metaplectic group
E860123
The metaplectic group is a double cover of the symplectic group that plays a central role in number theory and representation theory, particularly through its connection to theta functions and automorphic forms.
All labels observed (3)
| Label | Occurrences |
|---|---|
| metaplectic group canonical | 4 |
| metaplectic cover | 1 |
| symplectic group Sp(2n,ℝ) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10389292 — 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: metaplectic group Context triple: [Weil representation, definedOn, metaplectic group]
-
A.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
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B.
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.
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C.
Weil representation
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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D.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
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E.
Brauer group
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
- 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: metaplectic group Target entity description: The metaplectic group is a double cover of the symplectic group that plays a central role in number theory and representation theory, particularly through its connection to theta functions and automorphic forms.
-
A.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
B.
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.
-
C.
Weil representation
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.
-
D.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
-
E.
Brauer group
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
double cover
ⓘ
mathematical group ⓘ topological group ⓘ |
| actsOn |
Schwartz space of a symplectic vector space
ⓘ
space of theta functions ⓘ |
| arisesFrom |
Maslov index
ⓘ
square root of the determinant line bundle ⓘ |
| covers | symplectic group ⓘ |
| hasCenter | cyclic group of order 2 ⓘ |
| hasDimension | same dimension as the symplectic group ⓘ |
| hasLieAlgebra | symplectic Lie algebra ⓘ |
| hasNontrivialElementInKernelOfCoveringMap | element of order 2 ⓘ |
| hasRepresentation |
Weil representation
ⓘ
oscillator representation ⓘ |
| isAnalogOf | spin group for the symplectic group ⓘ |
| isCentralExtensionOf |
symplectic group
ⓘ
linked to:
symplectic group Sp(2n,R)
|
| isConstructedVia |
Stone–von Neumann theorem
ⓘ
central extension of the symplectic group by μ₂ ⓘ projective representation of the symplectic group ⓘ |
| isDefinedOver |
global fields
ⓘ
local fields ⓘ p-adic fields ⓘ real numbers ⓘ |
| isDoubleCoverOf |
symplectic group
ⓘ
linked to:
symplectic group Sp(2n,R)
|
| isNontrivialCoverWhen | dimension of underlying symplectic space is at least 2 ⓘ |
| isRelatedTo |
Heisenberg group
ⓘ
linked to:
Heisenberg Lie algebra
Maslov class ⓘ metaplectic cover ⓘ
linked to:
metaplectic group
metaplectic representation ⓘ spin group ⓘ |
| isUsedIn |
Fourier analysis on symplectic vector spaces
ⓘ
Langlands program ⓘ Shimura correspondence ⓘ Siegel modular forms ⓘ Weil representation ⓘ automorphic forms ⓘ covering groups in the Langlands program ⓘ geometric quantization ⓘ half-integral weight modular forms ⓘ harmonic analysis ⓘ metaplectic forms ⓘ microlocal analysis ⓘ number theory ⓘ oscillator representation ⓘ quantization ⓘ representation theory ⓘ theory of automorphic L-functions ⓘ theta correspondence ⓘ theta functions ⓘ |
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: metaplectic group Description of subject: The metaplectic group is a double cover of the symplectic group that plays a central role in number theory and representation theory, particularly through its connection to theta functions and automorphic forms.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
semisimple Lie groups
linked to: metaplectic group
linked to: metaplectic group
subject linked to:
Weil’s "Sur certains groupes d’opérateurs unitaires"