A Higgs bundle over an algebraic curve is a vector bundle with a twisted endomorphism. An important question is to calculate the volume of the groupoid of Higgs bundles over nite elds. In 2014, Olivier Schiffmann succeeded in finding the corresponding generating function and together with Mozvogoy reduced the problem to counting pairs of a vector bundle and a nilpotent endomorphism. It was generalized recently by Anton Mellit to the case of Higgs bundles with regular singularities. An important step in Mellit's calculations is the case of
and 2 marked points, which allows him to relate the corresponding generating function with the Macdonald polynomials.
It is a natural question to generalize Mellit's calculations to arbitrary reductive groups. In my work, I consider the case of
with 2 points for an arbitrary split connected reductive group
over
. Firstly, I give an explicit formula for the number of
-rational points of generalized Steinberg varieties of
. Secondly, for each principal $G$-bundle over
, I give an explicit formula counting the number of triples consisting of parabolic structures at 0 and
and compatible nilpotent sections of the associated adjoint bundle.
