Lemma 11

Let $G_1$ and $G_2$ be groups, and let $W_1$ and $W_2$ be K-vector spaces. We endow $W_1($resp. $W_2)$ with an action of $G_1($resp. $G_2)$ given by a group homomorphism $\rho_1: G_1\rightarrow$ Aut$_K(W_1) ($resp. $\rho_2: G_2\rightarrow$ Aut$_K(W_2))$. We consider the group homomorphism

$\rho_1\otimes \rho_2: G_1\times G_2\longrightarrow$ Aut$_K(W_1\otimes_KW_2)$

defined by the relation $(\rho_1\otimes \rho_2)(g_1, g_2)(w_1\otimes w_2) =\rho_1(g_1)(w_1)\otimes \rho_2(g_2)(w_2)$ for $g_1\in G_1,g_2\in G_2,w_1\in W_1$, and $w_2\in W_2$. Then the linear mapping from $W^{G_1}_1\otimes_KW^{G_2}_2$ to $W_1\otimes_KW_2$ given by the tensor product of the canonical injections induces an isomorphism of K-vector spaces from $W_1^{G_1}\otimes_KW^{G_2}_2$ to $(W_1\otimes_KW_2)^{G_1\times G_2}$.

This follows from Lemma 1 of VIII, p. 213 applied to the K[G]-modules $M_1= M_2= K$ endowed with the trivial action of $G, N_1= W_1$, and $N_2= W_2$.