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Ultimately, I'm interested in non-Archimedean local fields $F$, and am happy to take $\text{char}(F)=0$, and just focus on split $PGL_n(F)$. I suspect that the theory "more less" reduces to (smooth) representations of $GL_n(F)$ whose irreducible constituent have trivial central character. Maybe there's even an equivalence of categories?

It seems to me that there would be some things to check though, such as making sure one can pass between representations of the groups and preserve smoothness, etc. Also, I would like to know/hope that the Levis and parabolic induction "play nice" with each other (take that vague statement as you will).

Has anyone worked this out somewhere?

clathratus
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Kristaps John Balodis
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1 Answers1

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The statement for finite groups is: Let $G$ be a finite group, and let $Z\subset G$ be a normal subgroup. Then there is an equivalence of categories $$G-\mathrm{Rep}^Z\cong G/Z-\mathrm{Rep},$$ where the left category is the category of $G$-representation such that $Z$ acts trivially. The functor sends a $G/Z$-representation $V$ to its inflation to $G$ (making $g\in G$ act by $\overline g\in G/Z$) and sends a $G$-representation $V$ with trivial $Z$-action to the $G/Z$-representation $V$, where the action of $\overline g\in G/Z$ is given by $g\in G$, which is independent of the choice of lift since $Z$ acts trivially.

The argument readily extends to your situation: there is an equivalence $$\mathrm{GL}_n(F)-\mathrm{Rep}^{Z,\mathrm{sm}}\cong \mathrm{PGL}_n(F)-\mathrm{Rep}^{\mathrm{sm}},$$ where the left category is the category of smooth $\mathrm{GL}_n(F)$-representations such that $Z$ acts trivially. Moreover, there is a compatibility between parabolic inductions, and it preserves admissibility (which are easy to check, and I leave it to you as an exercise).

Kenta S
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  • Ok, I managed to convince myself that at least smoothness is preserved. Do you have a reference for the compatibility of parabolic inductions? To be honest, I don't even know what the Levis of PGL are (maybe M/Z for a Levi M of G?) Unlike GL, Sp and SO, I'm having a hard time finding any detailed references on anything to do with PGL, least of all its p-adic rep theory. I'm sure it's a good exercise to do this myself, but I'm trying to write a paper, and I doubt people want to read such proofs if they're "out there", and I can't in good conscience leave it un-referenced. – Kristaps John Balodis Jun 14 '23 at 17:18
  • Yes, Levi subgroups (resp., Borel, resp., parabolic subgroups) of $PGL$ are reductions mod $Z$ of Levis (resp., Borel, resp., parabolics) of $GL$. – Kenta S Jun 14 '23 at 17:21