Suborbits of subspaces of type (m,k) under finite singular general linear groups

Suppose denotes the (n+l)-dimensional vector space over a finite field Fq and denotes the corresponding singular general linear group. All the subspaces of type (m,k) form an orbit under , denoted by M(m,k;n+l,n). Let Λ be the set of all the orbitals of . Then (M(m,k;n+l,n),Λ) is a symmetric association scheme. In this paper, we determine all the orbitals and the rank of , calculate the length of each suborbit. Finally, we compute all the intersection numbers of the symmetric association scheme (M(m,k;n+l,n),Λ), where k=1 or k=l-1.