Waldspurger’s involution and lifting of characters

Devlin, Stephen; Schultz, Jason

doi:10.1090/S0002-9939-06-08522-4

Waldspurger’s involution and lifting of characters
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by Stephen Devlin and Jason Schultz PDF

Proc. Amer. Math. Soc. 135 (2007), 911-919 Request permission

Abstract:

We show that Adams’s $L$-packet of discrete series representations of the two-fold metaplectic cover of $SL_2$ coincides with Waldspurger’s local near equivalence class.

References

Jeffrey Adams, Characters of covering groups of $\textrm {SL}(n)$, J. Inst. Math. Jussieu 2 (2003), no. 1, 1–21. MR 1955205, DOI 10.1017/S147474800300001X
Yuval Z. Flicker, Automorphic forms on covering groups of $\textrm {GL}(2)$, Invent. Math. 57 (1980), no. 2, 119–182. MR 567194, DOI 10.1007/BF01390092
Yuval Z. Flicker and David A. Kazhdan, Metaplectic correspondence, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 53–110. MR 876160
Stephen S. Gelbart, Weil’s representation and the spectrum of the metaplectic group, Lecture Notes in Mathematics, Vol. 530, Springer-Verlag, Berlin-New York, 1976. MR 0424695
Stephen Gelbart, Roger Howe, and Ilya Piatetski-Shapiro, Uniqueness and existence of Whittaker models for the metaplectic group, Israel J. Math. 34 (1979), no. 1-2, 21–37 (1980). MR 571393, DOI 10.1007/BF02761822
S. Gelbart and I. Piatetski-Shapiro, Some remarks on metaplectic cusp forms and the correspondences of Shimura and Waldspurger, Israel J. Math. 44 (1983), no. 2, 97–126. MR 693355, DOI 10.1007/BF02760615
Stephen Gelbart and I. I. Piatetski-Shapiro, Distinguished representations and modular forms of half-integral weight, Invent. Math. 59 (1980), no. 2, 145–188. MR 577359, DOI 10.1007/BF01390042
S. Gelbart and I. Piatetski-Shapiro, On Shimura’s correspondence for modular forms of half-integral weight, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Institute of Fundamental Research, Bombay, 1981, pp. 1–39. MR 633657
D. A. Kazhdan and S. J. Patterson, Metaplectic forms, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 35–142. MR 743816
D. A. Kazhdan and S. J. Patterson, Towards a generalized Shimura correspondence, Adv. in Math. 60 (1986), no. 2, 161–234. MR 840303, DOI 10.1016/S0001-8708(86)80010-X
J.-P. Labesse and R. P. Langlands, $L$-indistinguishability for $\textrm {SL}(2)$, Canadian J. Math. 31 (1979), no. 4, 726–785. MR 540902, DOI 10.4153/CJM-1979-070-3
David Manderscheid, Waldspurger’s involution and types, J. London Math. Soc. (2) 70 (2004), no. 3, 567–585. MR 2096864, DOI 10.1112/S0024610704005721
R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pacific J. Math. 157 (1993), no. 2, 335–371. MR 1197062
J. Schultz, Lifting of characters on $\widetilde {SL}(2)$ and $SO(1,2)$ over a non-archimedean local field, Ph.D. Thesis, University of Maryland (1998).
J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, 1–132 (French). MR 577010
Jean-Loup Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219–307 (French). MR 1103429, DOI 10.1515/form.1991.3.219
André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI 10.1007/BF02391012