Exercise 9, § 1

We retain the notations of the preceding exercise, and assume further that $\mathfrak{g}$ is nilpotent.

a) Show that the basis $(e_1, . . . , e_n)$ can be chosen so that, for any pair $(i, j)$, $[e_i, e_j]$ is a linear combination of the $e_h$ for $h >$ sup($i, j$). Assume from now on that the $e_i$ satisfy this condition. For $i= 1, . . . , n$ choose a power $q(i)$ of $p$ such that ad($e_i$)$^{q(i)}= 0$, and put $z_i=e_i^{q(i)},A =k[z_1, . . . , z_n]$, cf. Exerc. 8. b) Let $\rho :\mathfrak{g}\rightarrow \mathfrak{g}\mathfrak{l}(V)$ be a linear representation of $\mathfrak{g}$. Assume that $\rho (e_i)$ is nilpotent for $i= 1, . . . , n$. Show that $\rho (x)$ is nilpotent for all $x\in \mathfrak{g}$. (Argue by induction on $n=$ dim $\mathfrak{g}$ and reduce to the case where $\rho$ is irreducible. Show that, in this case, $\rho (e_n) = 0$ and apply the induction hypothesis.)

c) Let $\rho_1:\mathfrak{g}\rightarrow \mathfrak{g}\mathfrak{l}(V_1)$ and $\rho_2:\mathfrak{g}\rightarrow \mathfrak{g}\mathfrak{l}(V_2)$ be two linear representations of $\mathfrak{g}$. Assume that $V_1$ and $V_2$ are $\not= 0$, and that $V_1= V^{\lambda_1}(\mathfrak{g}),V_2= V^{\lambda_2}(\mathfrak{g})$, where $\lambda_1$ and $\lambda_2$ are two functions on $\mathfrak{g}$, cf. no. 3. Show that, if $\lambda_1(e_i) =\lambda_2(e_i)$ for $i= 1, . . . , n$, then $\lambda_1=\lambda_2$ and there exists a non-zero $\mathfrak{g}$-homomorphism from $V_1$ to $V_2$ (apply $b$) to the $\mathfrak{g}$-module $V =\mathscr{L}(V_1,V_2)$ and use Engel’s theorem to show that V contains a non-zero $\mathfrak{g}$-invariant element). Deduce that if, in addition, $V_1$ and $V_2$ are simple, they are isomorphic.

d) Assume that $k$ is algebraically closed. Let R be the set of equivalence classes of irreducible representations of $\mathfrak{g}$. If $\rho \in R$, put

where $x_{\rho}(i)$ is the unique eigenvalue of $\rho (e_i)$. Show that $\rho \rightarrow x_{\rho}$ is a bijection from R to $k^n$. (Injectivity follows from $c$), and surjectivity from Exerc. $8d).)$ Deduce the following consequences:

(i) For any maximal ideal $\mathfrak{m}$ of A, the quotient of $U/\mathfrak{m}U$ by its radical is a matrix algebra.

(ii) The degree of any irreducible representation of $\mathfrak{g}$ is a power of $p$ (this follows from (i) and the fact that $[U/\mathfrak{m}U :k]$ is a power of $p$).

(iii) Every homomorphism from A to $k$ extends uniquely to a homomorphism from C to $k$ (use the fact that $C/\mathfrak{m}C$ is contained in the centre of $U/\mathfrak{m}U$, which is a local $k$-algebra with residue field $k$).

(iv) There exists an integer $N\geq 0$ such that $x^{p^N}\in A$ for all $x\in C$ (this follows from (iii)).$^6$