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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
##### Citations:
Zbl 1183.22006; Zbl 0753.43002
Full Text:
##### References:
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