Journal of Lie Theory

Let $G$ be a connected, simply connected, exponential solvable Lie group. The irreducible unitary representations of $G$ may be obtained by the Kirillov-Bernat orbit method. Let $l \in \frak g^*$, $\frak p$ a Pukanszky polarization associated to $l$, $P= \exp {\frak p}$, $\chi_l$ the corresponding character of $P$ and $\pi_l = \hbox {\rm ind}_P^G \chi_l$ the associated unitary representation. We show through an example that not all the functions of ${\cal C}_c^{\infty}(G/P,G/P,\chi_l)$ (${\cal C}^{\infty}$-functions with compact support on $G/P \times G/P$ satisfying a certain covariance condition) are kernel functions of some operator of the form $\pi_l(f)$, $f\in L^1(G)$, even if the polarization is well chosen. This contradicts a result of H. Leptin [J. Reine Angew. Math. 494 (1998) 1--34]). But if the polarization $\frak p$ is an ideal of $\frak g$, then the result of Leptin is true, the corresponding retract from ${\cal C}^{\infty}_c(G/P,G/P,\chi_l)$ into $L^1(G)$ exists and a construction algorithm of the function $f$ may be indicated.