Stable base change C/R of certain derived functor modules
Let \({\mathcal G}\) be a connected complex reductive algebraic group defined over \({\mathbb{R}}\). Let G(\({\mathbb{C}})\) be its complex points and G(\({\mathbb{R}})\) be its real points. Let \(\Gamma =\{1,\sigma \}\) be the Galois group of \({\mathbb{C}}\) over \({\mathbb{R}}\). Suppose \(\Pi\) is an irreducible representation of G(\({\mathbb{C}})\) which is \(\sigma\)-stable, i.e. equivalent to \(\Pi\) \(\circ \sigma\), then \(\Pi\) extends to a representation of G(\({\mathbb{C}})\rtimes \Gamma\). Suppose that \(g\in G({\mathbb{C}})\) is such that \(g\cdot \sigma (g)\) is conjugate to an element Ng\(\in G({\mathbb{R}})\) which is regular and semisimple. Let \(\chi_{\Pi}\) be the character of \(\Pi\). Then \textit{L. Clozel} has shown [Ann. Sci. Ec. Norm. Supér., IV. Sér. 15, 45-115 (1982; Zbl 0516.22010)] that if \(\pi\) is the direct sum of all irreducible representations of G(\({\mathbb{R}})\) in a tempered L-packet and \(\chi_{\pi}\) is its character then there exists \(\Pi\) as above such that \(\chi_{\Pi}(g\times \sigma)=\pm \chi_{\pi}(Ng)\) and \(\Pi\) is given by Langlands functoriality applied to \(\pi\). In this paper we extend this result to the non-tempered enlarged packets (of derived functor modules) defined by \textit{J. Adams} and \textit{J. E. Johnson} [Compos. Math. 64, 271-309 (1987; Zbl 0647.22008)].
connected complex reductive algebraic group
scientific article; zbMATH DE number 4099582
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