Exercise 6, § 1

The notations are those of Exerc. 12 of Chap. I, §3. In particular, $\mathfrak{g}$ is a Lie algebra, M a $\mathfrak{g}$-module, and $H^p(\mathfrak{g},M) = Z^p(\mathfrak{g},M)/B^p(\mathfrak{g},M)$ is the cohomology space of degree $p$ of $\mathfrak{g}$ with values in M.

a) Show that $B^p(\mathfrak{g},M)$ and $Z^p(\mathfrak{g},M)$ are stable under the natural representation $\theta$ of $\mathfrak{g}$ on the space of cochains $C^p(\mathfrak{g},M)$. It follows that there is a representation of $\mathfrak{g}$ on $H^p(\mathfrak{g},M)$. Show that this representation is trivial (use the formula $\theta =di+id$, loc. cit.).

b) Let $\overline{k}$ be an algebraic closure of $k$. Let $x\in \mathfrak{g}$ and let $x_M$ be the corresponding endomorphism of M. Let $\lambda_1, . . . , \lambda_n$ (resp. $\mu_1, . . . , \mu_m$) be the eigenvalues (in $k$) of ad$_{\mathfrak{g}}x$ (resp. of $x_M$), repeated according to their multiplicity. Show that the eigenvalues of the endomorphism $\theta (x)$ of $C^p(\mathfrak{g},M)$ are the $\mu_j-(\lambda_{i_1}+\cdots +\lambda_{i_p})$, where $1\leq j\leq m$ and

Deduce, using a), that $H^p(\mathfrak{g},M) = 0$ if none of the $\mu_j-(\lambda_{i_1}+\cdots +\lambda_{i_p})$ is zero.

c) Assume that the representation $\mathfrak{g}\rightarrow$ End(M) is faithful, and that $x$ satisfies the condition:

for all $j_1, . . . , j_p, k_1, . . . , k_{p+1}\in 1, m$.

Show that we then have $H^p(\mathfrak{g},M) = 0$ (remark that the eigenvalues $\lambda_i$ of ad$_{\mathfrak{g}}x$ are of the form $\mu_j-\mu_k$, and apply $b$)$)$.

$^3$ This proof was communicated to us by G. SELIGMAN.