TITLE: p-adic representations of a quaternion algebra over a local field
QUESTION [10 upvotes]: How to determine a complete set of isomorphism class representatives of the irreducible algebraic representations of $D^{\times}/F$ (where $D$ is a quaternion algebra over a local field $F/\mathbb{Q} _p$) on $E$ (also a finite extension of $\mathbb{Q} _p$)?
Answer for other (algebraic) groups would also be welcome as well as any references to the literature.

REPLY [3 votes]: If $E$ is an algebraic closure of $F$, then $D\otimes_F E\simeq M_2(E)$.  (In fact this is also true if $E$ is taken to be, say, the unramified quadratic extension field of $F$.)  We get an algebraic representation
$$\phi\colon D^\times\hookrightarrow (D\otimes E)^\times=\text{GL}_2(E).$$
And then for each $a\geq 0$ and $b\in \mathbf{Z}$ we get the representation $\text{Sym}^{a}\phi\otimes (\det\phi)^b$.  My feeling is that these exhaust the irreducible algebraic representations of $D^\times$, but I'm afraid I don't have a proof at the ready.
As the other answerers show, the question of classifying the admissible representations of $D^\times$ (with complex coefficients) is a far more subtle issue!