Lie algebra modules which are locally finite over the semi-simple part

For a finite-dimensional Lie algebra $\mathfrak{L}$ over $\mathbb{C}$ with a fixed Levi decomposition $\mathfrak{L} = \mathfrak{g} \oplus \mathfrak{r}$ where $\mathfrak{g}$ is semi-simple, we investigate $\mathfrak{L}$-modules which decompose, as $\mathfrak{g}$-modules, into a direct sum of simple finite-dimensional $\mathfrak{g}$-modules with finite multiplicities. We call such modules $\mathfrak{g}$-Harish-Chandra modules. We give a complete classification of simple $\mathfrak{g}$-Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak{g} = \mathfrak{sl}_2$, and for the Schr\"{o}dinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright's and Arkhipov's completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak{g}$-Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak{sl}_2$ and the Schr\"{o}dinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak{g}$-Harish-Chandra modules.