Differential geometry and quantization on a locally compact group

For an arbitrary locally compact group $G$, we describe the structure of the Lie algebra $\chi(G)$ of vector fields, the exterior algebra $\Lambda(G)$ of differential forms, and the Poisson algebra of symbols on $G$ polynomial with respect to the momenta. A continuous left-invariant $qp$-quantizaton is constructed, giving rise to a one-to-one correspondence between symbols and differential operators on $G$. It is demonstrated that neither of the other two classical quantizations, namely, the $pq$ and Weyl quantizations, can be constructed on an infinite group $G$ if the same properties are to be retained.