the one we used to define the $c_\chi$ above in case $\chi\in X({{\mathbb R}}G)_\pm$. So then $L$ preserves $c_\chi$ when defined. Hence in all cases $L$ lands in the corresponding factor of the decomposition given by Equation (\[eqn:rdec\]) so that it is contained in the image of ${{\mathbb R}}G$ in ${\operatorname{End}}(F)$.

\[rem:\] This lemma can fail when a symplectic representation occurs with multiplicity $2$. To see this, let $V$ be an irreducible ${{\mathbb R}}G$-module which is of quaternionic type. This means that the ${{\mathbb R}}$-algebra $k:={\operatorname{End}}_{{{\mathbb R}}G}(V)$ is isomorphic to ${{\mathbb K}}$ and the image of ${{\mathbb R}}G$ in ${\operatorname{End}}_{{\mathbb R}}(V)$ is equal to ${\operatorname{End}}_k(V)$. Let $s: V\times V\to {{\mathbb R}}$ be a $G$-invariant positive definite quadratic form. Then $L:={\operatorname{O}}(V, s)$ is clearly not contained in the image of ${{\mathbb R}}G$. The ${{\mathbb C}}G$-module $F:={{\mathbb C}}\otimes_{{\mathbb R}}V$ is a direct sum of two copies of a symplectic irreducible ${{\mathbb C}}G$-module, but both $G$ and $L$ have the same subspace of invariants in ${\operatorname{Sym}}^2(V_{{\mathbb C}})$, namely the line spanned by the tensor determined by $s$.

Before we begin the proof of Theorem \[endo\], we recall that for a polarized abelian variety $A$, the polarization determines in the algebra ${\operatorname{End}}_{{\mathbb Q}}(A)$ an 