THE LAPLACIAN ON AN AFFINE HOMOGENEOUS SPACE

The Laplacian on homogeneous spaces

The solution of the eigenvalue problem of the Laplacian on a general homogeneous space G/H is given. Here G is a compact, semi-simple Lie group, H is a closed subgroup of G, and the rank of H is equal to the rank of G. It is shown that the multiplicity of the lowest eigenvalue of the Laplacian on G/H is just the degeneracy of the lowest Landau level for a particle moving on G/H in the presence ...

More On Embedding an Affine Space in a Vector Space

for all a1 . . . , am ∈ E, all v1, . . . , vm ∈ −→E , and all λ1, . . . , λm ∈ R. Furthermore, for λi = 0, 1 ≤ i ≤ m, we have f̂(v1 +̂ λ1a1, . . . , vm +̂ λmam) = λ1 · · ·λmf(a1 + λ−1 1 v1, . . . , am + λ−1 m vm). Proof . Let us assume that f̂ exists. We first prove by induction on k, 1 ≤ k ≤ m, that f̂(a1, . . . , vi1 , . . . , vik , . . . , am) = fS(vi1 , . . . , vik), for every S ⊆ {1, . . . ,m},...

Zero - Entropy Affine Maps on Homogeneous Spaces

Positive eigenfunctions of the Laplacian andconformal densities on homogeneous trees

In this paper, we study some asymptotic aspects of the positive eigenfunctions of the combinatorial Laplacian associated to a homogeneous tree. The results are inspired by results of Dennis Sullivan concerning ?harmonic functions on the hyperbolic spaces H I n and contained in the paper Sul]. Let k be an integer 3 and X the homogeneous tree of degree k, that is, the unique simply connected simp...

On the Schwartz Space of the Basic Affine Space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G/U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct certain canonical G-invariant space S(X) (called the Schwartz space of X) of functions on X , which is an extension of the space of smooth compactly supported functions on X . We show that the s...