Let $F$ be a non Archimedean local field with ring of integers $\mathcal{O}_F$. Let $P$ be a standard proper parabolic subgroup of $GL_n(F)$. Let $M$ be the standard Levi subgroup of $P$ and $\sigma$ a supercuspidal representation of $M$ and $N$ be the unipotent radical of $P$. Let $s=[M,\sigma]$ be a Bernstein component of $GL_n(F)$. Is it possible to chose a semi-simple type $(J_s,\lambda_s)$ for $s$ such that $J_s\cap N=N(\mathcal{O}_F)$?
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1$\begingroup$ I think that the semisimple types considered by Bushnell and Kutzko in e.g. "The admissible dual of GL(N) via compact open subgroups" do not satifsy this property. But by inducing them to well chosen parahoric subgroups you can expect that the induced types do satisfy it. I try to make a more complete answer next week. $\endgroup$– Paul BroussousMar 7, 2014 at 14:12
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