@article{8636225,
  abstract     = {{For a connected simply connected nilpotent Lie group G with Lie algebra g and unitary dual G one has (a) a global quantization of operator-valued symbols defined on Gx G, involving the representation theory of the group, (b) a quantization of scalar-valued symbols defined on Gxg *, taking the group structure into account and (c) Weyl-type quantizations of all the coadjoint orbits . |.. G . We show how these quantizations are connected, in the case when flat coadjoint orbits exist. This is done by a careful new analysis of the composition of two different types of Fourier transformations, interesting in itself. We also describe the concrete form of the operator-valued symbol quantization, by using Kirillov theory and the Euclidean version of the unitary dual and Plancherel measure. In the case of the Heisenberg group, this corresponds to the known picture, presenting the representation theoretical pseudo-differential operators in terms of families ofWeyl operators depending on a parameter. For illustration, we work out a couple of examples and put into evidence some specific features of the case of Lie algebras with one-dimensional center. When G is also graded, we make a short presentation of the symbol classes Sm., d, transferred from G x G to G x g * by means of the connection mentioned above.}},
  author       = {{Măntoiu, M and Ruzhansky, Michael}},
  issn         = {{1050-6926}},
  journal      = {{JOURNAL OF GEOMETRIC ANALYSIS}},
  keywords     = {{Nilpotent group,Lie algebra,Coadjoint orbit,Pseudo-differential operator,Symbol,Weyl calculus,INVARIANT PSEUDODIFFERENTIAL-OPERATORS,CALCULUS}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{2823--2861}},
  title        = {{Quantizations on Nilpotent Lie groups and algebras having flat coadjoint orbits}},
  url          = {{http://doi.org/10.1007/s12220-018-0096-1}},
  volume       = {{29}},
  year         = {{2019}},
}

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