THE ACTION OF FINITE ORTHOGONAL GROUPS IN CHARACTERISTIC 2 ON THE SET OF ANISOTROPIC LINES

We prove that the permutation representation of the finite orthogonal group Ωε(n, q), where ε = + or −, on the set of anisotropic lines is multiplicity-free, if q is a power of 2 and n 6 is even. This result is established by giving a description of orbitals of this action. The rank of this action is (q2 + 2q)/2 if ε = + and n = 6, and (q2 + 2q + 2)/2 otherwise. Moreover, we compute the subdegrees of the orbitals of Ωε(n, q).