LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART

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