SL2(R) [Paperback]

SL2(R) gives the student an introduction to the infinite dimensional representation theory of semisimple Lie groups by concentrating on one example - SL2(R). This field is of interest not only for its own sake, but for its connections with other areas such as number theory, as brought out, for example, in the work of Langlands. The rapid development of representation theory over the past 40 years has made it increasingly difficult for a student to enter the field. This book makes the theory accessible to a wide audience, its only prerequisites being a knowledge of real analysis, and some differential equations.This book introduces the infinite dimensional representation theory of semisimple Lie groups by concentrating on one example - SL2(R). The contents are accessible to a wide audience, requiring only a knowledge of real analysis, and some differential equations.SL2(R) gives the student an introduction to the infinite dimensional representation theory of semisimple Lie groups by concentrating on one example - SL2(R). This field is of interest not only for its own sake, but for its connections with other areas such as number theory, as brought out, for example, in the work of Langlands. The rapid development of representation theory over the past 40 years has made it increasingly difficult for a student to enter the field. This book makes the theory accessible to a wide audience, its only prerequisites being a knowledge of real analysis, and some differential equations.I General Results.- 1 The representation on Cc(G).- 2 A criterion for complete reducibility.- 3 L2 kernels and operators.- 4 Plancherel measures.- II Compact Groups.- 1 Decomposition over K for SL2(R).- 2 Compact groups in general.- III Induced Representations.- 1 Integration on coset spaces.- 2 Induced representations.- 3 Associated spherical functions.- 4 The kernel defining the induced representation.- IV Spherical Functions.- 1 Bi-invariance.- 2 Irreducibility.- 3 The spherical property.- 4 Connl£¡