Balasubramanian, Kumar; Bhand, Ajit Dualizing involutions on the metaplectic \(\operatorname{GL}(2)\). (English) Zbl 1441.22027 J. Pure Appl. Algebra 225, No. 1, Article ID 106479, 14 p. (2021). 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. Cited in 1 Document MSC: 22E50 Representations of Lie and linear algebraic groups over local fields Keywords:metaplectic groups; dualizing involutions; genuine representations PDFBibTeX XMLCite \textit{K. Balasubramanian} and \textit{A. Bhand}, J. Pure Appl. Algebra 225, No. 1, Article ID 106479, 14 p. (2021; Zbl 1441.22027) Full Text: DOI References: [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) · Zbl 0348.22011 [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 1494.20074 [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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.