Nilpotent Lie Groups: Structure and Applications to Analysis

Nilpotent Lie Groups: Structure and Applications to Analysis

Roe W. Goodman, "Nilpotent Lie Groups: Structure and Applications to Analysis"
1977 | pages: 219 | ISBN: 3540080554 | PDF | 3,9 mb

These notes are based on lectures given by the author during the Winter semester 1975/76 at the University of Bielefeld. The goal of the lectures was to present some of the recent uses of nilpotent Lie groups in the representation theory of semi-simple Lie groups, complex analysis, and partial differential equations. A complementary objective was to describe certain structural aspects of simply-connected nilpotent Lie groups from a "global" point of view (as opposed to the convenient but often unenlightening induction-on-dimension treatment).
The unifying algebraic theme running through the notes is the use of fi ltrations; indeed, nilpotent Lie algebras are characterized by the property of admitting a positive, decreasing filtration. The basic analytic tool is a homogeneous norm, which replaces the usual Euclidean norm and gives a "non- isotropic" measurement of distances. One obtains a filtration on the algebra of germs of C°° functions at a point by measuring the order of vanishing in terms of the homogeneous norm. This in turn induces a filtration on the Lie algebra of vector fields and on the associative algebra of differential operators. To construct (approximate) inverses for certain differential operators, one uses integral
operators whose order of singularity along the diagonal is measured via the homogeneous norm.

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