Abstract: Let X be a smooth projective curve over a finite field. Consider the moduli stack classifying pairs (E,Phi), where E is a vector bundle over X, Phi is a nilpotent endomorphism of E. This is a stack of
infinite type, and, in fact, of infinite volume, that is, counting points (E,Phi) of this stack with weight 1/Aut(E,Phi), we get a divergent series. However, it is easy to stratify the stack by natural substacks whose volumes are finite. These volumes are relatively easy to calculate; this is an important step in O.~Schiffman's counting of Higgs bundles on curves. Vector bundles of rank r can be interpreted as principal GL(r)-bundles. One can define a similar stack for any algebraic group G: the stack classifying pairs (E,Phi), where E is a principal G-bundle over X, Phi is a nilpotent section of the adjoint vector bundle. For example, if G is the orthogonal group, one considers vector bundles with a quadratic form in each fiber.  We explain how to stratify the stack by locally closed substacks of finite volume, and address the question of counting points of this stack.