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I am currently following the MIT lie algebra notes (for fun, not for h.w.!) and I am stuck on a step in one of the main theorems of the lecture. The notes are found here, https://math.mit.edu/classes/18.745/classnotes.html , and the question is with regards to theorem 6.6 of lecture 6. The theorem is as follows:

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and π its representation on a finitedimensional vector space V , over an algebraically closed field $\mathbb{F}$ of characteristic 0. Let $\mathfrak{h}$ be a nilpotent subalgebra of $\mathfrak{g}$. Then the following equalities hold.

$$V = \bigoplus_{\lambda \in \mathfrak{h}*}V^{\mathfrak{h}}_{\lambda} \ \ \ (2)$$

$$ \pi(\mathfrak{g}^{\mathfrak{h}}_{\alpha}) \ V^{\mathfrak{h}}_{\lambda} \subset V^{\mathfrak{h}}_{\lambda+ \alpha} \ \ \ (3)$$

where $V^{\mathfrak{h}}_{\lambda}$ and $\mathfrak{g}^{\mathfrak{h}}_{\alpha}$ are the generalised weight spaces w.r.t to the rep $\pi$ and the adjoint rep.

The proof of this theorem is given below:

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I am stuck on the establishment of (2), I don't get what induction is being done on. Any help would be greatly appreciated!!

SheldonCooper
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  • See this post for a proof, too. – Dietrich Burde Mar 02 '23 at 19:24
  • @DietrichBurde I believe that this link takes me to a proof that given any $a\in \mathfrak{g}$ we can decompose the vector space into a direct sum of generalized eigenspaces w.r.t a single operator $\pi(a) \in End(V)$, my question is about using this result and extending it to the case of a decomposition of generalized weight spaces w.r.t a niltpotent subalgebra. – SheldonCooper Mar 03 '23 at 14:41
  • The post there proves $\mathfrak{g}=\oplus_{\lambda\in\Sigma(x)}\mathfrak{g}{\lambda}(x)$ for the generalized eigenspaces $\mathfrak{g}{\lambda}(x)$, and the same works for your more general version $(2)$, i.e., for $V = \oplus_{\lambda \in \mathfrak{h}*}V^{\mathfrak{h}}_{\lambda}$. – Dietrich Burde Mar 03 '23 at 14:46

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