# Harish-Chandra Modules of PSL_2($\mathbb{R})$)

I am sorry to bother you with this question but i can't figure out this myself (and mathstackexchange didnt 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 $PSL_2(\mathbb{C})$ representation are exactly the $SL_2(\mathbb{C})$ representations where $\pm I$ is fixed.

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