Harish-Chandra modules of $\mathrm{PSL}_2(\mathbb{R})$

Question

I am sorry to bother you with this question but I can't figure out this myself (and Mathematics Stack Exchange didn't help).

Is the category of Harish-Chandra Modules of $PSL_2(\mathbb{R})$ equivalent to the category of Harish-Chandra Modules of $SL_2(\mathbb{R})$ with even $K$-types?

The motivation of this question comes from the finite-dimensional case, the finite-dimensional $PSL_2(\mathbb{C})$-representations being exactly the $SL_2(\mathbb{C})$-representations where $\{\pm I\}$ acts trivially.

Answer 1

Yes, the central character of the even $K$-types is trivial, and of the odd ones is the sign character.

Also, this can easily be seen from the classification of irreducible representation on Hilbert spaces.

More generally, the category of rep on Hilbert spaces of a locally compact group $G$ with trivial central character coincides with that of the rep theory on Hilbert spaces of the locally compact group $G/Z$, here $Z$ being the center of $G$. So although, the categories are not semisimple, one can always decompose with respect to the center.

Quote from Wikipedia: In 1973, Lepowsky showed that any irreducible (g,K)-module X is isomorphic to the Harish-Chandra module of an irreducible representation of G on a Hilbert space....(http://en.wikipedia.org/wiki/Harish-Chandra_module)