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| Howe correspondence | |
|---|---|
| Name | Howe correspondence |
| Othernames | theta correspondence |
| Field | Representation theory |
| Introduced | 1970s |
| Founder | Roger Howe |
| Related | Weil representation, dual reductive pairs, theta series |
Howe correspondence
The Howe correspondence is a framework in representation theory relating pairs of representations of classical groups through the action of the metaplectic group and the Weil representation. Originating in the work of Roger Howe and influenced by earlier results of André Weil and developments around theta series, it provides a systematic method to match representations of dual reductive pairs such as (Sp(2n), O(V)) or (U(p,q), U(r,s)). The theory has deep connections to the Langlands program, automorphic forms, and number-theoretic constructions like the Siegel modular form and the theta correspondence in automorphic representation theory.
The correspondence began as a local and global mechanism to pair irreducible admissible representations of two groups forming a dual reductive pair inside a larger symplectic or metaplectic group. Initial motivations came from explicit theta lifting in the work of André Weil on the Weil representation and from classical theta series studied by Carl Gustav Jacobi and Srinivasa Ramanujan. Influential subsequent contributions include those of Roger Howe, Stephen Rallis, Binyong Sun, Chen-Bo Zhu, and Herb Gross who clarified multiplicity questions and the scope of the correspondence for real, p-adic, and global fields.
The historical thread begins with André Weil's construction of the oscillator representation of the metaplectic group and classical investigations of theta functions by Martin Eichler and Igor Shafarevich. Roger Howe formulated the duality conjecture in the 1970s, synthesizing ideas from Harish-Chandra's harmonic analysis and the algebraic theory of classical groups as treated by Claude Chevalley and Armand Borel. Subsequent progress involved local nonarchimedean fields via the work of Bernard Dwork, Ilya Piatetski-Shapiro, and Stephen Kudla, while global automorphic aspects were developed by James Arthur, Piatetski-Shapiro, Robert Langlands, and David Ginzburg. Recent breakthroughs addressing the Howe duality conjecture and multiplicity-one properties were achieved by Wee Teck Gan, Gordan Savin, Binyong Sun, and Chen-Bo Zhu.
Formally, one starts with a symplectic vector space over a local field and considers the associated metaplectic double cover of the symplectic group Sp(2n). Inside this group one locates a dual reductive pair (G, G') such as (O(V), Sp(W)) or (GL_n, GL_m), following the classification by Roger Howe and E. Cartan's classical groups. The Weil representation, constructed by André Weil and later algebraically interpreted by Harish-Chandra frameworks and Joseph Bernstein, realizes an action of the metaplectic group on a Schwartz–Bruhat space. Upon restricting this representation to G × G', one obtains a correspondence: for an irreducible admissible representation π of G, its theta lift θ(π) is a (possibly zero) representation of G' defined by maximal quotients or maximal π-isotypic subspaces. The notion of Howe duality predicts multiplicity-one behavior and a bijection between certain packets of irreducible representations, with formalism refined using the machinery of Jacquet modules, Bernstein center, and the theory of tempered representations developed by Harish-Chandra and Wilfried Schmid.
Local theory distinguishes between real (archimedean) and p-adic (nonarchimedean) fields. Over real fields, techniques from Harish-Chandra theory, the Langlands classification, and the work of David Vogan are central. Over p-adic fields, the methods of Colin Bushnell, Guy Henniart, and Jules Tits inform the classification of admissible representations and the behavior of theta lifts. Globally, automorphic representations on adelic groups like GL_n(𝔸), Sp(2n,𝔸), and O(V,𝔸) are connected via theta series and the integral transforms pioneered by Roger Howe and Stephen Rallis. Global theta lifts produce instances of functoriality predicted by the Langlands program and tie into the theory of L-functions studied by Jacquet, Shalika, and Friedrichs.
The theta correspondence is the manifestation of the Howe correspondence via explicit theta kernels built from the Weil representation. The oscillator representation provides an explicit Schwartz kernel on which both members of a dual pair act; integrating automorphic forms against this kernel yields theta lifts between automorphic representations of groups like Sp(2n), O(n), and U(p,q). Key contributors to the structural study of the Weil representation include André Weil, Roger Howe, Stephen Rallis, and Stephen S. Kudla. Analytic tools involve the theory of Eisenstein series by Eisenstein/Langlands and analytic continuation techniques used in the work of Stephen Gelbart and Ilya Piatetski-Shapiro.
Concrete instances include the classical Shimura correspondence linking half-integral weight modular forms (arising from SL_2) to integral weight forms on GL_2, explicated by Goro Shimura. The Saito–Kurokawa lifts between Siegel modular forms and elliptic modular forms were analyzed by Hisaaki Kurokawa and Toshikazu Saito. Explicit local correspondences have been computed for small rank dual pairs, including work on GL_1 × GL_2 and O(2) × Sp(2), with computational techniques contributed by Wee Teck Gan and Gordan Savin. In the nonarchimedean setting, explicit models use lattice filtrations studied in the tradition of James Milne and Martin Kneser.
The Howe correspondence provides explicit instances of functorial transfers predicted by the Langlands program and supplies concrete tools for constructing non-tempered and CAP representations examined by Henryk Iwaniec, Dorian Goldfeld, and Cogdell. It has been used to prove cases of the Gan–Gross–Prasad conjectures developed by Wee Teck Gan, Binyong Sun, and Chen-Bo Zhu, and features in the study of period integrals by Yuri Flicker and Joseph Bernstein. The correspondence interfaces with the local Langlands correspondences formulated by Robert Langlands, Pierre Deligne, and Michael Harris, and with the trace formula methods of James Arthur and Werner Müller.