Cohomology of finite groups of Lie type
Homological Methods in Representation Theory
March 31, 2008 02:00 PM to 03:00 PM
Speakers:
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Abstract: |
Let $G$ be a reductive algebraic group over a field $k$ of
prime characteristic $p$ which is split over the prime field ${\mathbb F}_p$.
Let $\Fr : G \to G$ denote the Frobenius map. Then the fixed points
of the $r$th iterate of the Frobenius map, denoted $G({\mathbb F}_{p^{r}})$,
is a finite Chevalley group. The question of interest in this talk is to
determine the least $i > 0$ such that the cohomology group
$\operatorname{H}^i(G({\mathbb F}_{p^{r}}),k) \neq 0$.
Quillen showed that $\opH^i(GL_n(\mathbb{F}_{p^r}),k) = 0$ for
all $0 < i < r(p - 1)$ for all $n$. In that work, he noted that for
all finite Chevalley groups there exist constants $C$ depending on the
root system and the prime such that
$\operatorname{H}^i(G({\mathbb F}_{p^{r}}),k) = 0$ for $0 < i < Cr$.
However, no explicit value of $C$ is given except for $G = SL_2$ (and $p$
odd) in which case one can take $C = (p-1)/2$. Further, it was not shown
whether these vanishing ranges were sharp. Indeed, in the case of $SL_2$,
one can see that these bounds are not sharp in general. Later work by
Friedlander and by Hiller extended Quillen's results and found vanishing
ranges for groups of all types. Since then, few if any results have been
obtained in this direction.
The goal here is to exploit techniques developed by Bendel, Nakano and
the presenter which relate $\opH^i(\gfpr,k)$ to extensions over $G$.
In particular, for a group $G$ of classical type and $r=1$, under the
assumption that $p > h$ (the Coxeter number), we improve on the vanishing
ranges of Hiller and in many cases find sharp bounds.
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