The multiplicity function of mixed representations on completely solvable Lie groups (Q2454932)

Let \(G= \exp{\mathfrak g}\) be a completely solvable Lie group with Lie algebra \({\mathfrak g}\). Let \(A\), \(H\) be closed connected subgroups of \(G\), \(\sigma\) a unitary representation of \(H\) and \(\pi\) that of \(G\). The canonical central decompositions of the induced representation \(\text{Ind}^G_H\sigma\) and the restricted representation \(\pi|_A\) are now well-known in terms of the orbit method [cf. \textit{R. L. Lipsman}, Trans. Am. Math. Soc. 313, No. 2, 433--473 (1989; Zbl 0683.22009)]. Here, the authors continue their study of mixed representations, namely an up-down representation \(\rho(G,H,A,\sigma)= (\text{Ind}^G_H\sigma)|_A\) and a down-up representation \(\rho(G,H, \pi)= \text{Ind}^G_H(\pi|_H)\). In their previous work [Russ. J. Math. Phys. 8, No. 4, 422--432 (2001; Zbl 1186.22011)] and a joint paper with \textit{A. Ghorbel} [Publ. Mat., Barc. 46, No. 1, 179--199 (2002; Zbl 1015.22004)], they gave detailed descriptions of the multiplicity function of these mixed representations when \(G\) is nilpotent. In the paper under review, they extend the results to completely solvable Lie groups. That is to say, they show that the multiplicities of mixed representations are uniformly infinite or finite and bounded. Necessary and sufficient conditions for the multiplicities to be finite are given using pseudo-algebraic geometry.