Invariant differential operators and central Fourier multipliers on exponential Lie groups

Let $G$ be an exponential solvable Lie group. By definition $G$ is $\ast$-regular if $ker_{L^1(G)}\pi$ is dense in $ker_{C^\ast(G)}\pi$ for all unitary representations $\pi$ of $G$. Boidol characterized the $\ast$-regular exponential Lie groups by a purely algebraic condition. In this article we will focus on non-$\ast$-regular groups. We say that $G$ is primitive $\ast$-regular if the above density condition is satisfied for all irreducible representations. Our goal is to develop appropriate tools to verify this weaker property. To this end we will introduce Duflo pairs $(W,p)$ and central Fourier multipliers $\psi$ on the stabilizer $M=G_fN$ of representations $\pi=\mcK(f)$ in general position. Using Littlewood-Paley theory we will derive some results on multiplier operators $T_\psi$ which might be of independent interest. The scope of our method of separating triples $(W,p,\psi)$ will be sketched by studying two significant examples in detail. It should be noticed that the methods 'separating triples' and 'restriction to subquotients' suffice to prove that all exponential solvable Lie groups of dimension $\le 7$ are primitive $\ast$-regular.