The author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l, this bound is proportional to l 3 , and for rank less than or equal to 11 much higher. The small rank cases are based on extensive computer calculations.

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Let G be a finite group of Lie type in odd characteristic defined over a field with q elements. We prove that there is an absolute (and explicit) constant c such that, if G is a classical matrix group of dimension n 2, then at least c/ log(n) of its elements are such that some power is an involution with fixed point subspace of dimension in the interval [n/3, 2n/3). If G is exceptional, or G is classical of small dimension, then, for each conjugacy class C of involutions, we find a very good lower bound for the proportion of elements of G for which some power lies in C.