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Berezin transform on compact Hermitian symmetric spaces. (English) Zbl 0920.22008
A. Unterberger and H. Upmeier [Math. Commun. Phys. 164, 563-597 (1994; Zbl 0843.32019)] studied the Berezin transform on the irreducible noncompact Hermitian symmetric space $$D = G/K$$ and obtained the spectral decomposition of the Berezin transform operator on $$L^2 (D)$$ under the irreducible decomposition of $$L^2(D)$$ into irreducible representations of $$G$$. In this paper the author considers the analogous but more difficult case of determining the spectrum of the Berezin transform on $$L^2(X)$$, where $$X=G^*/K$$ is the compact dual of $$G/K$$ by decomposing $$L^2(X)$$ into the irreducible representations of $$G^*$$. As applications the author obtains the expansion of powers of the canonical polynomial in terms of the spherical polynomials of the symmetric space $$G^*/K$$ and determines the irreducible decomposition of the tensor products of irreducible representations of $$G^*$$.

MSC:
 22E46 Semisimple Lie groups and their representations 47B38 Linear operators on function spaces (general)
Zbl 0843.32019
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