Notation in Kac Problem 3.2 I'm working through Kac's book, "Infinite Dimensional Lie Algebras", and have come across some notation I find confusing.
Here, $e$, $f$, and $h$ are the Cartan generators of $\mathfrak{sl}_{2}(\mathbb{C})$, and $\mathfrak{h}$ is the Cartan subalgebra.
In problem 3.2, p.42, it asks to show: If $f$ is locally nilpotent on an $\mathfrak{sl}_{2}(\mathbb{C})$-module $V$, then $V^{e}$ is $\mathfrak{h}$-diagonalizable.
I'm not entirely sure what $V^{e}$ refers to in this context, and I couldn't find it defined in the notation index.  My guess is it's something like $V^{e} = \bigoplus_{k=1}^{\infty} e^{k}V$, but I don't immediately see how that definition would solve the problem. A clarification or confirmation would be most welcome!
Thanks!

Presumably $$V^e=\{v \in V \ | \ ev=0 \}$$ is the Lie algebra version of fixed points---thus if $V$ is generated by $V^e$ and locally nilpotent for $f$ it is a quotient of a sum of Verma modules.
