More specifically, is it true that a representation of $\dim < p+1$ of the algebraic group $SL\_2(\mathbb{F}\_p)$ is always completely reducible? (of course above this dimension there are non completely reducible examples)


More general results that might help in this direction are also welcome.


Thanks



Are low dimensional modular representations of SL2(Fp) completely reducible?
The essential work in this direction was published from 1994 on by J.-P. Serre
and J.C. Jantzen, concerning both algebraic groups and related finite groups
of Lie type. Related papers by R. Guralnick and G.J. McNinch followed. There are uniform dimension bounds for complete reducibility, stricter in rank 1. For a finite group of simple type over a field of $q$ elements in characteristic $p$, Jantzen's upper bound is $p$ for rank at least 2 but $p-2$ in your case. The best I can do is list a few references:


MR1635685 (99g:20079) 20G05 (20G40)
Jantzen, Jens Carsten (1-OR)
Low-dimensional representations of reductive groups are semisimple.
Algebraic groups and Lie groups, 255–266, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ.
Press, Cambridge, 1997.


MR1753813 (2001k:20096) 
McNinch, George J.(1-NDM)
Semisimple modules for finite groups of Lie type. 
J. London Math. Soc. (2) 60 (1999), no. 3, 771--792. 


MR1717357 (2000m:20018) 20C20
Guralnick, Robert M. (1-SCA)
Small representations are completely reducible.
J. Algebra 220 (1999), no. 2, 531–541.


