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Stratonovich-Weyl correspondence for compact semisimple Lie groups. (English) Zbl 1218.22008
Let \(G\) be a compact semisimple Lie group with Lie algebra \(\mathfrak{g}\) and \(\pi_{\lambda}\) a unitary irreducible representation of \(G\) with highest weight \(\lambda\). The representation \(\pi_{\lambda}\) is induced by the character \(\chi\) of the subgroup \(H\) of \(G\), where \(H\) is the centralizer of a torus, and is realized on a finite-dimensional Hilbert space \(\mathcal{H}\) of holomorphic functions defined on dense open subsets of \(M=G/H\), in fact global sections of the \(G_{\mathbb{C}}\)-homogeneous line bundle \(L_{\chi}\) associated by the character \(\chi\) to the principal \(H\)-bundle.
In a previous paper [Math. Scand. 105 , 66–84 (2009; Zbl 1183.22006)], the author has calculated the Berezin symbols \(S_{\lambda}(\pi_{\lambda})\), \(g\in G\), and \(S_{\lambda}(d\pi_{\lambda}(X))\), \(X\in\mathfrak{g}\). In the paper under review, the author calculates the symbol \(W_{\lambda}(d\pi_{\lambda})\), where \(W_{\lambda}\) is the Stratonovich-Weyl correspondence for \((G,\pi_{\lambda},M)\) [cf., H. Figueroa, J. M. Gracia-Bondía and J. C. Várilly, J. Math. Phys. 31, No. 11, 2664–2671 (1990; Zbl 0753.43002)] using the polar decomposition of the Berezin calculus \(S\), \(W = B^{-1/2}S\), where \(B=SS^{\star}\) is the Berezin transform.
The result is expressed as \(W_{\lambda}(d\pi_{\lambda}(X))=a_{\lambda}S_{\lambda} (d\pi_{\lambda}(X))\), where \(a_{\lambda}\) is a constant. The author determines the constant \(a_{\lambda}\) in the case of a compact complex Grassmann manifold. Also the behavior of \(a_{m\lambda}\) and \(W_{m\lambda}(d\pi_{m\lambda}(X))\) as \(m\rightarrow\infty\) have been investigated, computing a Hua type integral on the Grassmannian.

MSC:
22E46 Semisimple Lie groups and their representations
81S10 Geometry and quantization, symplectic methods
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
32M10 Homogeneous complex manifolds
43A77 Harmonic analysis on general compact groups
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