Whitney numbers of projective space over \(\mathbb{R},\mathbb{C},\mathbb{H}\) and the \(p\)- adics (Q1347260)

The Whitney numbers of a finite projective space are the number of flats in a given dimension, so in the Desarguesian case they equal the Gaussian coefficients. The author defines and computes Whitney numbers for some infinite classical projective spaces, namely for the real, complex and quaternionic projective spaces and for projective spaces over the \(p\)- adics. Instead of cardinality he uses measures on the underlying vector spaces, the \(p\)-adic measure for the latter geometries, and Haar measures invariant under the general linear group for the other geometries [cf. \textit{L. Santaló}, `Integral geometry and geometric probability' (1976; Zbl 0342.53049)].