Character Tables of Parabolic Subgroups of the Chevalley Groups of TypeG₂

Dade's invariant conjecture for the Chevalley groups of type G2 in the defining characteristic

“…Let χ = χ(x) be an irreducible character of G 2 (q) labeled by the parameter x [4]. Then χ(x) α = χ(px) (which will be verified later).…”

“…Thus χ α t = χ if and only if (s) α t G = (s) G , namely, s α t = s w for some w ∈ W , where (s) G is the conjugacy class of G containing s. Thus χ α t = χ if and only if s ∈ C T (α t w −1 ), namely, s is a regular element of G 2 (p t ), since C T (α t w −1 ) is a maximal torus of G 2 (p t ). But a regular element s of G 2 (p t ) labels an irreducible character ψ = ψ s,1 of G 2 (p t ) such that its parameter y (see [4]) lies in X(p t ). It follows that C X(p a ) (H ) X p t as H -sets.…”

“…If L ∈ {G, W, P , Q}, then the character table of L is given in [4,6]. We will follow the same notation.…”

“…We will follow the same notation. Let X = X(q) be a parameter set given by [4], and let U O. Denote by C X (U ) the set of fixed-points of X under the action of U . Lemma 3.1.…”

“…Dade's invariant conjecture has been verified for the groups G 2 (q) in the defining characteristic when q = 2 a or 3 a [2] and non-defining characteristics [3]. Our goal in this paper is to verify Dade's invariant conjecture for the group G 2 (p a ) in the defining characteristic when p = 2 or 3, using the character tables of G 2 (q) and its Borel and parabolic subgroups [4,6]. Together with [12], this completes the verification of the conjecture for any blocks with positive defect of these groups.…”