Let $A$ be an associative $\mathbb{C}$-algebra and let $\mathfrak{r}\subset A$ be a finite dimensional Lie subalgebra of $A$ (for the commutator bracket) and assume that $\mathfrak{r}=\mathfrak{n}\rtimes \mathbb{C}\delta$ such that
- The adjoint action of $\mathfrak{n}$ on $A$ is locally-nilpotent (meaning that for each $a\in A$ there exists a positive integer $k$ depending on $a$ such that for any $x_1,\dots,x_k\in\mathfrak{n}$ we have that $(\mathrm{ad}x_1\cdots\mathrm{ad}x_k)a=0$).
- The action of $\mathrm{ad}\delta$ on $A$ is locally-finite and the eigenvalues of $\mathrm{ad}\delta$ on $\mathfrak{n}$ are integers $>0$.
Let $\mathfrak{U}$ be the (unital) subalgebra of $A$ generated by $\mathfrak{n}$ and $\mathfrak{U}_+$ the non-unital subalgebra of $A$ generated by $\mathfrak{n}$. If $M$ is an $A$-module finitely generated over $\mathfrak{U}$, then the quotients $M/\mathfrak{U}_+^kM$ are finite dimensional vector spaces and we have a natural sequence $$ M/\mathfrak{U}_+M \stackrel{\theta_1}{\longleftarrow} M/\mathfrak{U}_+^2M \stackrel{\theta_2}{\longleftarrow} M/\mathfrak{U}_+^3M \stackrel{\theta_3}{\longleftarrow} \cdots. $$ Let $\lambda\in \mathbb{C}$ and denote by $[M/\mathfrak{U}_+^kM]_\lambda$ the generalized eigenspace of the induced action of $\mathrm{ad}\delta$ on $M/\mathfrak{U}_+^kM$ with generalized eigenvalue $\lambda$. In paragraph 5.2 in the article On primitive ideals by Victor Ginzburg, it is stated that hypothesis 2 above implies that for all $k$ sufficiently large the maps $\theta_k$ induce isomorphisms $$ [\theta_k]_\lambda : [M/\mathfrak{U}_+^{k+1}M]_\lambda\to [M/\mathfrak{U}_+^kM]_\lambda, $$ in other words, the sequence of generalized eigenspaces stabilizes. This is a key step in the construction of what he calls the Jacquet functor.
My question is: Why do we have this stability property?
I can see that from hypothesis 2 the generalized eigenvalues of $\mathrm{ad}\delta$ on $\mathfrak{U}_+^k$ are integers $\geq k$, but I cannot see how this can control the generalized eigenspaces on the given sequence of quotients. Now, as the functor $[\cdot]_\lambda$ is exact (by elementary linear algebra) the claim made is equivalent to show that for $k\gg 0$ $$ [\mathfrak{U}_+^kM/\mathfrak{U}_+^{k+1}M]_\lambda = 0, $$ which in turn is equivalent to show that, for $k\gg 0$, the complex number $\lambda$ is not an eigenvalue of $\mathrm{ad}\delta$ on $\mathfrak{U}_+^kM/\mathfrak{U}_+^{k+1}M$, because every generalized eigenspace contains at least one eigenspace. But I'm stuck here.
How can I prove this stability property?