$q$, leaves $F_q$ invariant. In our case this means that $L$ leaves invariant every subspace $F_q$ with $q\in ({\operatorname{Sym}}^2 F)^G$. Among the nonzero subspaces $F_q$ we thus obtain are those of the form $V_\chi\otimes \ell$ with $\chi\in X({{\mathbb C}}G)_+$ and $\ell$ any line in $F[\chi]$, $V_\chi\otimes P$ with $\chi \in X({{\mathbb C}}G)_-$ and $P$ any plane in $F[\chi]$, and $(V_\chi\otimes \ell)\oplus (V_{\chi^*}\otimes \ell')$ with $\chi\in X({{\mathbb C}}G)_o$ and $\ell\subset F[\chi]$ and $\ell'\subset F[\chi^*]$ arbitrary lines. Then our assumptions (i) and (ii) imply that for every $\chi\in X({{\mathbb C}}G)$ and every line $\ell$ in $F[\chi]$, $V_\chi\otimes\ell$ is an intersection of such subspaces (and hence preserved by $L$). It follows that $L$ acts trivially on each ${{\mathbb P}}(F[\chi])$, or rather, that $L$ is contained in $\prod_{\chi\in X({{\mathbb C}}G)} {\operatorname{GL}}(V_\chi)\otimes \mathbf{1}_{F[\chi]}$, so that $L$ is at least contained in the image of ${{\mathbb C}}G$ in ${\operatorname{End}}(F)$. If $F[\chi]=0$, then the factor ${\operatorname{GL}}(V_\chi)\otimes \mathbf{1}_{F[\chi]}$ is of course trivial, but otherwise $L$ will act on $V_\chi$ in such a way that it preserves $\alpha_\chi$.

Assume that $F[\chi]\not=0$. Since $L$ is compact, it leaves invariant an inner product in $V_\chi$. Since this inner product is also left invariant by $G$, we may assume that this is 