0-Cycles on Grassmannians as Representations of Projective Groups

Abstract

Let F be an infinite division ring, V be a left F-vector space, \(r\ge 1\) be an integer. We study the structure of the representation of the linear group \(\mathrm {GL}_F(V)\) in the vector space of formal finite linear combinations of r-dimensional vector subspaces of V with coefficients in a field. This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth if F is locally compact and non-discrete.