In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type $BC$. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds $G/K$ over one of the (skew) fields $\mathbb F= \mathbb R, \mathbb C, \mathbb H.$ We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of $K$-biinvariant functions on $G$.
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…
Calogero-Sutherland hyperbolic system and Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Steklov Mathematical Institute（斯捷克洛夫数学研究所）；Department of Mathematics, Technion（以色列理工学院数学系）；National Research University Higher School of Economics（国立研究高等经济学院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）
We express explicitly the Heckman-Opdam hypergeometric function for the root system of type A with a certain degenerate parameter in terms of the Lauricella hypergeometric function.
In this article, we conjecture exact estimates for the Weyl-invariant Opdam-Cherednik hypergeometric functions. We prove the conjecture for the root system $A_n$ and for all rank 1 cases. We provide other evidence that the conjecture might be true in general.
For a connected semisimple real Lie group $G$ of non-compact type, Wallach introduced a class of $K$-types called small. We classify all small $K$-types for all simple Lie groups and prove except just one case that each elementary spherical function for each small $K$-type $(π,V)$ can be expressed as a product of hyperbolic cosines and a Heckman-Opdam hypergeometric function. As an application, the inversion formula for the spherical transform on $G\times_K V$ is obtained from Opdam's theory on hypergeometric Fourier transforms.
We continue a program generalizing classical results from the analysis on symmetric cones to the Dunkl setting for root systems of type A. In particular, we prove a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions of type A and more generally, for the associated Cherednik kernel. This is achieved by analytic continuation from a Laplace transform identity for non-symmetric Jack polynomials which was stated, for the symmetric case, as a key conjecture in an unpublished manuscript of Macdonald (2013). Our proof for the Jack polynomials is based on Dunkl operator techniq…
The spherical functions of the noncompact Grassmann manifolds over the real or complex numbers or the quaternions with rank q and dimension parameter p can be seen as Heckman-Opdam hypergeometric functions of type BC, when the double coset space is identified with some Weyl chamber of type B. The associated double coset hypergroups may be embedded into a continuous family of commutative hypergroups with these hypergeometric functions as multiplicative functions with p in some continuous parameter range by a result of Rösler. Several limit theorems for random walks associated with these hypergr…
Laplace-type representation for some generalized spherical functions of type BC
在[M. Rösler和M.沃特整表示and Uniform Limits for Some Heckman-Opdam超超几何函数数of type BC，Transactions of the American Mathematical Society，Vol. 368，No. 8，6005-6032，2016.]，Rösler和Voit利用A型球函数给出了$BC$型广义球函数的一个公式.我们用这个公式精确地描述了相关的广义Abel变换的支撑。此外，我们得到了一个类似的公式的广义球函数的有理Dunkl设置。还推导出了交织算子V的支集。我们还表明，作为一个结果，拉普拉斯型表达式存在的广义球函数在三角Dunkl设置和合理的Dunkl设置。
The main aim is to obtain integral formulas for DAHA coinvariants and the corresponding inner products for any values of the DAHA parameters. In the compact case, our approach is similar to the procedure of ``picking up residues" due to Arthur, Heckman, Opdam and others; the resulting formula is a sum of integrals over double affine residual subtori. A single real integral provides the required formula in the noncompact case. As q tends to 0, our integral formulas result in the trace formulas for the corresponding AHA, which calculate the Plancherel measures for the spherical parts of the regu…
Let $V_k$ denote Dunkl's intertwining operator associated with some root system $R$ and multiplicity function $k$. For two multiplicities $k, k^\prime$ on $R$, we study the operator $V_{k^\prime,k} = V_{k^\prime}\circ V_k^{-1}$, which intertwines the Dunkl operators for multiplicity $k$ with those for multiplicity $k^\prime.$ While it is well-known that the operator $V_k$ is positive for nonnegative $k$, it has been a long-standing conjecture that its generalizations $V_{k^\prime,k}$ are also positive if $k^\prime \geq k \geq 0,$ which is known to be true in rank one. In this paper, we disprov…
