The twisted cohomological equation over the partially hyperbolic flow

Let $\mathbb{G}$ be a higher-rank connected semisimple Lie group with finite center and without compact factors. In any unitary representation $(\pi, \mathcal{H})$ of $\mathbb{G}$ without non-trivial $\mathbb{G}$-fixed vectors, we study the twisted cohomological equation $(X+m)f=g$, where $m\in\mathbb{R}$ and $X$ is in a $\mathbb{R}$-split Cartan subalgebra of $\text{Lie}(\mathbb{G})$. We characterize the obstructions to solving the cohomological equation, construct smooth solutions of the cohomological equation and obtain tame Sobolev estimates for $f$. We also study common solution to (the infinitesimal version of) the twisted cocycle equation $(X+m)g_1=(\mathfrak{v}+m_1)g_2$, where $\mathfrak{v}$ is nilpotent or in a $\mathbb{R}$-split Cartan subalgebra, $m,m_1\in\mathbb{R}$. This is the first paper studying general twisted equations. Compared to former papers, a new technique in representation theory is developed by Mackey theory and Mellin transform.