The authors deal with the problem of finding admissible vectors for monomial representations of simply connected exponential Lie groups. More concretely, the representations appearing in the article belong to the type of a monomial representation \(\tau\) of an exponential solvable Lie group \(G\) (with Lie algebra \(\mathfrak g\)) induced from a unitary character of a closed connected subgroup \(H\) of \(G\).
The authors use the description of the irreducible decomposition of \(\tau\) in terms of the coadjoint orbit picture [H. Fujiwara, Prog. Math. 82, 61–84 (1990; Zbl 0744.22010)] and the explicit Plancherel formula for \(G\) with coadjoint orbit parameters [B. N. Currey, Pac. J. Math. 219, No 1, 97–138 (2005; Zbl 1089.22008)] in order to obtain a simple necessary and sufficient condition so that \(\tau\) is a subrepresentation of the left regular representation of \(G\) in terms of the orbit picture.
The authors define the real affine variety \(A_\tau\) of all \(l\) in the dual Lie algebra \(\mathfrak g^*\) and also take into consideration the unique measure class \(\nu\) on the Borel space \(\hat G\) of equivalence classes of irreducible representations of \(G\) such that \[ \tau=\int_{\hat G}^\oplus m_\tau(\pi)\pi \text{d}\nu(\pi). \]
In the basis of these notions, the representation \(\tau\) is a subrepresentation of the left regular representation if and only if \(\nu\) is absolutely continuous with respect to the Plancherel measure class \(\mu\) determined by the regular representation.
The main result of the article is Theorem 3.4 which gives a criterion to determine when \(\nu\) is absolutely continuous with respect to \(\mu\) and, equivalently, when \(\tau\) is a subrepresentation of the regular representation. The result states that if \(H\) acts freely on some point of \(A_\tau\), then \(\nu\) is absolutely continuous with respect to \(\mu\). Otherwise, \(\nu\) is singular with respect to \(\mu\). Moreover, the authors also prove that the free action on some element of \(A_\tau\) involves the free action on a Zariski open subset of \(A_\tau\).
Moreover, the previous criterion is used in the article, in combination with Theorem 4.22 in [H. Führ, Abstract harmonic analysis of continuous wavelet transforms. Berlin: Springer (2005; Zbl 1060.43002)], to answer the question about the existence of admissible vectors in the representation \(\tau\). Indeed, they prove Corollary 3.6 that states if \(G\) is nonunimodular, then \(\tau\) has an admissible vector if and only if \(H\) acts freely on some \(l\in A_\tau\).
The article concludes with a conjecture about the nonexistence of admissible vectors for monomial representations of unimodular exponential solvable Lie groups since this fact happens in all the existing examples, although the free action of \(H\) on some point of \(A_\tau\) persists as a necessary condition for admissibility in the case of unimodular groups.