilinear symmetric form for which the Jack polynomials are mutually orthogonal. As background there are some results about general finite reflection groups and singular values in the context of standard modules of the rational Cherednik algebra.
We give a method of decomposing bundle-valued polynomials compatible with the action of the Lie group $Spin(n)$, where important tools are $Spin(n)$-equivariant operators and their spectral decompositions. In particular, the top irreducible component is realized as an intersection of kernels of these operators.
Let $R$ be a root system of type BC in $\mathfrak a=\mathbb R^r$ of general positive multiplicity. We introduce certain canonical weight function on $\mathbb R^r$ which in the case of symmetric domains corresponds to the integral kernel of the Berezin transform. We compute its spherical transform and prove certain Bernstein-Sato type formula. This generalizes earlier work of Unterberger-Upmeier, van Dijk-Pevsner, Neretin and the author. Associated to the weight functions there are Heckman-Opdam orthogonal polynomials of Jacobi type on the compact torus, after a change of variables they form an…
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 …
The random walk to be considered takes place in the d- spherical dual of the group U(n + 1), for a fixed finite dimensional irreducible representation d of U(n). The transition matrix comes from the three term recursion relation satisfied by a sequence of matrix valued orthogonal polynomials built up from the irreducible spherical functions of type d of SU(n + 1). One of the stochastic models is an urn model and the other is a Young diagram model.
In this paper, we give the expressions for the bounded spherical functions, or equivalently the spherical functions of positive type, for the free two-step nilpotent Lie groups endowed with the actions of orthogonal groups or their special subgroups. Next we deduce some results about the (Kohn) sub-Laplacian, and we compute the radial Plancherel measure.
In this paper, we give an alternative proof of separation of variables for scalar-valued polynomials $P:(\mathbb R^m)^k\to\mathbb C$ in the semistable range $m\geq 2k-1$ for the symmetry given by the orthogonal group $O(m)$. It turns out that uniqueness of the decomposition of polynomials into spherical harmonics is equivalent to irreducibility of generalized Verma modules for the Howe dual partner $sp(2k)$ generated by spherical harmonics. We believe that this approach might be applied to the case of spinor-valued polynomials and to other settings as well.
We describe the asymptotic behavior of the multivariate BC-type Jacobi polynomials as the number of variables and the Young diagram indexing the polynomial go to infinity. In particular, our results describe the approximation of the spherical functions of the infinite-dimensional symmetric spaces of type B,C,D or BC by the spherical functions of the corresponding finite-dimensional symmetric spaces. Similar results for the Jack polynomials were established in our earlier paper (Intern. Math. Res. Notices 1998, no. 13, 641-s682; arXiv:q-alg/9709011). The main results of the present paper were o…
