Definition[edit]

The fundamental group of the symplectic Lie group Sp2n(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2n(R) and called the metaplectic group.


The metaplectic group Mp2(R) is not a matrix group: it has no faithful finite-dimensional representations. Therefore, the question of its explicit realization is nontrivial. It has faithful irreducible infinite-dimensional representations, such as the Weil representation described below.


It can be proved that if F is any local field other than C, then the symplectic group Sp2n(F) admits a unique perfect central extension with the kernel Z/2Z, the cyclic group of order 2, which is called the metaplectic group over F. It serves as an algebraic replacement of the topological notion of a 2-fold cover used when F = R. The approach through the notion of central extension is useful even in the case of real metaplectic group, because it allows a description of the group operation via a certain cocycle.

G is a vector space over the reals of dimension n. This gives a metaplectic group that is a double cover of the Sp2n(R).

symplectic group

More generally G can be a vector space over any F of dimension n. This gives a metaplectic group that is a double cover of the symplectic group Sp2n(F).

local field

G is a vector space over the of a number field (or global field). This case is used in the representation-theoretic approach to automorphic forms.

adeles

G is a finite group. The corresponding metaplectic group is then also finite, and the central cover is trivial. This case is used in the theory of of lattices, where typically G will be the discriminant group of an even lattice.

theta functions

A modern point of view on the existence of the linear (not projective) Weil representation over a finite field, namely, that it admits a canonical Hilbert space realization, was proposed by . Using the notion of canonical intertwining operators suggested by Joseph Bernstein, such a realization was constructed by Gurevich-Hadani.[2]

David Kazhdan

Weil showed how to extend the theory above by replacing by any locally compact abelian group G, which by Pontryagin duality is isomorphic to its dual (the group of characters). The Hilbert space H is then the space of all L2 functions on G. The (analogue of) the Heisenberg group is generated by translations by elements of G, and multiplication by elements of the dual group (considered as functions from G to the unit circle). There is an analogue of the symplectic group acting on the Heisenberg group, and this action lifts to a projective representation on H. The corresponding central extension of the symplectic group is called the metaplectic group.


Some important examples of this construction are given by:

Heisenberg group

Oscillator representation

Metaplectic structure

Reductive dual pair

another double cover

Spin group

Symplectic group

Theta function

Howe, Roger; Tan, Eng-Chye (1992), Nonabelian harmonic analysis. Applications of SL(2,R), Universitext, New York: Springer-Verlag,  978-0-387-97768-3

ISBN

Lion, Gerard; Vergne, Michele (1980), The Weil representation, Maslov index and theta series, Progress in Mathematics, vol. 6, Boston: Birkhäuser

Weil, André (1964), "Sur certains groupes d'opérateurs unitaires", Acta Math., 111: 143–211, :10.1007/BF02391012

doi

Gurevich, Shamgar; Hadani, Ronny (2006), "The geometric Weil representation", Selecta Mathematica, New Series, :math/0610818, Bibcode:2006math.....10818G

arXiv

Gurevich, Shamgar; Hadani, Ronny (2005), Canonical quantization of symplectic vector spaces over finite fields, :0705.4556

arXiv

Weissman, Martin H. (May 2023). (PDF). Notices of the American Mathematical Society. 70 (5): 806–811. doi:10.1090/noti2687.

"What is ... a Metaplectic Group?"