Bounds of restriction of characters to submanifolds

Abstract: A fruitful approach to the study of concentration of Laplacian eigenfunctions on a compact manifold as the eigenvalue grows to infinity is to bound their restriction to submanifolds. In this paper we take this approach in the setting of a compact Lie group, and provide sharp restriction bounds of general Laplacian eigenfunctions as well as important special ones such as sums of matrix coefficients and in particular characters of irreducible representations of the group. We deal with two classes of submanifolds, namely, maximal flats and all of their submanifolds, and the conjugation-invariant submanifolds. We prove conjecturally sharp asymptotic $L^p$ bounds of restriction of general Laplacian eigenfunctions to maximal flats and all of their submanifolds for all $p\geq 2$. We also prove sharp asymptotic $L^p$ bounds of restriction of characters to maximal tori and all of their submanifolds for all $p>0$ and to the conjugation-invariant submanifolds for all $p\geq 2$, and of general sums of matrix coefficients to maximal flats and all of their submanifolds for all $p\geq 2$. In the appendix we present similar results for products of compact rank-one symmetric spaces.