Towards a generalized Shimura correspondence. (English) Zbl 0616.10023

Kazhdan, D. A.; Patterson, S. J.

Adv. Math. 60, 161-234 (1986).

The concept of the ”Shimura correspondence” is based on a correspondence discovered by G. Shimura between modular forms of half-integral and integral weight which was announced by him in 1972 at the Antwerp Conference on Modular Forms [cf. Ann. Math., II. Ser. 97, 440-481 (1973; Zbl 0266.10022)]. Shimura formulated his results in terms of L-functions. A different method, based on theta-functions, was found a little later by T. Shintani and S. Niwa.
All these results were reformulated in terms of the metaplectic covering groups of GL(2) by the reviewer and I. Piatetski-Shapiro, who also separated the correspondence into local and global parts [Automorphic forms, representation theory and arithmetic, Pap. Colloq., Bombay 1979, 1-39 (1981; Zbl 0495.10020)]. A quite different approach was initiated by Y. Z. Flicker, who used the trace formula, and described the local correspondence in terms of a comparison of characters [Invent. Math. 57, 119-182 (1980; Zbl 0431.10014)].
The paper under review is concerned exclusively with this latter method, directed toward a ”generalized Shimura correspondence” for n-fold metaplectic covers \(\tilde G\) of \(GL_ r(F)\), where r is arbitrary and F is a local field of characteristic zero. The goal of obtaining such a correspondence has been realized in recent work of Y. Z. Flicker and the first author [Publ. Math., Inst. Hautes Étud. Sci. 64, 53-110 (1986; see the following review)]. In the present paper, the correspondence is formulated in general, and a very special case of it proved using the trace formula.
The first half of the paper is concerned with preliminary local material on orbital integrals and spherical functions for the metaplectic group, the main result being that the unit elements of the Hecke algebras for \(\tilde G\) and \(GL_ r(F)\) have matching orbital integrals. The second half of the paper establishes the ”local” Shimura correspondence for certain exceptional representations \(V_ 0({\tilde \omega})\) which are components of (reducible) non-unitary principal series representations and which correspond to the trivial representation of the ordinary group \(GL_ r(F)\); as a corollary, the set of Whittaker models for these representations is analyzed following earlier work of the authors [ibid. 59, 35-142 (1984; Zbl 0559.10026)]. As already suggested, the proof is global and uses the trace formula.

Reviewer: S.Gelbart

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11F11	Holomorphic modular forms of integral weight
22E50	Representations of Lie and linear algebraic groups over local fields

Keywords:

trace formula; Shimura correspondence; local correspondence; comparison of characters; generalized Shimura correspondence; metaplectic covers; local field; principal series representations; Whittaker models

Citations:

Zbl 0616.10024; Zbl 0266.10022; Zbl 0495.10020; Zbl 0431.10014; Zbl 0559.10026

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