  We study the space of invariant generalized functions supported on an orbit
of the action of a real algebraic group on a real algebraic manifold. This
space is equipped with the Bruhat filtration. We study the generating function
of the dimensions of the filtras, and give some methods to compute it. To
illustrate our methods we compute those generating functions for the adjoint
action of $\mathrm{GL}_3(\mathbb{C})$. Our main tool is the notion of
generalized functions on a real algebraic stack, introduced recently by
Sakellaridis.
