  To investigate concentration of Laplacian eigenfunctions on a compact
manifold as the eigenvalue grows to infinity, an important route is to bound
their restriction to submanifolds. In this paper we take the route 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 matrix
coefficients and in particular characters of irreducible representations of the
group. We focus on two classes of submanifolds, namely, submanifolds of maximal
flats and the conjugation-invariant submanifolds. We prove conjecturally sharp
asymptotic $L^p$ bounds of restriction of general Laplacian eigenfunctions to
submanifolds of maximal flats for all $p\geq 2$. We also prove sharp asymptotic
$L^p$ bounds of restriction of characters to submanifolds of maximal tori for
all $p>0$, of general matrix coefficients to submanifolds of maximal flats for
all $p\geq 2$, and of characters to the conjugation-invariant submanifolds for
all $p\geq 2$.
