Abstract:

We establish sharp Lp estimates for restriction of characters of a compact Lie group to any submanifold of a maximal torus. As a consequence, we establish sharp Lp estimates for restriction of Laplacian eigenfunctions of a compact Lie group to any flat totally geodesic submanifold, which match the corresponding bounds on tori as established by Xiaoqi Huang and Cheng Zhang (but with very different reasons). Not surprisingly, the proof requires a detailed analysis of the Weyl character formula. We decompose any submanifold of a maximal torus into pieces according to the distance from the conjugate points, and estimate the Lp norm of the characters over each piece. In particular, we examine some combinatorics of the irreducible root systems which encode geometric information about the conjugate points. Similar eigenfunction restriction bounds hold for a product of spheres; all these results suggest a study of asymptotic Lp bounds of general Jacobi polynomials associated to root systems.

Speaker: Yunfeng Zhang, University of Cincinnati, Postdoc