On theta functions (Q2782798)

The theta functions considered here are holomorphic functions of a variable \(W\) with parameters \(\Omega\), \(S\), \(A\), \(B\), where \(W\) is a complex \(h\times g\)-matrix, \(\Omega\) is a point in the Siegel upper half plane of degree \(g\), \(S\) is a positive definite real symmetric \(h\times h\)-matrix, and the characteristic \((A,B)\) consists of two real \(h\times g\)-matrices \(A\) and \(B\). The author establishes isomorphisms between various spaces of theta functions. The main result is an isomorphism between a space of theta functions and a space of smooth functions on the Heisenberg group \(H= H_{\mathbb{R}}^{(g,h)}\) that satisfy certain conditions. The group \(H\) is embedded as a subgroup in the real symplectic group of degree \(g+h\). Finally, a so-called lattice representation of \(H\) on a Hilbert space of certain measurable functions on \(H\) is defined, and a relation with theta functions is established. The main references are \textit{J. Igusa}'s monograph on Theta functions (Springer, 1972; Zbl 0251.14016), and \textit{D. Mumford}'s Tata Lectures on Theta, Vol. I Birkhäuser (1983; Zbl 0509.14049), II (1983; Zbl 0549.14014), III written in conjunction with \textit{M. Nori} and \textit{P. Norman} (1991; Zbl 0744.14033).