## On the exceptional representations of $$GL_ n$$.(English)Zbl 0683.22013

Representation theory and number theory in connection with the local Langlands conjecture, Proc. Conf., Augsburg/FRG 1985, Contemp. Math. 86, 177-185 (1989).
[For the entire collection see Zbl 0656.00012.]
This is a short survey without proofs concerning the following part of local Langlands theory: Let $$E| F$$ be a quadratic extension of a p- adic field F, then any character of $$E^*$$ can be viewed as a character of the Weil group $$W_ E$$ and thus induces a two dimensional representation of the Weil group $$W_ F$$ of F. If the character $$\theta$$ is different from its Galois conjugate then this representation $$\Sigma(\theta)$$ is irreducible. Thus by Langlands theory this representation corresponds to an irreducible representation $$\Pi(\theta)$$ of $$G=GL_ 2(F)$$. One of the results presented here is that an arbitrary irreducible representation $$\Pi$$ is of the form $$\Pi(\theta)$$ for some $$\theta$$ if and only if $$\Pi$$ decomposes when restricted to $$G_{E| F}$$. Here $$G_{E| F}$$ denotes the subgroup consisting of those $$g\in G$$, whose determinant is a norm of E. Further one can reconstruct the character $$\theta$$ from $$\Pi(\theta)$$. The author shows how the data (E,$$\theta)$$ are related to $$(K,\kappa)$$, where K is a maximal compact mod center subgroup of G and $$\kappa$$ a representation of K for which $$Ind^ G_ K(\kappa)=\Pi (\theta)$$. A precise description of the range of the map $$(E,\theta)\mapsto \Pi (\theta)$$ is given.
Reviewer: A.Deitmar

### MSC:

 22E50 Representations of Lie and linear algebraic groups over local fields 11F33 Congruences for modular and $$p$$-adic modular forms

Zbl 0656.00012