## First cohomology for finite groups of Lie type: Simple modules with small dominant weights

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## Abstract:

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb {F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb {F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $\operatorname {H}^1(G(\mathbb {F}_q),L(\lambda ))$, where $L(\lambda )$ is the simple $G$-module of highest weight $\lambda$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $\lambda$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $\lambda$ is a minimal non-zero dominant weight.## References

- Henning Haahr Andersen,
*Extensions of modules for algebraic groups*, Amer. J. Math.**106**(1984), no. 2, 489–504. MR**737781**, DOI 10.2307/2374311 - M. Aschbacher and R. Guralnick,
*Some applications of the first cohomology group*, J. Algebra**90**(1984), no. 2, 446–460. MR**760022**, DOI 10.1016/0021-8693(84)90183-2 - Christopher P. Bendel, Daniel K. Nakano, and Cornelius Pillen,
*On comparing the cohomology of algebraic groups, finite Chevalley groups and Frobenius kernels*, J. Pure Appl. Algebra**163**(2001), no. 2, 119–146. MR**1846657**, DOI 10.1016/S0022-4049(01)00024-X - Edward Cline, Brian Parshall, and Leonard Scott,
*Cohomology of finite groups of Lie type. I*, Inst. Hautes Études Sci. Publ. Math.**45**(1975), 169–191. MR**399283**, DOI 10.1007/BF02684301 - E. Cline, B. Parshall, L. Scott, and Wilberd van der Kallen,
*Rational and generic cohomology*, Invent. Math.**39**(1977), no. 2, 143–163. MR**439856**, DOI 10.1007/BF01390106 - Leonard Evens,
*The cohomology of groups*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR**1144017** - Eric M. Friedlander,
*Weil restriction and support varieties*, J. Reine Angew. Math.**648**(2010), 183–200. MR**2774309**, DOI 10.1515/CRELLE.2010.083 - Eric M. Friedlander and Brian J. Parshall,
*Cohomology of infinitesimal and discrete groups*, Math. Ann.**273**(1986), no. 3, 353–374. MR**824427**, DOI 10.1007/BF01450727 - The GAP Group,
*GAP – Groups, Algorithms, and Programming, Version 4.4.12*, 2008. - Peter B. Gilkey and Gary M. Seitz,
*Some representations of exceptional Lie algebras*, Geom. Dedicata**25**(1988), no. 1-3, 407–416. Geometries and groups (Noordwijkerhout, 1986). MR**925845**, DOI 10.1007/BF00191935 - Robert M. Guralnick,
*The dimension of the first cohomology group*, Representation theory, II (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1178, Springer, Berlin, 1986, pp. 94–97. MR**842479**, DOI 10.1007/BFb0075290 - Robert M. Guralnick and Corneliu Hoffman,
*The first cohomology group and generation of simple groups*, Groups and geometries (Siena, 1996) Trends Math., Birkhäuser, Basel, 1998, pp. 81–89. MR**1644977**, DOI 10.1007/bf01214004 - Robert M. Guralnick and Pham Huu Tiep,
*First cohomology groups of Chevalley groups in cross characteristic*, Ann. of Math. (2)**174**(2011), no. 1, 543–559. MR**2811608**, DOI 10.4007/annals.2011.174.1.16 - James E. Humphreys,
*Introduction to Lie algebras and representation theory*, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR**499562** - James E. Humphreys,
*Modular representations of finite groups of Lie type*, London Mathematical Society Lecture Note Series, vol. 326, Cambridge University Press, Cambridge, 2006. MR**2199819** - Jens C. Jantzen,
*First cohomology groups for classical Lie algebras*, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 289–315. MR**1112165** - Jens Carsten Jantzen,
*Representations of algebraic groups*, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR**2015057** - Wayne Russell Jones,
*COHOMOLOGY OF FINITE GROUPS OF LIE TYPE*, ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–University of Minnesota. MR**2625445** - Wayne Jones and Brian Parshall,
*On the $1$-cohomology of finite groups of Lie type*, Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975) Academic Press, New York, 1976, pp. 313–328. MR**0404470** - A. S. Kleshchev and J. Sheth,
*On extensions of simple modules over symmetric and algebraic groups*, J. Algebra**221**(1999), no. 2, 705–722. MR**1728406**, DOI 10.1006/jabr.1998.8038 - A. S. Kleshchev and J. Sheth,
*Corrigendum: “On extensions of simple modules over symmetric and algebraic groups” [J. Algebra 221 (1999), no. 2, 705–722; MR1728406 (2001f:20091)]*, J. Algebra**238**(2001), no. 2, 843–844. MR**1823787**, DOI 10.1006/jabr.2000.8667 - Zongzhu Lin and Daniel K. Nakano,
*Complexity for modules over finite Chevalley groups and classical Lie algebras*, Invent. Math.**138**(1999), no. 1, 85–101. MR**1714337**, DOI 10.1007/s002220050342 - Saunders Mac Lane,
*Homology*, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR**1344215** - Daniel K. Nakano,
*Cohomology of algebraic groups, finite groups, and Lie algebras: Interactions and connections*, Lie Theory and Representation Theory, Survey of Modern Mathematics Volume II (edited by N. Hu, B. Shu and J.P. Wang), International Press, 2012, pp. 151–176. - Brian J. Parshall and Leonard L. Scott,
*Integral and graded quasi-hereditary algebras, II with applications to representations of generalized $q$-schur algebras and algebraic groups*, 2009. - Brian J. Parshall and Leonard L. Scott,
*Bounding Ext for modules for algebraic groups, finite groups and quantum groups*, Adv. Math.**226**(2011), no. 3, 2065–2088. MR**2739773**, DOI 10.1016/j.aim.2010.09.021 - Leonard L. Scott,
*Some new examples in 1-cohomology*, J. Algebra**260**(2003), no. 1, 416–425. Special issue celebrating the 80th birthday of Robert Steinberg. MR**1976701**, DOI 10.1016/S0021-8693(02)00667-1 - University of Georgia VIGRE Algebra Group,
*On Kostant’s theorem for Lie algebra cohomology*, Representation theory, Contemp. Math., vol. 478, Amer. Math. Soc., Providence, RI, 2009, pp. 39–60. University of Georgia VIGRE Algebra Group: Irfan Bagci, Brian D. Boe, Leonard Chastkofsky, Benjamin Connell, Bobbe J. Cooper, Mee Seong Im, Tyler Kelly, Jonathan R. Kujawa, Wenjing Li, Daniel K. Nakano, Kenyon J. Platt, Emilie Wiesner, Caroline B. Wright and Benjamin Wyser. MR**2513265**, DOI 10.1090/conm/478/09318 - Andrew Wiles,
*Modular elliptic curves and Fermat’s last theorem*, Ann. of Math. (2)**141**(1995), no. 3, 443–551. MR**1333035**, DOI 10.2307/2118559

## Additional Information

**University of Georgia VIGRE Algebra Group**- Affiliation: Department of Mathematics, University of Georgia. Athens, Georgia 30602-7403; Department of Mathematics, Statistics, and Computer Science, University of Wisconsin–Stout, Menomonie, Wisconsin 54751
- MR Author ID: 310155
- ORCID: 0000-0001-7984-0341
- Received by editor(s): October 21, 2010
- Received by editor(s) in revised form: July 3, 2011, and July 5, 2011
- Published electronically: October 12, 2012
- Additional Notes: The members of the University of Georgia VIGRE Algebra Group are Brian D. Boe, Adrian M. Brunyate, Jon F. Carlson, Leonard Chastkofsky, Christopher M. Drupieski, Niles Johnson, Benjamin F. Jones, Wenjing Li, Daniel K. Nakano, Nham Vo Ngo, Duc Duy Nguyen, Brandon L. Samples, Andrew J. Talian, Lisa Townsley, and Benjamin J. Wyser.
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**365**(2013), 1025-1050 - MSC (2010): Primary 20G10; Secondary 20G05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05664-9
- MathSciNet review: 2995382