Analysis on Compact Lie Groups

  • David ApplebaumEmail author
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 70)


We study the Laplacian from an analytic viewpoint as a self-adjoint operator with discrete eigenvalues given by the Casimir spectrum. This leads naturally to a study of Sobolev spaces, which are also characterised from a Fourier analytic viewpoint. We introduce Sugiura’s zeta function as a tool to study regularity of Fourier series on groups. In particular, we find conditions for absolute and uniform convergence, and for smoothness. Smoothness is characterised by means of the Sugiura space of rapidly decreasing functions defined on the space of highest weights, and we will utilise this in the next chapter to study probability measures on groups that have smooth densities.


Fourier Series Zeta Function Heat Kernel Compact Group Topological Vector Space 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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