Metaplectic Forms and Representations

where the image of A is in the center of G̃. If G and G̃ are topological groups, and if A is a discrete subgroup of the center of G̃, then we may think of G̃ as a cover of G, in the topological sense. So we will sometimes use the term covering group. Very often for us, A will be the group μn(F ) of n-th roots of unity, in a given field F . We will use this notation only if |μn(F )| = n. By a Metaplectic group we mean a central extension G̃(F ) of the group G(F ) of F -rational points of a reductive algebraic group over a local field F , by μn where μn = μn(F ). When we consider complex representations of a metaplectic group we will also encounter μn(C) and we will chose fix an isomorphism to identify μn(F ) = μn(C). We will usually need μn ⊂ F and often it will be convenient to assume μ2n ⊂ F . Alternatively, let F be a global field of characteristic zero or prime to n. Let A = AF be the adele ring. We then look for an extension G̃(A) of the adele group G(A) by μn. Typically the cover will split over G(F ). This means that we may consider G(F ) to be a subgroup of G̃(A), and so we may consider automorphic forms on metaplectic groups. To give a famous example, consider the Jacobi theta function