Abstract
Let 𝐺 be a connected reductive group defined over
F
q
\mathbb{F}_{q}
.
Fix an integer
M
≥
2
M\geq 2
, and consider the power map
x
↦
x
M
x\mapsto x^{M}
on 𝐺.
We denote the image of
G
(
F
q
)
G(\mathbb{F}_{q})
under this map by
G
(
F
q
)
M
G(\mathbb{F}_{q})^{M}
and estimate what proportion of regular semisimple, semisimple and regular elements of
G
(
F
q
)
G(\mathbb{F}_{q})
it contains.
We prove that, as
q
→
∞
q\to\infty
, the set of limits for each of these proportions is the same and provide a formula.
This generalizes the well-known results for
M
=
1
M=1
where all the limits take the same value 1.
We also compute this more explicitly for the groups
GL
(
n
,
q
)
\mathrm{GL}(n,q)
and
U
(
n
,
q
)
\mathrm{U}(n,q)
and show that the set of limits are the same for these two group, in fact, in bijection under
q
↦
-
q
q\mapsto-q
for a fixed 𝑀.
Bessel F-isocrystals for reductive groups
