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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 \(G^*\).

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