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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 allunitary representations $\pi$ of $G$. Boidol characterized the $\ast$-regularexponential Lie groups by a purely algebraic condition. In this article we willfocus on non-$\ast$-regular groups. We say that $G$ is primitive $\ast$-regularif the above density condition is satisfied for all irreduciblerepresentations. Our goal is to develop appropriate tools to verify this weakerproperty. To this end we will introduce Duflo pairs $(W,p)$ and central Fouriermultipliers $\psi$ on the stabilizer $M=G_fN$ of representations $\pi=\mcK(f)$in general position. Using Littlewood-Paley theory we will derive some resultson multiplier operators $T_\psi$ which might be of independent interest. Thescope of our method of separating triples $(W,p,\psi)$ will be sketched bystudying two significant examples in detail. It should be noticed that themethods 'separating triples' and 'restriction to subquotients' suffice to provethat all exponential solvable Lie groups of dimension $\le 7$ are primitive$\ast$-regular.

Invariant differential operators and central Fourier multipliers on exponential Lie groups