Induced Representations, Intertwining Operators and Transfer

Induced representations and intertwining operators give rise to distributions that represent critical terms in the trace formula. We shall describe a conjectural relationship between these distributions and the LanglandsShelstad-Kottwitz transfer of functions Our goal is to describe a conjectural relationship between intertwining operators and the transfer of functions. One side of the proposed identity is a linear combination of traces tr ( RP (πw) ◦ IP (π, f) ) , where IP (π, f) is the value of an induced representation at a test function f , and RP (πw) is a standard self-intertwining operator for the representation. Distributions of this sort are critical terms in the trace formula. The other side is defined by the transfer of f to an endoscopic group. The endoscopic transfer of functions represents a sophisticated theory, some of it still conjectural, for comparing trace formulas on different groups [L2]. This is a central theme in the general study of automorphic forms. There are a number of relatively recent ideas of Langlands, Shelstad and Kottwitz, which will have to be taken for granted in the statement of such an identity. On the other hand, induced representations and intertwining operators go back many years. They were lifelong preoccupations of George Mackey [M1]–[M5], whose investigations anticipated their future role as basic objects in modern representation theory. It is therefore fitting to devote the first part of the article to a historical introduction. I shall try to describe in elementary terms how Mackey’s initial ideas are reflected in some of the ways the later theory developed. I hope that this compensates in some measure for the technical and rather sketchy nature of the remaining part of the article. 1991 Mathematics Subject Classification. Primary 22E50, 22E30, 22E35; Secondary 22D10.