Hecke Algebras

Hecke Algebras Synopsis

Directed at mathematicians interested in representation theory or number theory (and automorphic forms in particular), this book provides an introduction to the algebraic theory of Hecke algebras. The first part deals with the basic isomorphism theorems for abstract Hecke algebras. The author generalizes Masschke's classical theorem to determine the structure of finite-dimensional Hecke algebras, then goes on to discuss in more detail several examples of Hecke algebras for finite groups. The second part, which is of a more number-theoretic nature, starts with an explicit description of the Hecke algebra associated with SL(2,Z), and then generalizes the results to the unimodular group of arbitrary degree. In addition, the author describes the structure of the Hecke algebra associated with the Siegel modular group and explicitly derives, for generators of the Hecke algebra, the commutation relation with the Siegel phi-operator. Readers should gain an understanding of the role of Hecke algebras in number theory and in representation theory.