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.



Existence of supercuspidal representations
Various special constructions make the point for particular groups, which tends to convince one about the general case. E.g., for $G=SL(2,\mathbb Q)$, a well-known (Shalika, Jacquet, c. 1970) idea is that inducing "supercuspidals" from $K=SL(2,\mathbb Z\_p)$ to $G$ is a finite direct sum of supercuspidals. The "level zero" cuspidals on $SL(2,\mathbb Z)$ factor through $SL(2,\mathbb Z/p)$. The irreducibles of the latter finite group are ascertained in various ways, and one finds that a significant fraction are "cuspidal", so compose to "supercuspidal" on $K$, and (with proof) induce to finite direct sums of supercuspidals on $G$.


The critical point is that induced repns (from Iwahori subgroups) do not exhaust the irreds of $K$. For $SL(2)$ this is easily proven by a counting argument, and/or by a theta-correspondence argument. For general groups, some of these choices persist.


