Lemma 1

Let $\mathfrak{n}$ be a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ consisting of nilpotent endomorphisms, and N the subgroup exp$\mathfrak{n}$ of $\mathbf{G}\mathbf{L}(V) ($§3, no. 1, Lemma 1).

(i) Let $\rho$ be a finite dimensional linear representation of $\mathfrak{n}$ on W, such that the elements of $\rho (\mathfrak{n})$ are nilpotent, $W'$ a vector subspace of W stable under $\rho ,\rho_1$ and $\rho_2$ the subrepresentation and quotient representation of $\rho$ defined by $W',\pi , \pi_1, \pi_2$ the representations of N compatible with $\rho , \rho_1, \rho_2($§3, no. 1). Then $\pi_1, \pi_2$ are the subrepresentation and quotient representation of $\pi$ defined by $W'$.

(ii) Let $\rho_1, \rho_2$ be finite dimensional linear representations of $\mathfrak{n}$ such that the elements of $\rho_1(\mathfrak{n})$ and $\rho_2(\mathfrak{n})$ are nilpotent, and $\pi_1, \pi_2$ the representations of N compatible with $\rho_1, \rho_2$. Then $\pi_1\otimes \pi_2$ is the representation of N compatible with $\rho_1\otimes \rho_2$.

(iii) Let $\rho_1, \rho_2$ be finite dimensional linear representations of $\mathfrak{n}$ on vector spaces $V_1,V_2$, such that the elements of $\rho_1(\mathfrak{n})$ and $\rho_2(\mathfrak{n})$ are nilpotent, $\rho$ the representation of $\mathfrak{n}$ on Hom(V$_1,V_2$) determined by $\rho_1, \rho_2$. Let $\pi_1, \pi_2$ be the representations of N compatible with $\rho_1, \rho_2$, and $\pi$ the representation of N on Hom(V$_1,V_2$) determined by $\pi_1, \pi_2$. Then $\pi$ is the representation of N compatible with $\rho$.

Assertion (i) is clear. Let $\rho_1, \rho_2, \pi_1, \pi_2$ be as in (ii). If $x\in \mathfrak{n}$, we have, since $\rho_1(x)\otimes 1$ and $1\otimes \rho_2(x)$ commute,

exp($\rho_1(x)\otimes 1 + 1\otimes \rho_2(x)$) $=$ exp($\rho_1(x)\otimes 1$). exp(1 $\otimes \rho_2(x)$)

= (exp $\rho_1(x)$)$\otimes 1.1\otimes$ (exp$\rho_2(x)$)

= (exp $\rho_1(x)$)$\otimes$ (exp$\rho_2(x)$)

$=\pi_1$(exp $x$)$\otimes \pi_2$(exp $x$)

$= (\pi_1\otimes \pi_2$)(exp $x$),

hence (ii). Let $\rho_1, \rho_2, \rho , \pi_1, \pi_2, \pi ,V_1,V_2$ be as in (iii). If $v_1\in$ EndV$_1$ and $v_2\in$ EndV$_2$, denote by $R_{v_1}$ and $L_{v_2}$ the maps $u \rightarrow uv_1$ and $u \rightarrow v_2u$ from Hom(V$_1,V_2$) to itself; these maps commute and $\rho (x)u= (L_{\rho_2(x)}-R_{\rho_1(x)})u$, so

exp$\rho (x).u=$ exp $L_{\rho_2(x)}$. exp $R_{-\rho_1(x)}.u$

$=\pi$(exp $x$)$.u$,

hence (iii).