Weil representation

E244847

automorphic representation (in a broad sense) object in representation theory projective unitary representation representation of a group

The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.

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All labels observed (4)

Label	Occurrences
Weil representation canonical	8
Schrödinger model of the Weil representation	1
Weil representation of SL_2 over local fields	1
Weil representation of the metaplectic group	1

How this entity was disambiguated

This entity first appeared as the object of triple T2228058 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Weil representation
Context triple: [André Weil, notableConcept, Weil representation]

A. The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
B. Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
C. Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
D. Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
E. Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Weil representation
Target entity description: The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.

A. The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
B. Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
C. Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
D. Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
E. Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
F. None of above. chosen

Statements (52)

Predicate	Object
instanceOf	automorphic representation (in a broad sense) ⓘ object in representation theory ⓘ projective unitary representation ⓘ representation of a group ⓘ
actsOn	Schwartz space of a symplectic vector space ⓘ Schwartz–Bruhat space ⓘ space of functions ⓘ
appearsIn	Weil’s "Sur certains groupes d’opérateurs unitaires" ⓘ
constructedVia	Fock model ⓘ Schrödinger model ⓘ linked to: Schrödinger formulation of quantum mechanics Stone–von Neumann theorem ⓘ action of the symplectic group on the Heisenberg group ⓘ
definedOn	double cover of the symplectic group ⓘ metaplectic group ⓘ symplectic group ⓘ
definedOver	finite fields ⓘ global fields ⓘ local fields ⓘ p-adic fields ⓘ real numbers ⓘ
hasAlternativeName	metaplectic representation ⓘ oscillator representation ⓘ
hasProperty	admits local and global versions ⓘ compatible with tensor products of symplectic spaces ⓘ gives a projective representation of the symplectic group ⓘ lifts to a genuine representation of the metaplectic group ⓘ realized on L^2-spaces in the Schrödinger model ⓘ splits into even and odd parts in many cases ⓘ
introducedBy	André Weil ⓘ
isAssociatedWith	Heisenberg group ⓘ linked to: Heisenberg Lie algebra non-degenerate symplectic form ⓘ symplectic vector space ⓘ
isFundamentalIn	automorphic forms ⓘ global theta lifting ⓘ local Langlands program (via theta correspondence) ⓘ linked to: Langlands program number theory ⓘ theory of theta functions ⓘ theta correspondence ⓘ
isLinearRepresentationOf	metaplectic group ⓘ
isProjectiveRepresentationOf	symplectic group ⓘ
isUnitary	true ⓘ
isUsedFor	Weil’s proof of the functional equation of L-functions (conceptually related) ⓘ construction of theta series ⓘ explicit formulas in the theory of modular forms ⓘ realization of correspondences between representations of different groups ⓘ studying dual reductive pairs ⓘ theta lifting between automorphic representations ⓘ
relatedTo	Fourier transform on a local field ⓘ Maslov index ⓘ Weil index ⓘ Weil representation of SL_2 over local fields ⓘ linked to: Weil representation metaplectic cover ⓘ