On some characters of \(GL_ n(2)\) (Q802016)

Let V be the vector space of dimension at least 3 over the field of two elements. Let G be the general linear group acting on V. Let \(\theta\) be the nonidentity constituent of the permutation character from G acting on the nonzero elements of V. Let \(\Delta\) be the set of all ordered pairs where the components are a hyperplane of V and a vector not in that hyperplane. Let \(\xi\) be the permutation character of G on \(\Delta\). The author finds the irreducible constituents of \(\xi\). In fact, \(\xi =1+2\theta +\phi +\psi\) where \(\phi\) is a character obtained by Steinberg and \(\psi\) is apparently new.