Proposition 2

Let $V,V',W$ be vector spaces. Let $r$ : S $\rightarrow$ End(V), $r': S\rightarrow$ End(V$'$) and $q: S\rightarrow$ End(W) be maps.

(i) Let $f: V\rightarrow W$ be a linear map such that $q(s)f(v) =f(r(s)v)$ for $s\in S$ and $v\in V$. Then, for all $\lambda \in P,f$ maps $V^{\lambda}(S)$ (resp. $V_{\lambda}(S)$) into $W^{\lambda}(S)$ (resp. $W_{\lambda}(S)$).

(ii) Let $B : V\times V'\rightarrow W$ be a bilinear map such that

for $s\in S,v\in V,v'\in V'$. Then, for all $\lambda , \mu\in P$, B maps $V^{\lambda}(S)\times {V'}^\mu(S)$ (resp. $V_{\lambda}(S)\times V'_\mu(S)$) into $W^{\lambda+\mu}(S)$ (resp. $W_{\lambda+\mu}(S)$).

(iii) Let $B : V\times V'\rightarrow W$ be a bilinear map such that

for $s\in S,v\in V,v'\in V'$. Then, for all $\lambda , \mu\in P$, B maps $V^{\lambda}(S)\times {V'}^\mu(S)$ (resp. $V_{\lambda}(S)\times V'_\mu(S)$) into $W^{\lambda \mu}(S)$ (resp. $W_{\lambda \mu}(S)$).

In case (i), $(q(s)-\lambda (s))^nf(v) =f((r(s)-\lambda (s))^nv)$ for $s\in S$ and $v\in V$, hence the conclusion. In case (ii),

for $s\in S,v\in V,v'\in V'$, hence by induction on $n$

$(q(s)-\lambda (s)-\mu(s))^nB(v, v) =\sum B((r(s)-\lambda (s))^iv,(r(s)-\mu(s))^jv)$.

$i+j=n$

The assertions in (ii) follow immediately. In case (iii), $(q(s)-\lambda (s)\mu(s))B(v, v') = B((r(s)-\lambda (s))v, r'(s)v')+B(\lambda (s)v,(r'(s)-\mu(s))v')$ for $s\in S,v\in V,v'\in V'$, hence by induction on $n$

= $\sum(n)B(\lambda (s)^j(r(s)-\lambda (s))^iv, r'(s)^i(r'(s)-\mu(s))^jv')$.

$i+j=n$

The assertions in (iii) follow immediately.