Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-11T03:54:55.318Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE SUPPORT VARIETIES FOR DEMAZURE MODULES

Part of: Linear algebraic groups and related topics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  19 March 2012

Benjamin F. Jones  and
Daniel K. Nakano
Benjamin F. Jones
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, USA (email: jonesbe@uwstout.edu)
Daniel K. Nakano*
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA (email: nakano@math.uga.edu)
*
✉For correspondence; e-mail: nakano@math.uga.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The support varieties for the induced modules or Weyl modules for a reductive algebraic group G were computed over the first Frobenius kernel G1 by Nakano, Parshall and Vella. A natural generalization of this computation is the calculation of the support varieties of Demazure modules over the first Frobenius kernel, B1, of the Borel subgroup B. In this paper we initiate the study of such computations. We complete the entire picture for reductive groups with underlying root systems A1 and A2. Moreover, we give complete answers for Demazure modules corresponding to a particular (standard) element in the Weyl group, and provide results relating support varieties between different Demazure modules which depend on the Bruhat order.

Keywords

support varietiescohomologyDemazure modules

MSC classification

Secondary: 17B56: Cohomology of Lie (super)algebras 20G10: Cohomology theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 3 , December 2011 , pp. 343 - 363
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The first-named author was supported in part by NSF VIGRE grant DMS-0738586, and the second-named author was supported in part by NSF VIGRE grant DMS-1002135.

References

[1]Andersen, H. H., ‘Schubert varieties and Demazure’s character formula’, Invent. Math. 79 (1985), 611–618.CrossRefGoogle Scholar
[2]Bendel, C. P., ‘Support varieties for infinitesimal algebraic groups’, PhD Thesis, Northwestern University, 1996.Google Scholar
[3]Benson, D. J., Representations and Cohomology I., Cambridge Studies in Advanced Mathematics, 30 (Cambridge University Press, Cambridge, 1991).Google Scholar
[4]Bjorner, A. and Brenti, F., Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231 (Springer, New York, 2005).Google Scholar
[5]Bourbaki, N., Groupes et Algèbres de Lie (Hermann, Paris, 1968).Google Scholar
[6]Carlson, J. F., Lin, Z., Nakano, D. K. and Parshall, B. J., ‘The restricted nullcone’, Contemp. Math. 325 (2003), 51–75.CrossRefGoogle Scholar
[7]Cline, E., Parshall, B. and Scott, L., ‘A Mackey imprimitivity theory for algebraic groups’, Math. Z. 182 (1983), 447–471.CrossRefGoogle Scholar
[8]Friedlander, E. M. and Parshall, B. J., ‘Support varieties for restricted Lie algebras’, Invent. Math. 86 (1986), 553–562.CrossRefGoogle Scholar
[9]Friedlander, E. M. and Pevtsova, J., ‘Π supports for modules over finite group schemes’, Duke Math. J. 139 (2007), 317–368.CrossRefGoogle Scholar
[10]Friedlander, E. M. and Suslin, A., ‘Cohomology of finite group schemes over a field’, Invent. Math. 127(2) (1997), 209–270.Google Scholar
[11]Humphreys, J. E., Reflection Groups and Coxeter Groups (Cambridge University Press, Cambridge, 1994).Google Scholar
[12]Jantzen, J. C., Representations of Algebraic Groups (Academic Press, New York, 1987).Google Scholar
[13]Melnikov, A., ‘On varieties in an orbital variety closure in semisimple Lie algebra’, J. Algebra 295 (2006), 44–50.CrossRefGoogle Scholar
[14]Nakano, D. K., Parshall, B. J. and Vella, D. C., ‘Support varieties for algebraic groups’, J. reine angew. Math. 547 (2002), 15–49.Google Scholar
[15]Polo, P., ‘Variétés de Schubert et excellentes filtrations’, Astérisque 173–174 (1989), 10–11, 281–311.Google Scholar
[16]Suslin, A., Friedlander, E. M. and Bendel, C. P., ‘Infinitesimal 1-parameter subgroups and cohomology’, J. Amer. Math. Soc. 10 (1997), 693–728.CrossRefGoogle Scholar
[17]Suslin, A., Friedlander, E. M. and Bendel, C. P., ‘Support varieties for infinitesimal group schemes’, J. Amer. Math. Soc. 10 (1997), 729–759.Google Scholar
[18] University of Georgia VIGRE Algebra Group, ‘Support varieties for Weyl modules over bad primes’, J. Algebra 312 (2007), 602–633.Google Scholar
[19]van der Kallen, W., ‘Longest weight vectors and excellent filtrations’, Math. Z. 201 (1989), 19–31.CrossRefGoogle Scholar