Theta functions and symplectic groups (Q793933)

In connection with \textit{A. Weil's} paper [Acta Math. 111, 143-211 (1964; Zbl 0203.033), Œuvres, Vol. III, 1-69 (1980)], the following topics are discussed, for a locally compact commutative group \(G\). (i) The replacement of the Schwartz-Bruhat functions by a certain function space containing them; this space, a Segal algebra with basic functorial properties, was introduced by \textit{H. G. Feichtinger} [Monatsh. Math. 92, 269-289 (1981; Zbl 0461.43003)]. (ii) The definition of the Hilbert space \(L^ 2(G,\Gamma)\) of Weil's theta functions, relative to a closed subgroup \(\Gamma\) of \(G\), and the role of Poisson's formula in establishing Weil's isomorphism between \(L^ 2(G,\Gamma)\) and \(L^ 2(G)\). (iii) The considerations of theta functions for the product \(G\times G\) which are used to show that Weil's group \(B_ 0(G)\) may be realized as a group of unitary operators in \(L^ 2(G\times G)\); this shows the connection between theorems 1 and 4 in Weil's paper.