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 $PSL_2(\mathbb{C})$ representation are exactly the $SL_2(\mathbb{C})$ representations where $\pm I$ is fixed.

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.

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.

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 $PSL_2(\mathbb{C})$ representation are exactly the $SL_2(\mathbb{C})$ representations where $\pm I$ is fixed.

I am sorry to bother you with this question but i can't figure out this myself (and mathstackexchange didtnt 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.

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.