# Are all irreducible supercuspidal representation induced from compact-mod-center subgroups?

Let $G$ be a reductive group over a local non-archimedean field $F$.

Can every irreducible supercuspidal representation of $G(F)$ be realized as the induction from an open subgroup, which is compact modulo the center?

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It is known for GL(N) and SL(N) (Bushnell and Kutzko), for classical groups when the residue characteristic is not $2$ and when no quaternionic algebra is involved (Stevens), for GL(N) of a division algebra (Stevens and Sécherre), for a general reductive group when the residue characteristic of $F$ is large enough (Kim, Yu, ...).

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So in the last statement, the reductive group is defined over a global field a priori? –  plusepsilon.de Jul 2 '12 at 16:44
No, in all statements G is simply assumed to be defined over a local field. –  Paul Broussous Jul 3 '12 at 5:23
How does one define then large enough? –  plusepsilon.de Jul 3 '12 at 8:20
Roughly speaking large enough is so that all the field extensions involved are tamely ramified. –  mander Jul 3 '12 at 13:25