Berezin-type operators on the cotangent bundle of a nilpotent group

Abstract

We define and study coherent states, a Berezin–Toeplitz quantization and covariant symbols on the product $$\\varXi \\,{:}{=}\\,{\\mathsf {G}}\\times \\mathfrak {g}^\\sharp $$Ξ:=G×g♯ between a connected simply connected nilpotent Lie group and the dual of its Lie algebra. The starting point is a Weyl system codifying the natural canonical commutation relations of the system. The formalism is meant to complement the quantization of the cotangent bundle $$T^\\sharp {\\mathsf {G}}\\cong {\\mathsf {G}}\\times \\mathfrak {g}^\\sharp $$T♯G≅G×g♯ by pseudo-differential operators, to which it is connected in an explicit way. Some extensions are indicated, concerning $$\\tau $$τ-quantizations and variable magnetic fields.