2018/01/31 by Marius Măntoiu, M. Mantoiu, Maximiliano Sandoval +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Physics Problems #Algebra over a field #Compact group #Equivalence (formal languages) #Lie group #Locally compact space #Mathematical Analysis and Transform Methods #Mathematics #Pure mathematics #Second-countable space #Type (biology) #Unimodular matrix #Unitary state #math.FA #math.GR #msc:22D10 #msc:22D25 #msc:46L65 #msc:47G30
paper · pdf · doi:10.1007/s00605-019-01312-7
published as Monatshefte für Mathematik 190 (2019) 559-58 · 25 pages
openalex publication_date 2019/06/18 · arxiv created 2020/08/11 · arxiv updated 2020/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In recent papers and books, a global quantization has been developed for unimodular groups of type I. It involves operator-valued symbols defined on the product between the group G and its unitary dual \widehatG, composed of equivalence classes of irreducible representations. For compact or for graded Lie groups, this has already been developed into a powerful pseudo-differential calculus. In the present article we extend the formalism to arbitrary locally compact groups of type I, making use of the Fourier theory of non-unimodular second countable groups. The unitary dual and its Plancherel measure being quite abstract in general, we put into evidence situations in which concrete forms are available. Kirillov theory and parametrizations of large parts of \widehatG allow rewriting the basic formulae in a manageable form. Some examples of completely solvable groups are worked out.