Let $(X,L)$ be a polarized abelian variety over $k=\overline{k}$, and let $K(L)$ be the kernel of the isogeny $X\to X^\vee$ that sends $x$ to $t_x^*L\otimes L^{-1}$. The theta group $\mathscr{G}(L)$ of $(X,L)$ is a central extension $$0\to k^\times\to\mathscr{G}(L)\to K(L)\to0.$$ Now, $\mathscr{G}(L)$ acts on $H^0(X,L)$, and we get an irreducible representation $$\mathscr{G}(L)\to \mbox{GL}(H^0(X,L)),$$ which can be seen to descend to a projective representation $K(L)\to\mbox{PGL}(h^0(L)-1,k)$. The nice thing about this projective representation is that, if $\varphi:X\to\mathbb{P}^{h^0(L)-1}$ is the map associated to $|L|$, then for all $z\in K(L)$ and $x\in X$, we have that $z\cdot\varphi(x)=\varphi(x+z)$. Using this, it is easily verified that $\mbox{span}\{\varphi(K(L))\}$ is all $\mathbb{P}^{h^0(L)-1}$.

We have that $K(L)=K_1\oplus K_2$, where $K_2$ is the symplectic complement to $K_1$ using the Weil pairing.

My question is the following: Is it true that the projective representation restricted to $K_1$ (or $K_2$) is also irreducible? Put into other terms, is the linear span of $\varphi(K_1)$ all of $\mathbb{P}^{h^0(L)-1}$?