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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Weil representation canonical | 2 |
| Weil representation of SL_2 over local fields | 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 formulation of quantum mechanics ⓘ
surface form:
Schrödinger model
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 Lie algebra
ⓘ
surface form:
Heisenberg group
non-degenerate symplectic form ⓘ symplectic vector space ⓘ |
| isFundamentalIn |
automorphic forms
ⓘ
global theta lifting ⓘ Langlands program ⓘ
surface form:
local Langlands program (via theta correspondence)
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 self-linksurface differs ⓘ
surface form:
Weil representation of SL_2 over local fields
metaplectic cover ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Weil representation Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Weil representation of SL_2 over local fields