## The exceptional representations of $$GL_ 2$$.(English)Zbl 0545.22018

This paper characterizes the set of exceptional supercuspidal representations of GL(2,F), F a local field of residual characteristic p, by veryfying lemma 4.2.2 of the author [Ann. Math., II. Ser. 112, 381-412 (1980; Zbl 0469.22013)]. The author first parametrizes irreducible ramified supercuspidal representations in terms of certain induced representations $$\pi$$ depending on the following data: a positive integer n, a lattice flag in $$F\oplus F$$, a certain kind of 2$$\times 2$$ matrix b with entries in F, a character of $$F[b]^{\times}$$ and a character of $$F^{\times}$$. Using a counting argument, he characterizes which of these are Weil representations. The final result is that $$\pi$$ is exceptional (i.e. not a Weil representation) if and only if $$2(n+1)\leq 3 \deg(F(b)/F)$$ and the polynomial $$X^ 3-(tr b)X^ 2+(\det b)$$ is irreducible over F. In particular, if $$p\neq 2$$, GL(2,F) has no exceptional representations.
Reviewer: A.Ash

### MSC:

 2.2e+51 Representations of Lie and linear algebraic groups over local fields

Zbl 0469.22013
Full Text:

### References:

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