Invariant harmonic analysis and Selberg zeta functions (B06 #)

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Noncommutative harmonic analysis may be characterised as the study of group representationsinduced in spaces of functions, or their generalisations, by an action on the underlyingspace. A central problem is the decomposition of such representations into irreducible components.This is a version of spectral decomposition, since the set of equivalence classes ofirreducible representations may be thought of as the spectrum of the group algebra. In thisproject, we deal with representations of topological groups in complex Hubert spaces andtheir decoposition as direct sums or integrals.More precisely, we consider reductive linear algebraic groups over local fields, mainlyover the real numbers, and over adele rings. In the latter case, the representations in thespaces of automorphic forms can be studied by means of the Arthur-Selberg trace formula,which relates the spectral data with geometric data of the underlying moduli space. Oneof our aims is to study the asymptotic distribution of those geometric data, which have anumber-theoretic interpretation, in new situations. For this kind of applications, we haveto extend the trace formula to test functions of noncompact support.The terms in the trace formula encoding geometric data also contain distributions onthe adelic group which can be split into components living on its local factors. Transferidentities between those local distributions on related groups are critical for progress in theLanglands programme. Langlands' recent vision on how to go beyond the present methodsof endoscopy suggests that one would need to determine the exact Fourier transforms ofthose distributions. We are going to expand our partial results obtained in the real case.This will also enable us to obtain new instances of the functional equations and determinantformulas for zeta functions of Selberg's type.