CBMS Regional Conference Series in Mathematics 1985; 76 pp; softcover Number: 59 ISBN10: 0821807099 ISBN13: 9780821807095 List Price: US$23 Member Price: US$18.40 All Individuals: US$18.40 Order Code: CBMS/59
 This book introduces a systematic new approach to the construction and analysis of semisimple \(p\)adic groups. The basic construction presented here provides an analogue in certain cases of the HarishChandra homomorphism, which has played an essential role in the theory of semisimple Lie groups. The book begins with an overview of the representation theory of GL\(_n\) over finite groups. The author then explicitly establishes isomorphisms between certain convolution algebras of functions on two different groups. Because of the form of the isomorphisms, basic properties of representations are preserved, thus giving a concrete example to the correspondences predicted by the general philosphy of Langlands. The first chapter, suitable as an introduction for graduate students, requires only a basic knowledge of representation theory of finite groups and some familiarity with the general linear group and the symmetric group. The later chapters introduce researchers in the field to a new method for the explicit construction and analysis of representations of \(p\)adic groups, a powerful method clearly capable of extensive further development. Readership Table of Contents  A Hecke algebra approach to the representations of GL\(_n(F_q)\)
 Hecke algebras for GL\(_n\) over local fields: introduction
 HarishChandra homomorphism in the unramified anisotropic case
