Some characters of orthogonal groups over the field of two elements

Let G n , or G denote the (2n+1)-dimensional orthogonal group 02n+1(2) over the field F 2 of two elements, which is isomorphic with the symplectic group Sp2n (2), let A n or A and B n or B denote the maximal full orthogonal subgroups 02n (2, −) and 02n (2, +) of G n , and let A′ n or A′ and B′ n or B′ denote the subgroups of A n or B n of of index 2 which are simple commutator subgroups (with the exception of B′2 of order 36). Let 1 A G and 1 A −G denote the characters of G induced by the trivial 1-character l A of A and by the alternating character ī A of A whose value is +1 in the subgroup A′ and −1 in its second coset A′τ. Let A n ∩ B n = D n or D, and A′ n ∩ B′ n = D′ n or D′ Then D′ n is isomorphic with G n −1 and has index 4 n (4 n −1)/2 in G n . We denote certain factors of this index by a n = 2 n +1, b n = 2 n − 1.