  We prove that the invariant part, with respect to a compact group action
satisfying certain condition, of the direct image of a Nakano positive
Hermitian holomorphic vector bundle over a bounded pseudoconvex domain is
Nakano positive. We also consider the action of the noncompact group
$\mathbb{R}^m$ and get the same result for a family of tube domains, which
leads to a new method to the matrix-valued Prekopa's theorem originally proved
by Raufi. The two main ingredients in our method are H\"ormander's $L^2$ theory
of $\bar\partial$ and the recent work of Deng-Ning-Zhang-Zhou on
characterization of Nakano positivity of Hermitian holomorphic vector bundles.
