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Weil representation, Howe duality, and the theta correspondence. (English) Zbl 0820.11027
Murty, M. Ram (ed.), Theta functions: from the classical to the modern. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 1, 105-127 (1993).
This expository paper concerns Howe duality and the theta correspondence. Topics covered include: the Heisenberg group and the Stone-von Neumann theorem; the Weil representation and the metaplectic group; the definition and classification of dual reductive pairs; the Howe duality conjecture; Howe’s duality theorems in the archimedean case; the non- archimedean spherical case; seesaw dual pairs; the adelic theta correspondence; and open questions.
For the entire collection see [Zbl 0777.00021].

11F27 Theta series; Weil representation; theta correspondences
11F37 Forms of half-integer weight; nonholomorphic modular forms
22E50 Representations of Lie and linear algebraic groups over local fields
11-02 Research exposition (monographs, survey articles) pertaining to number theory
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings