This booklet begins with the fundamental conception of topological teams, harmonic research, and unitary representations. It then concentrates on geometric constitution, harmonic research, and unitary illustration idea in commutative areas. these areas shape a simultaneous generalization of compact teams, in the community compact abelian teams, and riemannian symmetric areas. Their geometry and serve as thought is an more and more energetic subject in mathematical study, and this e-book brings the reader as much as the frontiers of that learn quarter with the hot classifications of weakly symmetric areas and of Gelfand pairs. half 1, "General idea of Topological Groups", is an advent with many examples, together with all the typical semisimple linear Lie teams and the Heisenberg teams. It offers the development of Haar degree, the invariant fundamental, the convolution product, and the Lebesgue areas. half 2, "Representation concept and Compact Groups", offers heritage at a marginally greater point. in addition to the fundamentals, it comprises the Mackey Little-Group approach and its program to Heisenberg teams, the Peter-Weyl Theorem, Cartan's maximum weight idea, the Borel-Weil Theorem, and invariant functionality algebras. half three, "Introduction to Commutative Spaces", describes that quarter as much as its fresh resurgence. round capabilities and linked unitary representations are constructed and utilized to harmonic research on $G/K$ and to uncertainty ideas. half four, "Structure and research for Commutative Spaces", summarizes riemannian symmetric house thought as a rôle version, and with that orientation delves into fresh study on commutative areas. the consequences are specific for areas $G/K$ of nilpotent or reductive style, and the hot constitution and type thought will depend on these situations. components 1 and a pair of are obtainable to first-year graduate scholars. half three takes a little bit analytic sophistication yet often is obtainable to graduate scholars. half four is meant for mathematicians starting their examine careers in addition to mathematicians attracted to seeing simply how a ways you will select this unified view of algebra, geometry, and research.
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