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Metaplectic representation and Howe conjectures. (Représentation métaplectique et conjectures de Howe.) (French) Zbl 0634.10026
Sémin. Bourbaki, 39ème année, Vol. 1986/87, Exp. No. 674, Astérisque 152/153, 85-99 (1987).
[For the entire collection see Zbl 0627.00006.]
This survey discusses ideas, results and conjectures concerning the metaplectic representation.
§1 Theta functions; relation between classical and adelic formulation,
§2 Metaplectic group, metaplectic representation; relation with theta functions.
§3 Dual reductive pairs and metaplectic representation.
§4 The conjectures of Howe. For a dual reductive pair $$(U_1,U_2)$$ one may form the inverse images $$(\tilde{U}_1)$$ and $$(\tilde{U}_2)$$ in the metaplectic group. One studies how the metaplectic representation associates representations of $$(\tilde{U}_2)$$ to representations of $$(\tilde{U}_1)$$. In the local case it is conjectured that irreducibility is conserved in some sense. The local cases may be combined to give a global correspondence between representations of adele groups, which is conjectured to conserve the property of being automorphic.
§5 Results of Kudla, in the local non-archimedean case, which shows that the correspondence is compatible with parabolic induction.
§6 Results of Howe. Proof of the local conjecture in some cases.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11F33 Congruences for modular and $$p$$-adic modular forms
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