Finite dimensional irreducible representations of quasisplit p-adic groups For split groups over a $p$-adic field, every irreducible smooth (complex) representation is either infinite-dimensional or one-dimensional. Is it true for quasisplit groups that split over an unramified extension, or quasisplit groups in general? Unfortunately I don't have a good feel for examples, since the books I consult for learning the representation theory of $p$-adic groups usually stick to groups like $GL_n$.
 A: See Prop. 3.9 of http://math.stanford.edu/~conrad/JLseminar/Notes/L2.pdf for an optimal affirmative answer (no quasi-split condition needed: any connected reductive group over any non-archimedean local field, with the minimal necessary isotropicity hypotheses).
A: This is an editted version, as my previous answer had an error.
Every $p$-adic group is totally-disconnected locally compact (tdlc).
For the class of tdlc group, every continuous finite dimensional complex representation is through a discrete quotient (that is, the kernel contains an open subgroup). More generally, every continuous homomorphism from a tdlc group to a Lie group (eg $\text{GL}_n(\mathbb{C})$) has this property. To see this, recall that every tdlc group has a compact open subgroup (van Dantzig theorem) and observe that the image of such a group should be a compact (hence closed, hence Lie) tdlc subgroup in the target, so it must be finite.
Let $G$ be the $k$ points of a reductive $k$-algebraic group which has no anisotropic factor. The group that you consider are such,
and an affirmative answer to your question will follow from the fact that every discrete quotient of $G$ is abelain. A way to see this is to consider the group generated by all unipotent subgroups of $G$, $G^+$, and to see that every open normal subgroup of $G$ contains $G^+$ and $G/G^+$ is commutative. This is discussed in the work of Borel-Tits and nicely summerized in Margulis book "Discrete subgroups of semisimple Lie groups" Chapter I.1.5. I recommend reading it.
