Abstract
AbstractLet G be a split connected reductive group over $$\mathbb {F}_q$$
F
q
and let $$\mathbb {P}^1$$
P
1
be the projective line over $$\mathbb {F}_q$$
F
q
. Firstly, we give an explicit formula for the number of $$\mathbb {F}_{q}$$
F
q
-rational points of generalized Steinberg varieties of G. Secondly, for each principal G-bundle over $$\mathbb {P}^1$$
P
1
, we give an explicit formula counting the number of triples consisting of parabolic structures at 0 and $$\infty $$
∞
and a compatible nilpotent section of the associated adjoint bundle. In the case of $$GL_{n}$$
G
L
n
we calculate a generating function of such volumes re-deriving a result of Mellit.
Publisher
Springer Science and Business Media LLC
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