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Symbols in Berezin quantization for representation operators


The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.


Lie groups and Lie algebras, pseudo-orthogonal groups, representations of Lie groups, para-Hermitian symmetric spaces, Berezin quantization, covariant and contravariant symbols

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[1] V.F. Molchanov, N.B. Volotova, S.V. Tsykina, O.V. Grishina, “Polynomial quantization”, Tambov University Reports. Series: Natural and Technical Sciences, 14:6-3 (2009), 1443-1474 (In Russian). [2] S.V. Tsykina, “Symbols in polynomial quantization: explicit formulas”, Tambov University Reports. Series: Natural and Technical Sciences, 23:124 (2018), 838-845 (In Russian). [3] V.F. Molchanov, “Quantization on para-Hermitian symmetric spaces”, Contemporary Mathematical Physics, American Mathematical Society. Translations: Series 2, 175, eds. R. L. Dobrushin, R.A. Minlos, M.A. Shubin, A.M. Vershik, 1996, 81-95. [4] V.F. Molchanov, N.B. Volotova, “Finite-dimensional analysis and polynomial quantization on a hyperboloid of one sheet”, Tambov University Reports. Series: Natural and Technical Sciences, 3:1 (1998), 65–78.



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