## 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:

 2.2e+51 Representations of Lie and linear algebraic groups over local fields 2.2e+36 Analysis on $$p$$-adic Lie groups

Zbl 0634.22011
Full Text:

### References:

 [1] J. Arthur and L. Clozel, Base change for $$\operatorname{GL} (n)$$, preprint. · Zbl 0666.22004 [2] J. N. Bernstein, Le ”centre” de Bernstein, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1 – 32 (French). Edited by P. Deligne. · Zbl 0599.22016 [3] J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive \?-adic groups, J. Analyse Math. 47 (1986), 180 – 192. · Zbl 0634.22011 [4] I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive \?-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441 – 472. · Zbl 0412.22015 [5] Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. · Zbl 0443.22010 [6] W. Casselman, Characters and Jacquet modules, Math. Ann. 230 (1977), no. 2, 101 – 105. · Zbl 0337.22019 [7] L. Clozel, Characters of non-connected reductive $$p$$-adic groups, preprint. · Zbl 0629.22008 [8] Laurent Clozel, Sur une conjecture de Howe. I, Compositio Math. 56 (1985), no. 1, 87 – 110 (English, with French summary). · Zbl 0599.22015 [9] Seminar on the trace formula, Institute for Advanced Study, Princeton, N. J., 1984. [10] Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ, 1990. · Zbl 0724.11031 [11] Jonathan D. Rogawski, Representations of \?\?(\?) and division algebras over a \?-adic field, Duke Math. J. 50 (1983), no. 1, 161 – 196. · Zbl 0523.22015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.