p-adic representations of a quaternion algebra over a local field - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:26:16Z http://mathoverflow.net/feeds/question/56738 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56738/p-adic-representations-of-a-quaternion-algebra-over-a-local-field p-adic representations of a quaternion algebra over a local field Przemyslaw Chojecki 2011-02-26T14:19:50Z 2011-03-01T17:17:23Z <p>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 $\mathbb{Q} _p$)?</p> <p>Answer for other (algebraic) groups would also be welcome as well as any references to the literature.</p> http://mathoverflow.net/questions/56738/p-adic-representations-of-a-quaternion-algebra-over-a-local-field/56742#56742 Answer by Joël Cohen for p-adic representations of a quaternion algebra over a local field Joël Cohen 2011-02-26T15:21:44Z 2011-02-26T15:21:44Z <p>I think you may find a description of representation theory of $D^{\times}$ in the following work of E.W. Zink :</p> <p>Ernst-Wilhelm Zink. Representation filters and their application in the theory of local fields. J. Reine Angew. Math., 387 :182–208, 1988.</p> <p>Ernst-Wilhelm Zink. Representation theory of local division algebras. J. Reine Angew. Math., 428 :1–44, 1992.</p> http://mathoverflow.net/questions/56738/p-adic-representations-of-a-quaternion-algebra-over-a-local-field/56760#56760 Answer by Paul Broussous for p-adic representations of a quaternion algebra over a local field Paul Broussous 2011-02-26T19:31:32Z 2011-02-26T19:31:32Z <p>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 :</p> <p>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.</p> <p>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)).</p> <p>(I don't give any precise references for you may easily find them with Mascinet.)</p> <p>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.</p> <p>Yu, Jiu-Kang Construction of tame supercuspidal representations. J. Amer. Math. Soc. 14 (2001), no. 3, 579–622.</p> <p>Kim, Ju-Lee Supercuspidal representations: an exhaustion theorem. J. Amer. Math. Soc. 20 (2007), no. 2, 273–320.</p> <p>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 :</p> <p>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. </p> <p>This notion allows to develop a general strategy to construct all representations of a given reductive group.</p> http://mathoverflow.net/questions/56738/p-adic-representations-of-a-quaternion-algebra-over-a-local-field/57016#57016 Answer by Jared Weinstein for p-adic representations of a quaternion algebra over a local field Jared Weinstein 2011-03-01T17:17:23Z 2011-03-01T17:17:23Z <p>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 <code>$$\phi\colon D^\times\hookrightarrow (D\otimes E)^\times=\text{GL}_2(E).$$</code> And then for each $a\geq 0$ and $b\in \mathbf{Z}$ we get the representation <code>$\text{Sym}^{a}\phi\otimes (\det\phi)^b$</code>. 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.</p> <p>As the other answerers show, the question of classifying the admissible representations of $D^\times$ (with complex coefficients) is a far more subtle issue!</p>