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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:

22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 0469.22013
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References:

[1] P. Gérardin and P. Kutzko : Facteurs locaux pour GL(2) . Ann. Scient. Éc. Norm. Sup. 13 (1980) 349-384. · Zbl 0448.22015
[2] H. Jacquet and R.P. Langlands : Automorphic forms on GL(2) . Lecture Notes in Mathematics 114 (1970). · Zbl 0236.12010
[3] P. Kutzko : On the supercuspidal representations of Gl 2, I . Amer. J. Math. 100 (1978) 43-60. · Zbl 0417.22012
[4] P. Kutzko : The irreducible imprimitive local galois representations of prime dimension . J. Algebra 57 (1979) 101-110. · Zbl 0438.12007
[5] P. Kutzko : The Langlands conjecture for Gl2 of a local field . Ann. of Math. 112 (1980) 381-412. · Zbl 0469.22013
[6] A. Nobs : Die irreduziblen Darstellungen von GL2(Zp), insbesondere Gl2(Z2) . Math. Ann. 229 (1977) 113-133. · Zbl 0342.20020
[7] A. Weil : Sur certaines groupes d’operateurs unitaires . Acta Math. 111 (1964) 143-211. · Zbl 0203.03305
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