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Humphreys. Linear algebraic groups, volume 21 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1975. [31] G. Kempf. Instability in invariant theory. Ann. of Math. (2), 108(2):299– 316, 1978. [32] B. N. Kimel’feld and E. B. Vinberg. Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups. Func. Anal. , 12(3):168–174, 1978. [33] F. Knop. The Luna-Vust theory of spherical embeddings. In S. Ramanan, editor, Proc. Hyderabad Conf. on Algebraic Groups, pages 225– 249.

Bialynicki-Birula, and G. Hochschild. Extensions of representations of algebraic linear groups. Amer. J. , 85:131–144, 1963. [49] M. Nagata. On the fourteenth problem of Hilbert. In Proc. Internat. Congress Math. 1958, pages 459–462. Cambridge Univ. Press, 1960. [50] A. L. Onishchik. Transitive compact transformation groups. Amer. Math. Soc. , 55:153–194, 1966. [51] R. S. Palais and T. E. Stewart. The cohomology of differentiable transformation groups. Amer. J. , 83(4):623–644, 1961. [52] V. L.

Denote the image of this embedding by C [M ]. 2. A function f ∈ C(M ) is called spherical if the linear span Kf is finite-dimensional. More generally, for a linear action of a Lie group K on vector space V , a vector v ∈ V is spherical if Kv is finite-dimensional. Denote by Vsph the subspace of all spherical vectors in V . 3. The algebra C [M ] coincides with C(M )sph . Proof. Any regular function is contained in a finite-dimensional invariant subspace. Conversely, any complex finite-dimensional representation of K is completely reducible and any irreducible component may be considered as a simple G-module.

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