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Let $F$ be a finite extension of $\mathbb{Q}_p$ and let $G=\operatorname{GL}_2(F)$ (or $\operatorname{GL}_n$ or any reductive group). I consider smooth representations of $G$ in characteristic $p$. Because we don't have the Hecke algebra in this setting we can't prove by the usual argument that the Jacquet functor and the $\Gamma$-invariant functor are exact.

I know from Barthel–Livne that those functors are indeed not exact, but I don't have any examples. I try to take the sequence $$0\to\mathbf{1}_G\to\operatorname{Ind}_B^G\mathbf{1}_B\to\mathrm{St}\to0$$ but it doesn't seem to give an example for $\Gamma$-invariant functor. Maybe this sequence gives an example of the not exactness of Jacquet functor but I don't know how to compute this.

Any help will be appreciated.

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  • $\begingroup$ Here it is better to form a derived Jacquet module. The nice paper (arxiv.org/pdf/2204.11581.pdf) discusses this in detail. See section 5 for some sample calculations in the case of $\mathrm{GL}_2(\mathbf{Q}_p)$. $\endgroup$
    – Satan's Minion
    Commented Jan 30 at 13:57

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