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Title An Intrinsic Theory of Smooth Automorphic Representations
Publication Date
Degree PhD
Discipline/Department Mathematics
Degree Level doctoral
University/Publisher The Ohio State University
Abstract Our goal in this paper is to lay the foundation for a theory of smooth automorphic forms and representations on local and adelic reductive groups G that does not rely on the K-finite or classical unitary theories. Specifically, we consider forms that are smooth but not necessarily K-finite, that is, whose orbits under a maximal compact subgroup are not necessarily contained in finite dimensional spaces. To do so, we introduce spaces of automorphic forms of "fixed type" à la Borel and Jacquet, imposing locally convex topologies to gain control of these now infinite dimensional spaces. Along the way we present a novel treatment of group norms on G, emphasizing their compatibility both with the structure of G as a reductive group and a restricted product of reductive groups over local fields.Motivated by the work of Wallach in the 1980s, we relate the asymptotic behavior of our representations to the growth of the automorphic forms that generate them. We show that our smooth representations behave analogously to the Casselman-Wallach globalizations of representations of real reductive groups. We then introduce a new algebra of "Schwartz functions" on G and demonstrate it serves the same role for smooth representations as the Hecke algebra does for their K-finite counterparts.Ultimately, our aim is to illustrate how smooth representations serve the same role for K-finite automorphic representations as Casselman-Wallach globalizations do for Harish-Chandra modules. Our tensor product theorem for smooth automorphic representations completes the picture of these representations as generalizations of Casselman-Wallach globalizations. Therein we show the "factors" of such a representation (ϖ, V) at the archimedean places are precisely the Casselman-Wallach globalizations corresponding to archimedean factors in the underlying Harish-Chandra module of K-finite vectors in V.
Subjects/Keywords Mathematics; analytic number theory; automorphic representation theory; Schwartz functions; Casselman-Wallach
Contributors Cogdell, James (Advisor)
Language en
Rights unrestricted ; This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.
Country of Publication us
Format application/pdf
Record ID oai:etd.ohiolink.edu:osu1524174589105537
Repository ohiolink
Date Indexed 2021-01-29
Grantor The Ohio State University

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