Titlebook: An Introduction to Automorphic Representations; With a view toward t Jayce R. Getz,Heekyoung Hahn Textbook 2024 Springer Nature Switzerland

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Archimedean Representation Theory,In this chapter, we introduce the main players in the representation theory of real Lie groups. In particular, we define admissible representations, .-modules and infinitesimal characters. The chapter ends with a brief discussion of the Langlands classification.

连词

Automorphic Forms,In this chapter, we define automorphic forms and use them to give the general definition of an automorphic representation. We then explain the relationship between discrete automorphic representations and automorphic representations.

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Unramified Representations,In this chapter, we describe the classification of unramified representations of reductive groups over non-Archimedean local fields. Along the way, we discuss the Satake isomorphism and the Langlands dual group.

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Non-Archimedean Representation Theory,In this chapter, we explain how general admissible representations are built up out of supercuspidal representations via the process of parabolic induction.

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The Cuspidal Spectrum,The cuspidal spectrum of . decomposes discretely into a Hilbert space direct sum with finite multiplicities. We give a proof of this fact in this chapter. We also prove that cuspidal automorphic forms are rapidly decreasing in the number field case and are compactly supported in the function field case.