2002/07/19 by Ilka Agricola, Agricola, Ilka, Roe Goodman +1
Mathematics · #17B66 #53C35 #58J70 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Representation Theory (math.RT) #math.RT #msc:17B66 #msc:53C35 #msc:58J70
paper · pdf · doi:10.48550/arxiv.math/0207161
Latex2e, 18 pages
openalex publication_date 2002/07/19 · arxiv created 2003/03/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
When G is a complex reductive algebraic group and G/K is a reductive symmetric space, the decomposition of \C[G/K] as a K-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of Kostant-Rallis for the linearized problem (the harmonic decomposition of the isotropy representation). To obtain a constructive version of Richardson's results, this paper studies the infinite dimensional Lie algebra \X(G/K)K of K-invariant regular algebraic vector fields using the geometry of G/K and the K-spherical representations of G. Assume G is semisimple and simply-connected and let \J be the algebra of K biinvariant functions on G. An explicit set of free generators for the localization \X(G/K)Kψ is constructed for a suitable ψ∈ \J. A commutator formula is obtained for K-invariant vector fields in terms of the corresponding K-covariant maps from G to the isotropy representation of G/K. Vector fields on G/K whose horizontal lifts to G are tangent to the Cartan embedding of G/K into G are called flat. When G is simple and simply connected, it is shown that every element of \X(G/K)K is flat if and only if K is semisimple. The gradients of the fundamental characters of G are shown to generate all conjugation-invariant vector fields on G. These results are applied in the case of the adjoint representation of G = \SL(2,\C) to construct a conjugation invariant differential operator whose kernel furnishes a harmonic decomposition of \C[G].