Some Aspects of Positive Kernel Method of Quantization

We discuss various aspects of positive kernel method quantization the one-parameter groups $\tau_t \in \mbox{Aut}(P,\vartheta)$ automorphisms a $G$-principal bundle $P(G,\pi,M)$ with fixed connection form $\vartheta$ on its total space $P$. show that generator $\hat{F}$ unitary flow $U_t = e^{it \hat{F}}$ being $ is realized by generalized Kirillov-Kostant-Souriau operator whose domain consists sections some vector over $M$, which are defined suitable kernel. This applied to case when $G=GL(N,\mathbb{C})$ and $M$ non-compact Riemann surface leads arbitrary holomorphic $\tau_t^{hol} \mbox{Aut}(P,\vartheta)$. For above case, we present integral decompositions kernels $P\times P$ invariant respect flows $\tau_t^{hol}$ in terms spectral measure $\hat{F}$. These generalize ones given Bochner theorem for $\mathbb{C} \times \mathbb{C}$ translations complex plane.