Covariant affine integral quantization(s) (Q2814207)

The manuscript is a continuation of work on covariant integral quantization on a Lie group which was introduced earlier by \textit{H. Bergeron} and the first author [Ann. Phys. 344, 43--68 (2014; Zbl 1343.81157)]. The general setting is as follows: Let \(G\) be a Lie group with left Haar measure \(d\mu\). Suppose \(U\) is an irreducible representation of \(G\) on a Hilbert space \(\mathcal{H}\). Then we define an operator associating to a function or distribution on \(G\) by \(f\mapsto A_f=\int_G f(g)U(g)MU(g)^*d\mu(g)\) for a given bounded operator \(M\) on \(\mathcal{H}\) that satisfies \(c_M\int_G U(g)MU(g)^*d\mu(g)=I\) for a normalization constant \(c_M\).NEWLINENEWLINEIf \(G\) is the Heisenberg group, \(U\) the Schrödinger representation and \(M\) a rank-one operator, then the operator \(A_f\) is a localization operator and if \(M\) is a density operator, then \(A_f\) is a so-called mixed-state localization operator, see the results in [the reviewer and \textit{E. Skrettingland}, J. Math. Pures Appl. (9) 118, 288--316 (2018; Zbl 1486.47086)].NEWLINENEWLINEThe main goal of this paper is to define covariant integral quantization for affine groups.