Conical Distributions on the Space of Flat Horocycles

2009/04/09 by Fulton B. Gonzalez, Gonzalez, Fulton B.
Mathematics · #43A85 #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Functional Analysis (math.FA) #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.0904.1559

openalex publication_date 2009/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G₀=K\ltimes\mathfrak p be the Cartan motion group associated with a noncompact semisimple Riemannian symmetric pair (G, K). Let \frak a be a maximal abelian subspace of \mathfrak p and let \p=\a+\q be the corresponding orthogonal decomposition. A flat horocycle in \p is a G₀-translate of \q. A conical distribution on the space Ξ₀ of flat horocycles is an eigendistribution of the algebra \mathbb D(Ξ₀) of G₀-invariant differential operators on Ξ₀ which is invariant under the left action of the isotropy subgroup of G₀ fixing \q. We prove that the space of conical distributions belonging to each generic eigenspace of \mathbb D(Ξ₀) is one-dimensional, and we classify the set of all conical distributions on Ξ₀ when G/K has rank one.

Citations

• Invariant Affine Connections on Homogeneous Spaces

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