Let $G$ be a simple algebraic group over an algebraically closed field $K$, $V$ its rational $KG$-module, and $L$ a long root $SL_2$-subgroup in $G$. Assume that $[V,L]$ is an irreducible $KL$-module.
Has anyone tried to classify representations with this property? They look as a natural generalisation of classical quadratic pairs in sense of John Thompson, or of even more classical classification of groups generated by transvections.
What follows is not a complete solution: it would take too long to write it down.
Suppose first $V$ is infinitesimally irreducible (and rational) and let $\lambda=\sum_{i=1}^\ell a_i\varpi_i$ be the highest weight of $V$. We may assume that $L$ is generated by the unipotent root subgroups $U_{\pm\theta}$ where $\theta$ is the highest root of $\Phi$, the root system of $G$. Express the coroot $\theta^\vee$ via simple coroots of $\Phi^\vee$ as $\theta^\vee=\sum_{i=1}^\ell m_i\alpha_i^\vee$. Then $m_i\in\mathbb{Z}_{>0}$ for all $i$ and $\langle\lambda,\theta^\vee\rangle= \sum_{i=1}^\ell a_im_i$. We call this number $n(V)$. Let $X_+(V)$ be the set of all dominant weights of $V$. If we exclude ${\rm G}_2$ in characteristics $2$ and $3$ and ${\rm F}_4$, ${\rm B}_n$, ${\rm C}_n$ in characteristic $2$, then it is proved in an old paper of mine here that, as in the characteristic $0$ case, $X_+(V)=(\lambda-Q_+)\cap P_{++}$ where $Q_+=\{\sum_{i=1}^\ell r_i\alpha_i\,|\,\,r_i\in\mathbb{Z}_{\ge 0}\}$ and $P_{++}=\{\sum_{i=1}^\ell q_i\varpi_i\,|\,\,q_i\in\mathbb{Z}_{\ge 0}\}$. If $\mu=\sum_{i=1}^\ell r_i\varpi_i\in X_+(V)\setminus\{0\}$ has the property that $\mu+\theta\not\in X_+(V)$ then any nonzero vector in $V_\mu$ will give rise to a composition factor of $[V,L]$. The irreducibility of $[V,L]$ then yields $\sum_{i=1}^\ell r_im_i=n(V)$ (otherwise $[V,L]$ would have two non-isomorphic composition factors). This means that $V$ has very few dominant weights. At this point the problem becomes a routine exercise on root systems.
$\mathbf{Remark.}$ 1. If $\mu-q\theta,\ldots,\mu,\ldots, \mu+p\theta$ is a $\theta$-string of weights of $V$, then the above reasoning also shows that either $p=q=\langle \mu,\theta^\vee\rangle=0$ or $\langle \mu,\theta^\vee \rangle +2p=n(V)$ and $\langle\mu,\theta^\vee\rangle -2q=-n(V)$. Since $q-p= \langle \mu,\theta^\vee\rangle$ this yields that any $\theta$-string of weights of $V$ either has length $1$ or $n(V)+1$. This should narrow the possibilities for $\lambda$ even further.
In the excluded cases in types ${\rm B}_n$, ${\rm C}_n$, ${\rm F}_4$ and ${\rm G}_2$ the dominant weights of infinitesimally irreducible representations are also known.
The general case, for $V$ rational, can be reduced to the above case by using Steinberg's tensor product theorem. If $V$ is not rational (but still irreducible) one can use the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups; see ${\it Ann. Math}$., vol. 97, 499-571, 1973.
