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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

Citations:

Zbl 0656.00012