Consider a Riemannian symmetric space space X = G/K of non-compact type, where G is a connected, real, semi-simple Lie group, and K a maximal compact subgroup of G. Let X' be its Oshima compactification, and (π,C(X')) the regular representation of G on X'. In this thesis, we examine the convolution operators π(f), for rapidly decaying functions f on G, and characterize them within the framework of totally characteristic pseudo-differential operators, describing the singular nature of their Schwartz kernels. In particular, we obtain asymptotics for the heat and resolvent kernels associated to strongly elliptic operators on X'. Based on the de- scription of the Schwartz kernels we define a regularized trace for the operators π(f), yielding a distribution on G. We then show a regularity result for this distribution, and in fact prove a fixed-point formula for it, analogous to the Atiyah-Bott fixed-point formula for parabolically induced representations. Finally, we make some preliminary computations that suggest a possible development of scattering theory on symmetric spaces, and in the light of results earlier in the thesis, indicate some lines along which this could be done.
Strongly elliptic operatorsPseudodifferential and Fourier integral operators on manifoldsAnalysis on homogeneous spacesSpectral problems; spectral geometry; scattering theoryOshima compactificationAnalysis on real and complex Lie groupsSymmetric spacesRegularized traceScattering theoryHeat and other parabolic equation methods
Aprameyan Parthasarathy: Analysis on the Oshima compactification of a Riemannian symmetric space of non-compact type. : Philipps-Universität Marburg 2013-02-25. DOI: https://doi.org/10.17192/z2012.1058.
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