  Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote
integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$
that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$
consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$
are adjacent whenever their intersection has dimension $k-1$. Let $\partial$
denote the path-length distance function of $J_q(n,k)$. Pick vertices $x,y$ of
$J_q(n,k)$ such that $1<\partial(x,y)<k$. Let $\text{Stab}(x,y)$ denote the
subgroup of $GL(V)$ that stabilizes both $x$ and $y$. In this paper, we
investigate the orbits of $\text{Stab}(x,y)$ acting on the local graph
$\Gamma(x)$. We show that there are five orbits. By construction, these five
orbits give an equitable partition of $\Gamma(x)$; we find the corresponding
structure constants. In order to describe the five orbits more deeply, we bring
in a Euclidean representation of $J_q(n,k)$ associated with the second largest
eigenvalue of $J_q(n,k)$. 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.
