Spherical functions of fundamental \(K\)-types associated with the \(n\)-dimensional sphere (Q401260)

In this lengthy paper, the authors study irreducible spherical functions for which the first results date back to 1950 (Gel'fand; in a special case there is a one-to-one correspondence between all zonal spherical functions and a sequence of orthogonal polynomials).NEWLINENEWLINEAn important step was taken in a paper by \textit{F. A. Grünbaum} et al. [J. Funct. Anal. 188, No. 2, 350--441 (2002; Zbl 0996.43013)] and the present work is an extension of results from Chapter 5 in the authors' paper [J. Lie Theory 24, No. 1, 147--157 (2014; Zbl 1291.43010)]. NEWLINENEWLINENEWLINE\S1. Introduction [3 pages]. NEWLINENEWLINENEWLINE\S2. Preliminaries [\(6{1\over 2}\) pages]. NEWLINENEWLINENEWLINESpherical functions, root space structure of \(\mathbf{so}(n,\mathbb{C})\) (Lie algebra), special operators, Gel'fand-Tsetlin basis with irreducible representations for the chain of subgroups \(\mathrm{SO}(n-2)>\mathrm{SO}(n-3)>\cdots >\mathrm{SO}(3)>\mathrm{SO}(2)\).NEWLINENEWLINENEWLINE\S3. The differential operator \(\Delta\) [\(3{1\over 2}\) pages].NEWLINENEWLINENEWLINENEWLINENEWLINEThe operator works on \(\mathrm{SO}(n+1)\) and is given by NEWLINE\[NEWLINE\Delta=\sum_{j=1}^n\,I_{n+1,j}^2,NEWLINE\]NEWLINE where \(I_{n+1,j}=E_{j,n+1}-E_{n+1,j}\) and \(I_{ik}\) is a square matrix with \(1\) in the \(ik\) position and zeros elsewhere. NEWLINENEWLINENEWLINEStudy of eigenfunctions and eigenvalues in order to give insight into fundamental \(K\)-types associated with the pair \((G,K)= (\mathrm{SO}(n+1),\mathrm{SO}(n))\). NEWLINENEWLINENEWLINE\S4. The \(K\)-types which are \(M\)-irreducible [\(2{1\over 2}\) pages].NEWLINENEWLINENEWLINEHere \(K=\mathrm{SO}(n),\;M=\mathrm{SO}(n)\) with \(n=2\ell +1\); the main result isNEWLINENEWLINE{ Theorem 4.2}. The scalar valued functions \(H=h\), associated with the irreducible spherical functions on \(\mathrm{SO}(n+1),\;n=2\ell\), of \(\mathrm{SO}(n)\)-type \(m_n=(d,\ldots,d,\pm d)\in\mathbb{C}^{\ell}\), are parametrized by the integers \(w\geq d\) and explicitly given by NEWLINE\[NEWLINEh_w(y)=uy^d{}_2F_1\left(\begin{matrix} d-w,2\ell +d+w-1 \\ 2d+\ell;y\end{matrix}\right)NEWLINE\]NEWLINE where the constant \(u\) is determined by the condition \(h_w(1)=1\). NEWLINENEWLINENEWLINENEWLINENEWLINE\S5. The operator \(\Delta\) for the fundamental \(K\)-types [4 pages].NEWLINENEWLINENEWLINEHere explicit expressions for the cases \(K=\mathrm{SO}(2\ell)\) and \(K=\mathrm{SO}(2\ell +1)\) are given (fundamental \(K\)-modules, fundamental weights). NEWLINENEWLINENEWLINE\S6. The spherical functions of fundamental \(K\)-types [\(3{1\over 2}\) pages].NEWLINENEWLINE Role of \(2\times 2\) matrix-valued functions and their hypergeometric differential equation. Furthermore \(\Delta\)-eigenvalues of spherical functions and polynomial eigenfunction of the hypergeometric operator \textbf{D} arise, NEWLINE\[NEWLINE\mathbf{D}P=y(1-y)P''+(C-yU)P'-VP,NEWLINE\]NEWLINE with NEWLINE\[NEWLINEC=\left(\begin{matrix} (n/2+1) & 1 \\ 1 & (n/2+1)\end{matrix}\right),\;U=(n+2)I,\;V=\left(\begin{matrix} p & 0 \\ 0 & n-p\end{matrix}\right),NEWLINE\]NEWLINE and \(n\) is of the form \(2\ell\) or \(2\ell +1\) for \(\ell\in\mathbb{N}\) and \(1\leq p<\ell\).NEWLINENEWLINEThe main result states the eigenvalues polynomial eigenfunctions (from \(\mathbb{C}^2\)) to be \(-w(w+n+1)-p\) or \(-w(w+n+1)-n+p\) with \(w\in\mathbb{N}_0\) with the degree of the polynomial being \(w\) and the leading coefficient a multiple of \(\left(\begin{matrix}1\\ 0\end{matrix}\right)\) or \(\left(\begin{matrix} 0\\ 1\end{matrix}\right)\) respectively.NEWLINENEWLINENEWLINE\S7. The inner product [6 pages].NEWLINENEWLINE Using Haar measure and Schur's orthogonality relations the authors proveNEWLINENEWLINE{ Proposition 7.1.} If \(\Phi_1\), \(\Phi_2\) are two irreducible spherical functions of type \(\pi\in \tilde{K}\), then NEWLINE\[NEWLINE\langle\Phi_1,\Phi_2\rangle={(n--1)!!\over (n-2)!!}{2\over \omega_{*}}\sum_{i=1}^m\,(y(1-y))^{n/2-1}h_i(y) \overline{f_i(y)} dy,NEWLINE\]NEWLINE with \(\omega^{*}=\pi\) if \(n\) is even and \(\omega^{*}=2\) if \(n\) is odd. NEWLINENEWLINENEWLINE\S8. Matrix valued orthogonal polynomials [\(1{1\over 2}\) pages].NEWLINENEWLINE Construction of a sequence of matrix valued polynomials related to irreducible spherical functions of type \(\pi\in {\hat{\mathrm{SO}}}(n)\) of highest weight \(m_p=(1,\ldots,1,0,\ldots,0)\in\mathbb{C}^{\ell}\) with \(p\) ones. NEWLINENEWLINENEWLINE\S9. The \(\mathrm{SO}(2\ell +1)\)-type with heighest weight \(2\lambda_{\ell}\) [8 pages].NEWLINENEWLINE Spherical functions, hypergeometric sidderential equation and vector valued eigenfunctions.NEWLINENEWLINENEWLINEAppendix [\(1{1\over 2}\) pages].NEWLINENEWLINENEWLINEProof of Proposition 3.2.NEWLINENEWLINENEWLINEReferences [1 page].NEWLINENEWLINENEWLINEThe list contains 25 items.