On the number of conjugacy classes in finite groups of lie type

On the existence of a unipotent support for the irreducible characters of a finite group of Lie type

“…Finally, there exists a unique integer n ρ ≥ 1 such that the scalar product of ρ and R s E1 is given by ±n −1 ρ . We call n ρ the generic denominator in ρ (1). These definitions are justified by the fact that we have…”

“…In order to indicate this, we shall also denote it by a ρ and call it the generic q-part of ρ (1). Finally, there exists a unique integer n ρ ≥ 1 such that the scalar product of ρ and R s E1 is given by ±n −1 ρ .…”

Fixed point ratios in actions of finite exceptional groups of Lie type

“…We next consider the semisimple element s. We recall that each semisimple class in G σ may be associated with a pair (J, [w]), where J is a proper subset of Π ∪ {α 0 } (determined up to conjugacy in W ), W J is the subgroup of W generated by reflections in the roots in J, and [w] = W J w is a conjugacy class representative of N W (W J )/W J , as explained in [16,21,22]. This association has the following properties: If s ∈ G σ has class associated with (J, [w]), then s lies in T ww * −1 , and if we set t = s g ww * −1 ∈ T 0 , then …”

“…For example, the maximal parabolics of 2 E 6 (q) are labelled according to the root system F 4 : Thus P 1 , P 2 , P 3 , P 4 correspond respectively to the E 6 -parabolics P 2 , P 4 , P 35 , P 16 . P poly L = 2 E 6 (q) F 4 (q) P 1 f P,α (q) = q 6 − q 3 + 1 q 2 (q 3 − 1)…”

On degrees of irreducible Brauer characters

“…C G (t) = T ).) Let l denote the Lie rank of G. Applying 3.1 and 3.2 in [10] (see also [12]), we get…”