# Prove the Weyl's complete reducibility Theorem on finite-dimensional $\mathfrak{g}-modules$ by Kostant's $\mathfrak{n}$-cohomology result

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)).

But I have no idea about its proof.

Any suggestion will be very appreciated. Thank you very much.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $\mathfrak{n}$ be the usual upper nilradical and $\lambda$ be a dominant integral weight. Then the Kostant's $\mathfrak{n}$-cohomology result can be stated as follows:

$\text{H}^i(\mathfrak{n}, L(\lambda)) =\bigoplus_{w\in W,\text{ } \ell(w)=i} \mathbb{C}_{w(\lambda+\rho)-\rho}$

Question (1): How to use the Hochschild-Serre Spectral sequence to prove Kostant's $\mathfrak{n}$-cohomology result.

Question (2): How to derive the Weyl's Theorem from Kostant's $\mathfrak{n}$-cohomology result.

Thanks very much!

## migrated from mathoverflow.netAug 4 '14 at 15:25

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• I've suggested this question to be moved to math.stackexchange. I've also fixed some grammar errors. You should really specify which Weyl's theorem you mean in Q2, although it is obvious from the context. (Because if you do in the right way, then you are just one general fact about alternating sums of characters short of the answer. ;) ) – Vít Tuček Aug 4 '14 at 12:53
• Thank you very much for your comments. By the Weyl's theorem I mean the completely reducibility of a finite-dimensional $\mathfrak{g}$-module. Is it possible that we show this via Kostant $\mathfrak{n}$-cohomology result ? – user56730 Aug 4 '14 at 13:02