# Metaplectic correspondences and unitary representations

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Metaplectic correspondences and unitary representations COMPOSITIO MATHEMATICA JING-SONGHUANG Metaplectic correspondences and unitary representations Compositio Mathematica, tome 80, no 3 (1991), p. 309-322. <http://www.numdam.org/item?id=CM_1991__80_3_309_0> © Foundation Compositio Mathematica, 1991, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 309 Metaplectic correspondences and unitary representations JING-SONG HUANG* School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, U.S.A. Received 11 April 1990; accepted 11 March 1991 Compositio Mathematica 80: 309-322, 1991. (Ç) 1991 Kluwer Academic Publishers. Printed in the Netherlands. 1. Introduction It is observed in [H] that there is a correspondence from the irreducible unitary almost spherical representations (cf. Definition 5.1 of [H]) of the universal covering group of GL(n, R) to the irreducible unitary spherical representations of GL(n, R). When n is greater than or equal to 3, the universal covering group of GL(n, R) is just a double cover. We write G = GL(n, R) for the double cover of G = GL(n, R). We let B = MAN be a Borel subgroup of GL(n, R) and let B = MAN be the corresponding Borel subgroup of GL(n, R). We let £5 be the pin representation of M. Then we observe that the Langlands quotient of the principal series Indi( £5 0 v (D 1) is unitary if and only if the Langlands quotient of the principal series IndB(1 (D 2v (8) 1) is unitary. The correspondence is given by Langlands quotient of IndG(03B4 Q v Q 1) ~

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