
Journal of Lie Theory 17 (2007), No. 4, 869898 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 sinton@gmail.com The theory of a spherical Fourier transform for measures on certain projective limits of symmetric spaces of noncompact type is developed. Such spaces are introduced for the first time and basic properties of the spherical transform, including a LevyCramer 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 semisimple Lie groups {G_{j}} such that G_{j} is essentially the semisimple component of a parabolic subgroup of G_{j+1}. This class includes most of the classical families of Lie groups as well as infinite direct products of semisimple groups. Keywords: Heat kernel, heat equation, projective limit, inverse limit, symmetric spaces, spherical Fourier transform, Lie group. MSC: 43A85; 43A30 [ Fulltextpdf (298 KB)] for subscribers only. 