Related papers: Associated functions on $SO(p,q)$ Groups
Explicit expressions for zonal spherical functions of $SO(p,q)$ matrix groups are obtained using a generalized hypergeometric series of two variables.
Explicit expressions for associated spherical functions of $SO(p,q)$ matrix groups are obtained using a generalized hypergeometric series of two variables. In this paper, we present explicit expressions for zonal functions of de Sitter…
We give explicit models for spherical functions on $p$-adic symmetric spaces $X=H\backslash G$ for pairs of $p$-adic groups $(G,H)$ of the form $(\mathrm{U}(2r),\mathrm{U}(r)\times \mathrm{U}(r)),$ $(\mathrm{O}(2r),\mathrm{O}(r)\times…
We give a general expression of spherical functions on $p$-adic homogeneous spaces of $G$, based on data of $G$ and functional equations of spherical functions. Then, we show a unified method to obtain functional equations of spherical…
In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=({\mathrm{SO}}(n+1),{\mathrm{SO}}(n))$ in terms of matrix hypergeometric functions. The output of this description is…
A double covering of the proper orthochronous Lorentz group is understood as a complexification of the special unimodular group of second order (a double covering of the 3-dimensional rotation group). In virtue of such an interpretation the…
Homogeneous superspaces arising from the general linear supergroup are studied within a Hopf algebraic framework. Spherical functions on homogeneous superspaces are introduced, and the structures of the superalgebras of the spherical…
Matrix elements of spinor and principal series representations of the Lorentz group are studied in the basis of complex angular momentum (helicity basis). It is shown that matrix elements are expressed via hyperspherical functions…
This paper is devoted to the representations of the groups $SO (2,1)$ and $ISO (2,1)$. Those groups have an important role in cosmology, elementary particle theory and mathematical physics. Irreducible unitary representations of the…
The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of arbitrary type $\delta\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by…
We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$. The irreducible $K$ representations $(\pi,V)$ in the ${\rm U}(n)$ part are considered…
The main purpose of this paper is to compute all irreducible spherical functions on $G={SL}(2,{\mathbb C})$ of arbitrary type $\delta\in \hat K$, where $K={SU}(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a…
The representation theory of the quantum group su$_q(2)$ is used to introduce $q$-analogues of the Wigner rotation matrices, spherical functions, and Legendre polynomials. The method amounts to an extension of variable separation from…
This paper provides a classification of analytic actions of the semi-orthogonal group $\text{SO}^\circ(p,q)$, for $p,q \geq 3$, on closed, connected $(p+q-1)$-dimensional manifolds. Adapting Uchida's construction of $\text{SO}^\circ(p,q)$…
In this article, using the notion of group contraction, we obtain the spherical functions of the strong Gelfand pair $(\mathrm{M}(n),\mathrm{SO}(n))$ as an appropriate limit of spherical functions of the strong Gelfand pair…
Recently, various extensions and variants of Bessel functions of several kinds have been presented. Among them, the $(p,q)$-confluent hypergeometric function $\Phi_{p,q}$ has been introduced and investigated. Here, we aim to introduce an…
We consider (projectively) linearly sofic groups, i.e. groups which can be approximated using (projective) matrices over arbitrary fields, as a generalization of sofic groups. We generalize known results for sofic groups and groups which…
In this work we start by determining all irreducible spherical functions $\Phi$ 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…
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…
We show a connection formula for the $q$-confluent hypergeometric functions ${}_2\varphi_1(a,b;0;q,x)$. Combining our connection formula with Zhang's connection formula for ${}_2\varphi_0(a,b;-;q,x)$, we obtain the connection formula for…