On an extension of a theorem of Tunnell. (English) Zbl 0824.11035

Let \(k\) be a non-archimedean local field of characteristic \(\neq 2\), and \(K\) a quadratic extension of \(k\). \(x\to \overline {x}\) is the nontrivial automorphism of \(K\) over \(k\). For a character \(\theta\) of \(K^*\) denote by \(\overline {\theta}\) the character \(\overline {\theta} (x)= \theta (\overline {x})\). Denote by \(\omega_{K/k}\) the quadratic character of \(k^*\) associated by the class-field theory to \(K\). Put \(\text{GL} (2,k)^ += \{x\in \text{GL}(2,k) \mid \det(x)\in NK^*\}\) where \(NK^*\) is the subgroup of \(k^*\) of index 2 consisting of norms from \(K^*\). If \(D_ k\) is the unique quaternion division algebra over \(k\), put \(D_ k^{*+}\) the corresponding subgroup of index 2 in \(D_ k^*\). Clearly \(K^* \subset \text{GL} (2,k)^ +\) and also \(\subset D_ k^{*+}\). J. Tunnell [Am. J. Math. 105, 1277–1307 (1983; Zbl 0532.12015)] and H. Saito [Compos. Math. 85, 99–108 (1993; Zbl 0795.22009)] have described which characters of \(K^*\) appear in an irreducible representation of \(\text{GL}(2;k)\) (resp. of \(D_ k^ +\)). In this note the author yields an extension of the theorem of Tunnell. For a discrete series representation \(\pi\) of \(\text{GL}(2; k)\) denote by \(\pi'\) the representation of \(D^*_ k\) associated by Jacquet-Langlands to \(\pi\).
Theorem. Let \(\pi\) be an irreducible admissible representation of \(\text{GL}(2;k)\) associated to a character \(\theta\) of \(K^*\). Fix embeddings of \(K^*\) in \(\text{GL}(2; k)^ +\) and in \(D_ k^{*+}\), choose an additive character \(\psi\) of \(k\), and an element \(x_ 0\) of \(K^*\) with \(\text{tr} (x_ 0)=0\). Put \(\psi_ 0 (x)= \psi (\text{tr } {{-xx_ 0} \over 2})\). Then \[ \pi|_{\text{GL}(2,k)^ +}= \pi_ + \oplus \pi_ -, \qquad \pi'|_{D_ k^{*+}}= \pi_ +^ \prime \oplus \pi_ -^ \prime, \] wherein for every character \(\chi\) of \(K^*\) with \((\chi \theta^{-1} )|_{k^*}= \omega_{K/k}\)
\(\chi\) appears in \(\pi_ + \iff \varepsilon (\theta\chi^{-1}, \psi_ 0)= \varepsilon (\overline {\theta} \chi^{-1}, \psi_ 0)=1\);
\(\chi\) appears in \(\pi_ - \iff \varepsilon (\theta \chi^{-1}, \psi_ 0)= \varepsilon (\overline {\theta} \chi^{-1}, \psi_ 0) =-1\);
\(\chi\) appears in \(\pi_ +^ \prime \iff \varepsilon (\theta \chi^{- 1}, \psi_ 0) =1\) and \(\varepsilon (\overline {\theta} \chi^{-1}, \psi_ 0) =-1\);
\(\chi\) appears in \(\pi_ -^ \prime \iff \varepsilon (\theta x^{-1}, \psi_ 0) =-1\) and \(\varepsilon (\overline {\theta} \chi^{-1}, \psi_ 0)=1\).
For deriving these conditions on \(\varepsilon\)-factors the author proves an interesting lemma [Lemma 3.1].


11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
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