Abstract
Let $G$ be a simple Chevalley group defined over $\mathbb{F}_{q}$. We show that if $r$ does not divide $q$ and $k$ is an algebraically closed field of characteristic $r$, then very few irreducible $kG$-modules have nonzero $H^1(G,V)$. We also give an explicit upper bound for $\dim H^1(G,V)$ for $V$ an irreducible $kG$-module that does not depend on $q$, but only on the rank of the group. Cline, Parshall and Scott showed that such a bound exists when $r|q$. We obtain extremely strong bounds in the case that a Borel subgroup has no fixed points on $V$.
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ISSN = {0021-8693},
MRCLASS = {20C20},
MRNUMBER = {0427443},
MRREVIEWER = {G. Michler},
DOI = {10.1016/0021-8693(75)90181-7},
ZBLNUMBER = {0303.20008},
} -
[Sc]
L. L. Scott, "Some new examples in 1-cohomology," J. Algebra, vol. 260, iss. 1, pp. 416-425, 2003.
@article {Sc, MRKEY = {1976701},
AUTHOR = {Scott, Leonard L.},
TITLE = {Some new examples in 1-cohomology},
NOTE = {special issue celebrating the 80th birthday of Robert Steinberg},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {260},
YEAR = {2003},
NUMBER = {1},
PAGES = {416--425},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {20J06 (20G05 20G10)},
MRNUMBER = {1976701},
MRREVIEWER = {Christopher P. Bendel},
DOI = {10.1016/S0021-8693(02)00667-1},
ZBLNUMBER = {1019.20020},
}