Math 2.0

Topological groups, Lie groups Classical Lie groups Example

Metaplectic group

Builds on Symplectic groupFundamental groupCovering spaceSL(2,R)

Example

Let n≥1n \ge 1 and let Sp(2n,ℝ)\mathrm{Sp}(2n, \R) be the real symplectic group. It is a connected Lie group, and U(n)U(n) is a maximal compact subgroup of it, so its fundamental group is isomorphic to π1(U(n))≅ℤ\pi_1(U(n)) \cong \Z. The group ℤ\Z has exactly one subgroup of index 22, so Sp(2n,ℝ)\mathrm{Sp}(2n, \R) has exactly one connected two-sheeted covering, up to isomorphism. That covering space carries a unique Lie group structure for which the covering map is a homomorphism.

The metaplectic group Mp(2n,ℝ)\mathrm{Mp}(2n, \R) is this connected double cover of Sp(2n,ℝ)\mathrm{Sp}(2n, \R). So there is an exact sequence 1→{±1}→Mp(2n,ℝ)→Sp(2n,ℝ)→1.1 \to \set{\pm 1} \to \mathrm{Mp}(2n, \R) \to \mathrm{Sp}(2n, \R) \to 1. For n=1n = 1, Sp(2,ℝ)=\mathrm{Sp}(2, \R) = SL(2,R), and Mp(2,ℝ)\mathrm{Mp}(2, \R) is the connected double cover of SL(2,ℝ)SL(2, \R).

Remarks

Mp(2n,ℝ)\mathrm{Mp}(2n, \R) is not a matrix group: it has no faithful finite-dimensional representation. It does have a faithful unitary representation on the infinite-dimensional Hilbert space L2(ℝn)L^2(\R^n), the oscillator representation, also called the Segal–Shale–Weil representation. This representation does not come from a representation of Sp(2n,ℝ)\mathrm{Sp}(2n, \R); it is defined only up to sign on Sp(2n,ℝ)\mathrm{Sp}(2n, \R). The group Mp(2,ℝ)\mathrm{Mp}(2, \R) is the setting for modular forms of half-integral weight, such as the Jacobi theta function.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectCovering space, concept, no evidence levelCovering spaceconceptFundamental group, concept, no evidence levelFundamental groupconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptSymplectic group, object, no evidence levelSymplectic groupobjectVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial linear group, object, no evidence levelSpecial linear groupobjectUpper half-plane, object, no evidence levelUpper half-planeobjectSL(2,R), example, no evidence levelSL(2,R)exampleMetaplectic group, example, no evidence levelMetaplectic groupexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Covering space concept
  3. Fundamental group concept
  4. General linear group object
  5. Identity matrix object
  6. Matrix definition
  7. Möbius transformation concept
  8. Symplectic group object
  9. Vector space definition
  10. Square matrix definition
  11. Transpose definition
  12. Linear map definition
  13. Multilinear map definition
  14. Orthogonal matrix definition
  15. Alternating multilinear map definition
  16. Orthogonal group object
  17. Determinant definition
  18. Special orthogonal group object
  19. Special linear group object
  20. Upper half-plane object
  21. SL(2,R) example
  22. Metaplectic group example

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Metaplectic_group
About this page
Tag8B0JM2YDE2
URIm2://8B0JM2YDE2
Kindexample
WritingW0 stub
MSC22E46, 11F27, 22E15
Instance ofLie group
UsesSymplectic group, Fundamental group, Covering space, SL(2,R)
Same as
Changed2 November 2026

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