Metaplectic group

In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence. 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. In the case n = 1, the symplectic group coincides with the special linear group SL2(R). This group biholomorphically acts on the complex upper half-plane by fractional-linear transformations, is a real 2-by-2 matrix with the unit determinant and z is in the upper half-plane, and this action can be used to explicitly construct the metaplectic cover of SL2(R). The elements of the metaplectic group Mp2(R) are the pairs (g, ε), where g ∈ S L 2 ( R ) {displaystyle gin mathrm {SL} _{2}(mathbf {R} )} and ε is a holomorphic function on the upper half-plane such that ϵ ( z ) 2 = c z + d = j ( g , z ) {displaystyle epsilon (z)^{2}=cz+d=j(g,z)} . The multiplication law is defined by: That this product is well-defined follows from the cocycle relation j ( g 1 g 2 , z ) = j ( g 1 , g 2 ⋅ z ) j ( g 2 , z ) {displaystyle j(g_{1}g_{2},z)=j(g_{1},g_{2}cdot z)j(g_{2},z)} . The map