Classical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated with discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$. We study some basic results for this notion and present several classes of examples. It turns out that, for a given commutative hypergroup, the existence of a related continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We, in particular, show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.

We prove that all convolution products of pairs of continuous orbital measures in rank one, compact symmetric spaces are absolutely continuous and determine which convolution products are in $L^{2}$ (meaning that their density function is in $L^{2}$). We characterise the pairs whose convolution product is either absolutely continuous or in $L^{2}$ in terms of the dimensions of the corresponding double cosets. In particular, we prove that if $G/K$ is not $\text{SU}(2)/\text{SO}(2)$, then the convolution of any two regular orbital measures is in $L^{2}$, while in $\text{SU}(2)/\text{SO}(2)$ there are no pairs of orbital measures whose convolution product is in $L^{2}$.

We study the absolute continuity of the convolution ${\it\delta}_{e^{X}}^{\natural }\star {\it\delta}_{e^{Y}}^{\natural }$ of two orbital measures on the symmetric spaces $\mathbf{SO}_{0}(p,p)/\mathbf{SO}(p)\times \mathbf{SO}(p)$, $\mathbf{SU}(p,p)/\mathbf{S}(\mathbf{U}(p)\times \mathbf{U}(p))$ and $\mathbf{Sp}(p,p)/\mathbf{Sp}(p)\times \mathbf{Sp}(p)$. We prove sharp conditions on $X$, $Y\in \mathfrak{a}$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.

Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.

The Casselman–Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of $p$-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of $\text{G}{{\text{L}}_{n}}$, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y.Hironaka, and in effect reduce many such problems to straightforward calculations on the group.

In this paper, we consider the set $\text{S}=a\left( {{e}^{X}}K{{e}^{Y}} \right)$ where $a\left( g \right)$ is the abelian part in the Cartan decomposition of $g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of $\text{SL}\left( 3,\,\mathbf{F} \right)\,\text{where}\,\mathbf{F}\,=\,\mathbf{R},\,\mathbf{C}\,\text{or}\,\mathbf{H}$. In particular, we show that $\text{S}$ is convex.

We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.

For the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer polynomials, Hahn polynomials or continuous symmetric Hahn polynomials. We give a group theoretic explanation of this phenomenon and relate the properties of the polynomials p to the properties of the corresponding representation. The method is extended to the case of intertwining functions.

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.