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Trace Paley-Wiener theorem in the twisted case. (English) Zbl 0663.22011
Let \({\mathbb{G}}\) be a connected reductive algebraic group over a p-adic field F (of characteristic 0) and let \(G={\mathbb{G}}(F)\). Let \(\epsilon\) be an outer automorphism of \({\mathbb{G}}\) of finite order r of G. Let \(\pi\) be an irreducible admissible representation of G. Suppose that \(\pi\) is equivalent to \(\pi\) \(\circ \epsilon\), then there exists an intertwining operator \(\pi\) (\(\epsilon)\). In this paper a theorem of Paley-Wiener type is given which characterizes functions of the form \(\pi\) \(\to Tr(\pi (f)\pi (\epsilon))\) for some locally constant functions f on G with compact supports. The paper follows the work of J. Bernstein, P. Deligne and D. Kazhdan [J. Anal. Math. 47, 180-192 (1986; Zbl 0634.22011)].
Reviewer: V.F.Molchanov

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
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