Question

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.

Answer 1

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!

Answer 2

If you want a construction entirely compatible with Bushnell and Kutzko's theory of strata and simple characters (and that also works when $F$ has positive characteristic), you may refer to my PhD thesis :

Broussous, P. Extension du formalisme de Bushnell et Kutzko au cas d'une algèbre à division. (French) [Extension of the Bushnell-Kutzko formalism to the case of a division algebra] Proc. London Math. Soc. (3) 77 (1998), no. 2, 292–326.

For other reductive groups, there are basically two "schools". First Bushnell and Kutzko (GL(N), SL(N)) and the students of Bushnell (Shaun Stevens : classical groups), of Henniart (myself and Vincent Secherre : GL(m,D)), of Zink (Martin Grabitz : GL(m,D)).

(I don't give any precise references for you may easily find them with Mascinet.)

Second, you have the "american school", initiated by Roger Howe, it has entirely solved the construction of "tame" supercuspidal representations for a general reductive group. Howe itself did GL(n) a long time ago. The following papers solve the general case.

Yu, Jiu-Kang Construction of tame supercuspidal representations. J. Amer. Math. Soc. 14 (2001), no. 3, 579–622.

Kim, Ju-Lee Supercuspidal representations: an exhaustion theorem. J. Amer. Math. Soc. 20 (2007), no. 2, 273–320.

To finish I must add that Bushnell and Kutzko have defined the beautiful notion of "type" for Bernstein blocks of the category of smooth complex representations of a given reductive group :

Bushnell, Colin J.; Kutzko, Philip C. Smooth representations of reductive $p$-adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634.

This notion allows to develop a general strategy to construct all representations of a given reductive group.

Answer 3

I think you may find a description of representation theory of $D^{\times}$ in the following work of E.W. Zink :

Ernst-Wilhelm Zink. Representation filters and their application in the theory of local fields. J. Reine Angew. Math., 387 :182–208, 1988.

Ernst-Wilhelm Zink. Representation theory of local division algebras. J. Reine Angew. Math., 428 :1–44, 1992.