What is known about existence of quasi-invariant smooth function with some eigencharacter on Lie algebra of a reductive Lie group? Consider reductive Lie group $G$ and its Lie algebra $\mathfrak{g}$. We have the conjugate action of $G$ on $\mathfrak{g}$: $g\cdot X=gXg^{-1}$. The following classical result claims the existence of $G$-invariant bump function on $ \mathfrak{g} $ for invariant enighborhood near semisimple point:
For semisimple point $X\in\mathfrak{g}$ and an invariant open neighborhood $\Omega$ of $X$, then there exists a $G$-invariant smooth function $f\in C^\infty(\mathfrak{g})^G$ such that
(1) $0\leq f\leq1$, $f=1$ in an neighborhood of $X$
(2) $\operatorname{supp}(f)\subseteq\Omega$
Suppose now $ \eta:G\to\mathbb{C}^\times $ is a unitary character of $ G $. We call a function $f:G\to\mathbb{C}$ is quasi-invariant with eigencharacter $\eta$ if $ f(g\cdot X)=\eta(g)f(X) $ for $g\in G,X\in\mathfrak{g}$.
Can we construct a quasi-invariant smooth function on $\mathfrak{g}$ with eigencharacter $\eta$ still satisfies the above conditions?
