Weil representation

Weil representation
Name	Weil representation
Field	Representation theory, Number theory
Introduced	1930s
Founder	André Weil
Related	Metaplectic group, Symplectic group, Theta function

Weil representation The Weil representation is a fundamental unitary representation arising in the study of the symplectic group and the metaplectic double cover, introduced in analytic and arithmetic contexts by André Weil in the 1930s. It connects the works of André Weil, Hermann Weyl, John von Neumann, Erwin Schrödinger, and Harish-Chandra through links to theta functions, quadratic forms, and harmonic analysis on locally compact abelian groups. The representation plays a central role in the theory of automorphic forms, the Langlands program, and the study of theta correspondences involving groups such as GL_n, SL_2, and classical groups.

Definition and basic properties

The Weil representation is defined for a symplectic vector space V over a local or global field together with a nondegenerate alternating form and a choice of additive character; key figures in this development include André Weil, Hermann Weyl, Harish-Chandra, Roger Howe, and Stephen Gelbart. It is a projective representation of the symplectic group that lifts to a genuine representation of the metaplectic group; related objects studied by Ilya Piatetski-Shapiro, Srinivasa Raghavan Ramanujan (via theta), and Atle Selberg illuminate its arithmetic and spectral properties. Important invariants include the central character, Weil index, and compatibility with Fourier transform as in the work of Norbert Wiener, John Tate, and Jean-Pierre Serre.

Construction and models (Schrödinger, lattice, oscillator)

Concrete realizations arise in several classical models: the Schrödinger model realized on L^2-spaces links to Erwin Schrödinger, Paul Dirac, John von Neumann, David Hilbert, and Norbert Wiener; the lattice model uses self-dual lattices as in the theories of Carl Ludwig Siegel, Martin Kneser, Ernst Witt, and Louis Mordell; the oscillator (or Segal–Shale–Weil) model connects to Irving Segal, Daniel Shale, André Weil, and Hermann Weyl. These constructions employ Fourier transform operators tied to Joseph Fourier and quadratic Gauss sums studied by Carl Friedrich Gauss and Emil Artin; they also interact with theta series in the writings of Srinivasa Ramanujan and Carl Gustav Jakob Jacobi.

Relation to symplectic and metaplectic groups

The representation furnishes the canonical genuine representation of the metaplectic double cover of the symplectic group, central in the studies of Paul Dirac’s canonical commutation relations and the structure theory of Sp_{2n} and its covers as examined by Roger Howe, Harish-Chandra, Robert Langlands, and Hiroshi Saito. The metaplectic group appears in harmonic analysis on adele groups in the work of John Tate and Jacques Tits, and the lifting of projective factors involves cocycles studied by Murray Gerstenhaber and Beno Eckmann in group cohomology contexts. Relations to dual reductive pairs introduced by Roger Howe produce correspondences between representations of classical groups such as O_n, U_n, and Sp_{2n} with implications for the theta correspondence analyzed by Stephen Rallis and Ilya Piatetski-Shapiro.

Character formulas and explicit computations

Explicit character formulas for the Weil representation involve Gauss sums, Weil indices, and orbital integrals tied to harmonic analysis results of Harish-Chandra, Atle Selberg, and James Arthur. Computations over local fields rely on classification results of André Weil and explicit evaluation of quadratic exponential integrals as carried out by Emil Artin, John Tate, and Robert Langlands. In the finite field setting, characters relate to work of E. Lucas, Philip Hall, and Jean-Pierre Serre, while p-adic character expansions use Moy–Prasad filtrations and methods influenced by Allen Moy, Gopal Prasad, and Bernstein–Zelevinsky techniques; these yield explicit formulae utilized by Hervé Jacquet and Paul Garrett.

Applications in number theory and automorphic forms

The Weil representation underpins constructions of theta series and theta lifts central to the theories of André Weil, Srinivasa Ramanujan, Goro Shimura, and Kurt Gödel’s contemporaries in modular forms; it features prominently in the theta correspondence connecting automorphic representations of Sp_{2n}, O_n, and U_n as developed by Roger Howe and Stephen Rallis. In the Langlands program of Robert Langlands, the representation provides explicit kernels for integral representations of L-functions studied by James Arthur, Henryk Iwaniec, Wenguang Zhang, and Goro Shimura. Applications include constructions of cusp forms, lifting results in the work of S.-T. Yau’s circle, relations to the Gross–Prasad conjectures examined by Benedict Gross and Dipendra Prasad, and connections to arithmetic geometry studied by Pierre Deligne and Jean-Pierre Serre.

Generalizations and variants (p-adic, finite fields, higher rank)

Generalizations include p-adic Weil representations over local fields treated by John Tate, I. M. Gelfand, Pavel Etingof’s collaborators, and Weil’s successors; finite field analogues were systematically studied by Gerald Lusztig, Pierre Deligne, and Roger Howe. Higher-rank and derived-category variants interact with categorical representation theory as in the works of Joseph Bernstein, Alexander Beilinson, Vladimir Drinfeld, Edward Frenkel, and Dennis Gaitsgory, and they enter quantum versions connected to Max Planck’s foundational physics and deformation quantization studied by Flato, Sternheimer, and Murray Gerstenhaber. Modern developments relate the Weil representation to the relative trace formula studied by James Arthur and to categorical and geometric Langlands themes explored by Edward Frenkel and David Ben-Zvi.

Category:Representation theory