Local \(\epsilon\)-factors and characters of GL(2). (English) Zbl 0532.12015

Tunnell, Jerrold B.

Am. J. Math. 105, 1277-1307 (1983).

Let F be a locally compact non-Archimedean field and \(\pi\) an admissible irreducible representation of \(G=GL(2,F)\). Such a representation has a character \(ch_{\pi}\) which is a locally constant function of the set \(G_{reg}\) of regular semisimple elements in G. If L is a quadratic Galois algebra over F and T a torus in G corresponding to \(L^{\times}\), then the restriction of \(\pi\) to T is admissible and decomposes as a sum \(\sum b(\pi,\chi) \chi\) where the multiplicity \(b(\pi\),\(\chi)\) of the character \(\chi\) of \(F^{\times}\) is equal to 0 or 1. In this paper is presented a conjectural formula for \(b(\pi\),\(\chi)\) in terms of the \(\epsilon\)-factor of the lift of \(\pi\) to GL(2,L) twisted by \(\chi^{- 1}\). This conjecture was suggested by results of Waldspurger and is proved here in a great number of cases. However the method is not very enlightening since it consists of explicit computations of the \(\epsilon\)-factor terms and comparison with known formulae for the characters. The conjecture is proved in the case of principal or special representations. When \(\pi\) is supercuspidal, it is proved if L is split or if L has odd residue characteristic but the central character \(\omega_{\pi}\) of \(\pi\) is a square (however this last restriction is removable with more computations of the same kind). The case of even residue characteristic is more complicated, which is not surprising in view of Langlands correspondence with degree 2 representations of the Weil group of F. Computations are done in some cases and the conjecture proved when F has \({\mathbb{F}}_ 2\) for residue field, and \(\pi\) is supercuspidal and verifies \(a(\pi)=2\) or \(a(\pi)=3\), \(\omega_{\pi}=1\). A direct proof of the conjecture is most desirable.

Reviewer: G.Henniart

MSC:

11S37	Langlands-Weil conjectures, nonabelian class field theory
22E50	Representations of Lie and linear algebraic groups over local fields

Keywords:

formula for character of infinite dimensional representations of GL(2,K); admissible representations; characters; \(\epsilon\)-factors