Some orbits of a two-vertex stabilizer in a Grassmann graph

Let Fq denote a finite field with q elements. Let $n,k$ denote integers with n>2k≥ 6. Let V denote a vector space over Fq that has dimension n. The vertex set of the Grassmann graph Jq(n,k) consists of the k-dimensional subspaces of V. Two vertices of Jq(n,k) are adjacent whenever their intersection has dimension $k-1$. Let ∂ denote the path-length distance function of Jq(n,k). Pick vertices $x,y$ of Jq(n,k) such that 1<∂(x,y)<k. Let Stab(x,y) denote the subgroup of $GL(V)$ that stabilizes both x and y. In this paper, we investigate the orbits of Stab(x,y) acting on the local graph Γ(x). We show that there are five orbits. By construction, these five orbits give an equitable partition of Γ(x); we find the corresponding structure constants. In order to describe the five orbits more deeply, we bring in a Euclidean representation of Jq(n,k) associated with the second largest eigenvalue of Jq(n,k). By construction, for each orbit its characteristic vector is represented by a vector in the associated Euclidean space. We compute many inner products and linear dependencies involving the five representing vectors.