Some Characters of Orthogonal Groups Over the Field of Two Elements

Abstract

Let G _n , or G denote the (2n+1)-dimensional orthogonal group 0_2n+1(2) over the field F ₂ of two elements, which is isomorphic with the symplectic group Sp_2n (2), let A _n or A and B _n or B denote the maximal full orthogonal subgroups 0_2n (2, −) and 0_2n (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 ^G _A and 1 ^−G _A 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 4ⁿ(4ⁿ−1)/2 in G _n . We denote certain factors of this index by a _n = 2ⁿ+1, b _n = 2ⁿ − 1.