Geodesic
Unipotent elements of nonprime order in representations of the classical algebraic groups: two big Jordan blocks
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 193-241

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For irreducible rational representations of the classical algebraic groups in characteristic $p>2$ that are not equivalent to a composition of a group morphism and the standard representation, it is proved that usually the image of a unipotent element of order $p^{s+1}>p$ has at least two Jordan blocks of size $>p^s$; all exceptions are indicated explicitly. As a corollary, irreducible rational representations of these groups whose images contain unipotent elements with just one Jordan block of size $>1$ are classified. 
@article{ZNSL_2013_414_a11,
     author = {I. D. Suprunenko},
     title = {Unipotent elements of nonprime order in representations of the classical algebraic groups: two big {Jordan} blocks},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {193--241},
     publisher = {mathdoc},
     volume = {414},
     year = {2013},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_414_a11/}
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I. D. Suprunenko. Unipotent elements of nonprime order in representations of the classical algebraic groups: two big Jordan blocks. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 193-241. http://geodesic.mathdoc.fr/item/ZNSL_2013_414_a11/