Association schemes based on isotropic subspaces, Part 1

The subspaces of a given dimension in a finite classical polar space form the points of an association scheme. When the dimension is zero, this is the scheme of the collinearity graph of the space. At the other extreme, when the dimension is maximal, it is the scheme of the corresponding dual polar graph. These extreme cases have been thoroughly studied. In this article, the general case is examined and a detailed computation of the intersection numbers of these association schemes is initiated.