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 $\mathbb{P}^{h^0(L)-1}$?



Theta group representation
I think the answer is no due to the following counterexample (there might be a mistake somewhere though):


Take an elliptic curve $X$ with $L = O\_X(3P\_0)$ (characteristic $\neq 2, 3$), then $K$ will be the $3$-torsion subgroup and $K\_1$ will be generated by a $3$-torsion point $P$. Then in the embedding by $|L|$ of $X$ in $\mathbb{P}^2$, the points $P$, $-P$ and $P\_0$ are collinear, that is, $K\_1$ is mapped to a lower-dimensional linear subspace.


