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Given a compact Gelfand pair \((G,K)\) of rank one, the induction of the trivial
\(K\)-representation to \(G\)
decomposes multiplicity free as a \(G\)-module. In the harmonic analysis on \(G/K\)
this is reflected in the fact that the algebra of \(G\)-invariant differential
operators on \(G/K\) is commutative. This, in turn, brings a family of Jacobi
polynomials into the game, as simultaneous eigenfunctions for the \(G\)-invariant
differential operators on \(G/K\).
There are more irreducible \(K\)-representations \(\pi^{K}\) whose induction to \(G\)
decomposes
multiplicity free as a \(G\)-module. In fact, the triples \((G,K,\pi^{K})\) with
this property have been classified recently. In the case that \((G,K)\) is of rank
one, such a triple \((G,K,\pi^{K})\) gives rise to a family of matrix valued
orthogonal polynomials with properties that are similar to those of a family of
Jacobi polynomials. For some higher rank examples we find similar families of
orthogonal polynomials, now in several variables.
In this talk I will report on this research and on possible applications to the
harmonic
analysis on homogeneous vector bundles over non-compact homogeneous spaces, that
are subject to similar multiplicity free regime.
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