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In this article, we classify the multiplicity free systems and we give examples of families of matrix valued orthogonal polynomials that generalize Jacobi polynomials with geometric parameters. Let   $(G,H)$   be a reductive spherical pair and   $P\subset H$   a parabolic subgroup such that   $(G,P)$   is a spherical pair. The triples   $(G,H,P)$   are called multiplicity free systems. We fix one and let   $\mu\in P^{+}_{H}$   be a character of   $P$  . We denote by   $\pi^{H}_{\mu}=\mathrm{ind}_{P}^{H}(\mu)$   the Borel-Weil realization of the irreducible   $H$  -representation of highest weight   $\mu$   and consider the representation   $\mathrm{ind}_{P}^{G}(\mu)=\mathrm{ind}_{H}^{G}(\pi^{H}_{\mu})$   of   $G$  , called a multiplicity free induced representation of   $G$  . For three multiplicity free systems, the spectra are calculated explicitly. The spectra give rise to families of multi-variable matrix valued orthogonal polynomials which generalize families of Jacobi polynomials.

Multiplicity Free Induced Representations and Orthogonal Polynomials