Let $G$ be an unramified reductive group over $Q_p$. I want to prove that the group $G(Q_p)$ has a supercuspidal representation (complex coefficients).
I have been looking in many parts of the literature, and it seems that many people are convinced that it is true; however up to now I never saw it stated explicitly.
By the works of L. Morris I reduce to showing that a reductive group $M$ over $F_p$ has a cuspidal representation (L. Morris, level zero G-types, p 140).
So then, I should prove that $M(F_p)$ has a cuspidal representation. The article of Deligne–Lusztig provides such a representation when given a minisotropic torus $T$ in $M$ and a character $\chi$ of $T(F_p)$ which is in general position.
Let me recall that "character in general position" means that the rational Weil group acts freely on the character.
So now comes my doubt and question. Is it true that such a pair $(T, \chi)$ can always be found for all reductive groups $M$ over $F_p$ ?
I am "afraid" of "small" groups that have tori with "large" Weyl groups.
The supercuspidal representations that come from the above construction are of "level 0".
In the book of Carter (Finite Groups of Lie Type) I found a result pointing in this direction. Lemma 8.4.2 p. 281 (with an easy proof) shows that for $T$ given and $q$ sufficiently large, the torus $T(F_q)$ has a character in general position.