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Journal of Lie Theory 17 (2007), No. 4, 869--898
Copyright Heldermann Verlag 2007

The Spherical Transform on Projective Limits of Symmetric Spaces

Andrew R. Sinton
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel

The theory of a spherical Fourier transform for measures on certain projective limits of symmetric spaces of non-compact type is developed. Such spaces are introduced for the first time and basic properties of the spherical transform, including a Levy-Cramer type continuity theorem, are obtained. The results are applied to obtain a heat kernel measure on the limit space which is shown to satisfy a certain cylindrical heat equation. The projective systems under consideration arise from direct systems of semi-simple Lie groups {Gj} such that Gj is essentially the semi-simple component of a parabolic subgroup of Gj+1. This class includes most of the classical families of Lie groups as well as infinite direct products of semi-simple groups.

Keywords: Heat kernel, heat equation, projective limit, inverse limit, symmetric spaces, spherical Fourier transform, Lie group.

MSC: 43A85; 43A30

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