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MR2096864 (2005j:11037) 11F70 (22E50)
Manderscheid, David (1-IA)
Waldspurger's involution and types. (English summary)
J. London Math. Soc. (2) 70 (2004), no. 3, 567585.
Let F be a p-adic field, and let G denote the unique nontrivial two-fold cover of G = SL2(F).
Let V1 and V2 be orthogonal vector spaces associated to ternary quadratic forms with V1 split and
V2 anisotropic. Then (G, O(Vi)), i = 1, 2, are dual reductive pairs. If is a genuine irreducible
discrete series representation of G, then the results of J.-L. Waldspurger [in Festschrift in honor of
I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), 267324,
Weizmann, Jerusalem, 1990; MR1159105 (93h:22035); Forum Math. 3 (1991), no. 3, 219307;
MR1103429 (92g:11054)], together with the Jacquet-Langlands correspondence, attach a genuine
irreducible discrete series representation of G, denoted by W , to . This is the Waldspurger
involution of .
In this paper, under the assumption that p is odd, the author first proves that generically Wald-
spurger's involution takes supercuspidals to supercuspidals. When both and W are supercuspi-
dals, the main theorem of the paper parametrizes W in terms of types. In this case, W is obtained