zbMATH — the first resource for mathematics

Dualizing involutions on the metaplectic \(\operatorname{GL}(2)\). (English) Zbl 1441.22027
Summary: Let \(F\) be a non-Archimedean local field of characteristic zero. Let \(G=\operatorname{GL}(2, F)\) and \(\widetilde{G}=\widetilde{\operatorname{GL}}(2, F)\) be the metaplectic group. Let \(\tau\) be the standard involution on \(G\). A well known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of \(G\) to its contragredient. In such a case, we say that \(\tau\) is a dualizing involution. In this paper, we show that any lift of the standard involution to \(\widetilde{G}\) is also a dualizing involution.
22E50 Representations of Lie and linear algebraic groups over local fields
Full Text: DOI
[1] Gelbart, Stephen S., Weil’s Representation and the Spectrum of the Metaplectic Group, Lecture Notes in Mathematics, vol. 530 (1976), Springer-Verlag: Springer-Verlag Berlin-New York, MR 0424695 · Zbl 0365.22017
[2] Gel’fand, I. M.; Kajdan, D. A., Representations of the group \(\operatorname{GL}(n, K)\) where K is a local field, (Lie Groups and Their Representations, Proc.. Lie Groups and Their Representations, Proc., Summer School, Bolyai János Math. Soc., Budapest, 1971 (1975), Halsted: Halsted New York), 95-118, MR 0404534 (53 #8334)
[3] Harish-Chandra, A submersion principle and its applications, Proc. Indian Acad. Sci. Math. Sci., 90, 2, 95-102 (1981), MR 653948 · Zbl 0512.22010
[4] Kable, Anthony C., The main involutions of the metaplectic group, Proc. Am. Math. Soc., 127, 4, 955-962 (1999), MR 1610921 · Zbl 0912.19001
[5] Kazhdan, D. A.; Patterson, S. J., Metaplectic forms, Publ. Math. IHÉS, 59, 35-142 (1984), MR 743816 · Zbl 0559.10026
[6] Kazhdan, D. A.; Patterson, S. J., Towards a generalized Shimura correspondence, Adv. Math., 60, 2, 161-234 (1986), MR 840303 · Zbl 0616.10023
[7] Kubota, Tomio, On automorphic functions and the reciprocity law in a number field, (Lectures in Mathematics. Lectures in Mathematics, Department of Mathematics, Kyoto University, vol. 2 (1969), Kinokuniya Book-Store Co., Ltd.: Kinokuniya Book-Store Co., Ltd. Tokyo), MR 0255490 · Zbl 0231.10017
[8] Li, Wen-Wei, La formule des traces pour les revêtements de groupes réductifs connexes. II. Analyse harmonique locale, Ann. Sci. Éc. Norm. Supér. (4), 45, 5, 787-859 (2012), (2013), MR 3053009 · Zbl 1330.11037
[9] Mackey, George W., Les ensembles boréliens et les extensions des groupes, J. Math. Pures Appl. (9), 36, 171-178 (1957), MR 0089998 · Zbl 0080.02303
[10] Roche, Alan; Vinroot, C. Ryan, A factorization result for classical and similitude groups, Can. Math. Bull., 61, 1, 174-190 (2018), MR 3746483 · Zbl 06856975
[11] Serre, J.-P., A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7 (1973), Springer-Verlag: Springer-Verlag New York-Heidelberg, translated from the French, MR 0344216 · Zbl 0256.12001
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.