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 $F^{\natural}$ 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 ${\rm Env}\, \mathfrak g$ of the Lie algebra $\mathfrak g$ of $G.$ In this case symbols turn out to be polynomials on the Lie algebra $\mathfrak g.$
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) $G={\rm SL}(2,\mathbb R),$ $H$ — the subgroup of diagonal matrices, $G/H$ — a hyperboloid of one sheet in $\mathbb R^3;$
b) $G$ — the pseudoorthogonal group ${\rm SO}_0 (p,q),$ the subgroup $H$ covers with finite multiplicity the group ${\rm SO}_0 (p-1,q-1) \times {\rm SO}_0 (1,1);$ the space $G/H$ (a pseudo-Grassmann manifold) is an orbit in the Lie algebra $\mathfrak g$ of the group $G.$