Park City Lectures on Eigenfunctions, Lecture 5: Lp norms of ...

Highest weight spherical harmonics = Gaussian beam
Another important spherical harmonic is Yk k , which is the spherical
harmonic in H k with the largest eigenvalue of L 3 = 1 ∂
i ∂θ
, or in
other words the highest weight.
It is an example of a Gaussian beam along a closed geodesic– a
function like e iks e −ky 2
in Fermi normal coordinates on the
geodesic, with s arc-length and y the coordinate in the normal
direction. It is Gaussian in the normal direction.
Yk
k is the restriction of the harmonic polynomial(x 1 + ix 2 ) k (up to
normalization).