  We prove sufficient conditions in order to obtain a sharp G\aa rding
inequality for pseudo-differential operators acting on vector-valued functions
on compact Lie groups. As a consequence, we obtain a sharp G\aa rding
inequality for compact homogeneous vector bundles and compact homogeneous
manifolds. The sharp G\aa rding inequality is the strongest lower bound
estimate known to hold for systems on $\mathbb{R}^n$, and the aim of this paper
is to extend this property to systems on compact Lie groups and compact
homogeneous manifolds. Our results extend previous works by Lax and Nirenberg
[Comm. Pure Appl. Math., Vol. 8, 129-209, (1966)], and by Ruzhansky and Turunen
[J. Funct. Anal., Vol. 267, 144-172, (2011)]. As an application, we establish
existence and uniqueness of solution to a class of systems of initial value
problems of pseudo-differential equations on compact Lie groups and compact
homogeneous manifolds.
