Nonzero eigenfunctions annihilate equivariant BM on their nonvanishing loci

The set $D(f)$ is $T$-invariant because $f(t\cdot x)=\chi (t)f(x)$. On its Borel construction, define $s([e,x])=[e,f(x)]$ in the associated character line. This is well-defined: replacing $(e,x)$ by $(e\cdot t,t^{-1}\cdot x)$ replaces $f(x)$ by $f(t^{-1}\cdot x)=\chi (t{)}^{-1}f(x)$, exactly the right equivalence $(e,z)\cdot t=(e\cdot t,\chi (t{)}^{-1}z)$. Since $f$ never vanishes on $D(f)$, $s$ is a nowhere-zero section and trivializes that line bundle. The same formula trivializes its restriction to every $T$-invariant open $V\subseteq D(f)$. To ensure the preceding definition applies to $V$, write $D(f)=\mathrm{Spec}{A}_f$. Its coordinate ring is graded by the character lattice ${\mathbb{Z}}^r$, and the ideal $I(Z)$ of the invariant closed complement $Z=D(f)\setminus V$ is homogeneous. For each $x\in V$, choose $g\in I(Z)$ with $g(x)\mathrm{\ne }0$; a homogeneous component $g_{\lambda }\in I(Z)$ also satisfies $g_{\lambda }(x)\mathrm{\ne }0$. The principal open $D(g_{\lambda })$ is invariant, affine, contains $x$, and lies in $V$. Since $D(f)$ is noetherian, its open subset $V$ is quasi-compact, so finitely many such principal opens cover $V$. Thus $V$ has a finite invariant affine cover.

On each finite Borel approximation, the first Chern class of the trivialized line is zero. By [F2] this is the Euler class of the line, and by [F3] its cap action on the approximating Borel–Moore groups is zero. The stable comparison in [F1] therefore gives $u_{\chi }\cap \beta =0$ for every $\beta \in H_j^{\mathrm{B}\mathrm{M},T}(D(f);\mathbb{Q})$, and the identical argument holds on $V$.

Since $\chi$ is nontrivial, $u_{\chi }\in S_T$ is invertible after localization. For any class $\beta$, its localized image satisfies $\beta \mathrm{/}1=(u_{\chi }\cap \beta )\mathrm{/}{u}_{\chi }=0$ by step 2.1. Every homogeneous fraction is $\beta \mathrm{/}s=(\beta \mathrm{/}1)\mathrm{/}s$, so the localized total modules, and hence all their degree pieces, vanish on both $D(f)$ and $V$. If the opposite character convention is used, the class is $-u_{\chi }$, which is also in $S_T$ up to the unit $-1$ and has the same annihilation consequence.