This book presents a detailed study of the generalized Heisenberg group, a 2‑step nilpotent Lie group that plays an important role in the theory of smooth compactifications of the Siegel modular variety, number theory, and mathematical physics. The text explicates its connections with theta functions, the Weil representation, and the Schrödinger–Weil representation, emphasizing their structural and conceptual relationships.

Heisenberg groups, theta functions, and the Schrödinger–Weil representation occupy central positions in modern mathematics and mathematical physics. By providing explicit descriptions and clear explanations, this book aims to serve as a valuable resource for researchers working in these areas. The material is intended for both mathematicians and physicists interested in the interplay between representation theory, automorphic forms, and physical models.

Jae-Hyun Yang is a professor emeritus of Inha Univeristy.