Monomial representations of discrete type of an exponential solvable Lie group (Q2290607)

Let \(G=\exp(\mathfrak{g})\) be a connected and simply connected exponential solvable Lie group with Lie algebra \(\mathfrak{g}\), \(H= \exp(\mathfrak{h})\) a closed connected subgroup of \(G\), \(\chi= \chi_f\) a unitary character of \(H\), with \(f \in \mathfrak{g}^*\) vanishing in \([\mathfrak{h}, \mathfrak{h}]\) and \(\Gamma = f + \mathfrak{h}^{\perp }\). A monomial representation can be disintegrated as: \[\tau:= \operatorname{ind}_H^G\chi= \int_{\widehat{G}} m(\pi) \pi d\mu(\pi), \] the multiplicity function \(m(\pi)\) is either finite and uniformly bounded or uniformly equal to the infinity \(\mu\)-almost everywhere on \(\Gamma\). The measure \(\mu\) can be chosen in such a manner that for any \(\pi\) in the spectrum of \(\tau\), there exist some distributions \(a_\pi^k\) (\(1\leq k\leq m(\pi))\) such that the abstract Plancherel formula due to Penney is described for all \( \varphi \in C_c^\infty(G)\) by: \[\varphi_H^f(e)= \int_{\widehat{G}}\sum_{k=1}^{m(\pi)}\langle \pi(\varphi)a_\pi^k, a_\pi^k\rangle d\mu(\pi) \quad \text{where}\quad \varphi_H^f (g) = \int_H\varphi(gh)\chi_f(h)dh, ~ \forall g \in G. \] The Duflo problem says that the algebra \(D_\tau(G/H)\) is commutative if and only if generically on \(\Gamma= f +\mathfrak{h}^\perp\) the subspace \(\mathfrak{h}+ \mathfrak{g}(\ell)\) is Lagrangian with respect to the antisymmetric bilinear form \(B_\ell\) on \(\mathfrak{g}\). In this paper, the authors generalize some results for an exponential solvable Lie group, they study some problems when \(m(\pi)\) is finite or infinite of discrete type. They give an explicit formula for the Penney's distributions \(a_\pi^k\) which comes directly from the disintegration of the Dirac measure: \[\delta_\tau \simeq \int_{\hat{G}}^\oplus \left( \sum_{k=1}^{m(\pi)} a_\pi^k \right)d\mu(\pi).\] Their existence as generalized integrals of continuous functions is far from being evident, contrarily to the nilpotent cases, and the Plancherel formula holds for all \(\varphi \in C_c(G)\) with: \[ \varphi_H^f (g) = \int_H\varphi(gh)\chi_f(h)\Delta^{-1/2}_{H, G}(h)dh, \quad \text{where }\Delta_{H, G}(\exp(X) = e^{(\text{Tr ad}_{\mathfrak{g}/ \mathfrak{h}}X)},\ \forall X \in \mathfrak{h}. \] The authors study the commutativity Duflo problem for \(D_\tau(G/H)\) of \(G\)-invariant differential operators on the fiber space associated to the data \((H, \chi)\) over \(G/H\). They prove if \(\tau\) has multiplicities of discrete type and the algebra \(D_\tau(G/H)\) is commutative. They give, in particular, an example where this problem admits a negative solution in the framework of exponential solvable Lie groups. For the entire collection see [Zbl 1426.22001].