Bounds of restriction of characters to submanifolds

A fruitful approach to studying the concentration of Laplace--Beltrami eigenfunctions on a compact manifold, as the eigenvalue tends to infinity, is to bound their restriction to submanifolds. In this paper, we adopt this approach in the setting of compact Lie groups and provide sharp restriction bounds for general Laplace--Beltrami eigenfunctions, as well as for important special cases such as sums of matrix coefficients and, in particular, characters of irreducible representations. We prove sharp asymptotic $L^p$ bounds for the restriction of general Laplace--Beltrami eigenfunctions to maximal flats and all of their submanifolds, for all $p \geq 2$. Furthermore, we establish sharp asymptotic $L^p$ bounds for the restriction of characters to maximal tori and all of their submanifolds for all $p>0$, and to torus-generated conjugation-invariant submanifolds for all $p \geq 2$. We also obtain sharp $L^p$ bounds for the restriction of general sums of matrix coefficients to maximal flats and all of their submanifolds, for all $p \geq 2$.