×

Vector bundles associated to Lie algebras. (English) Zbl 1370.14018

Let \(\mathfrak{g}\) be a restricted Lie algebra. Let \(\mathbb{E}(r, \mathfrak{g})\) denote the projective variety of all elementary subalgebras of \(\mathfrak{g}\). For any representation of \(\mathfrak{g}\) and any \(G\)-stable locally closed subset \(X\) in \(\mathbb{E}(r, \mathfrak{g})\), in this paper the authors describe two different but equivalent constructions of \(G\)-equivariant coherent sheaves \(\mathcal{K}er^{j,X}(M)\) and \(\mathcal{I}m^{j,X}(M)\) on \(X\), which are called kernel and image sheaves correspondingly. The projective variety \(\mathbb{E}(r, \mathfrak{g})\) can be embedded into the Grassmannian \(\mathrm{Gr}(r,\mathfrak{g})\) of \(r\)-planes in \(\mathfrak{g}\). The Stiefel variety of \(r\)-points in \(\mathfrak{g}\) is a \(\mathrm{GL}_r\)-torsor over \(\mathrm{Gr}(r, \mathfrak{g})\). The pull-back to \(\mathbb{E}(r,\mathfrak{g})\) is basically a commuting variety \(\mathcal{C}_r(\mathcal{N}_p(\mathfrak{g}) )^\circ\) of \(r\) elements of \(\mathfrak{g}\) with trivial \(p\)-restriciton. One approach of the construction is using the descent of coherent sheaves on \(\mathcal{C}_r(\mathcal{N}_p(\mathfrak{g}) )^{\circ}\) to \(\mathbb{E}(r,\mathfrak{g})\). Another approach is simply by straightforward patching technique.
The authors showed that the associated coherent sheaves \(\mathcal{K}er^{j,X}(M)\) and \(\mathcal{I}m^{j,X}(M)\) are equivariant vector bundles on \(\mathbb{E}(r,\mathfrak{g})\) when \(M\) is a representation of constant radical or socle rank. When \(\mathfrak{g}\) is the Lie algebra of an algebraic group over an algebraically closed field of characteristic \(p\), and the representation \(M\) is induced from a rational representation of \(G\), the coherent sheaves \(\mathcal{K}er^{j,X}(M)\) and \(\mathcal{I}m^{j,X}(M)\) are just equivariant vector bundles on a \(G\)-orbit in \(\mathbb{E}(r,\mathfrak{g})\) that are known already, just some associated vector bundles of socle and radical modules of the stabilizer of the orbit.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L05 Formal groups, \(p\)-divisible groups
06B15 Representation theory of lattices
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Billey S. and Lakshmibai V., Singular loci of Schubert varieties, Progr. Math. 182, Birkhäuser, Boston 2000.; Billey, S.; Lakshmibai, V., Singular loci of Schubert varieties (2000) · Zbl 0959.14032
[2] Borel A., Linear algebraic groups, Grad. Texts in Math. 126, Springer, New York 1991.; Borel, A., Linear algebraic groups (1991) · Zbl 0726.20030
[3] Bourbaki N., Groupes et algebres de Lie, Chaps. 4, 5 et 6, Masson, Paris 1981.; Bourbaki, N., Groupes et algebres de Lie, Chaps. 4, 5 et 6 (1981) · Zbl 0483.22001
[4] Carlson J. F., Friedlander E. M. and Pevtsova J., Representations of elementary abelian p-groups and bundles on Grassmannians, Adv. Math. 229 (2012), 2985-3051.; Carlson, J. F.; Friedlander, E. M.; Pevtsova, J., Representations of elementary abelian p-groups and bundles on Grassmannians, Adv. Math., 229, 2985-3051 (2012) · Zbl 1260.20013
[5] Carlson J. F., Friedlander E. M. and Pevtsova J., Elementary subalgebras of Lie algebras, preprint 2014.; Carlson, J. F.; Friedlander, E. M.; Pevtsova, J., Elementary subalgebras of Lie algebras (2014) · Zbl 1392.17014
[6] Demazure M. and Gabriel P., Groupes algébriques, Tome I, North Holland 1970.; Demazure, M.; Gabriel, P., Groupes algébriques (1970) · Zbl 0203.23401
[7] Erdmann K. and Wildon M., Introduction to Lie algebras, Springer Undergrad. Math. Ser., Springer, Berlin 2006.; Erdmann, K.; Wildon, M., Introduction to Lie algebras (2006) · Zbl 1139.17001
[8] Friedlander E. M. and Parshall B., Cohomology of algebraic and related finite groups, Invent. Math. 74 (1983), 85-117.; Friedlander, E. M.; Parshall, B., Cohomology of algebraic and related finite groups, Invent. Math., 74, 85-117 (1983) · Zbl 0526.20035
[9] Friedlander E. M. and Pevtsova J., Generalized support varieties for finite group schemes, Doc. Math. Extra Vol. (2010), 197-222.; Friedlander, E. M.; Pevtsova, J., Generalized support varieties for finite group schemes, Doc. Math., Extra Vol., 197-222 (2010) · Zbl 1231.20043
[10] Friedlander E. M. and Pevtsova J., Constructions for infinitesimal group schemes, Trans. Amer. Math. Soc. 363 (2011), no. 11, 6007-6061.; Friedlander, E. M.; Pevtsova, J., Constructions for infinitesimal group schemes, Trans. Amer. Math. Soc., 363, 11, 6007-6061 (2011) · Zbl 1241.20051
[11] Hartshorne R., Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977.; Hartshorne, R., Algebraic geometry (1977) · Zbl 0367.14001
[12] Jantzen J. C., Representations of algebraic groups, 2nd ed., Math. Surveys Monogr. 107, American Mathematical Society, Providence 2003.; Jantzen, J. C., Representations of algebraic groups (2003) · Zbl 1034.20041
[13] Jantzen J. C., Nilpotent orbits in representation theory, Lie theory: Lie algebras and representations, Birkhäuser, Boston (2004), 1-211.; Jantzen, J. C., Nilpotent orbits in representation theory, Lie theory: Lie algebras and representations, 1-211 (2004) · Zbl 1169.14319
[14] McNinch G., Abelian reductive subgroups of reductive groups, J. Pure Appl. Algebra 167 (2002), 269-300.; McNinch, G., Abelian reductive subgroups of reductive groups, J. Pure Appl. Algebra, 167, 269-300 (2002) · Zbl 0999.20035
[15] Panin I., On the algebraic K-theory of twisted flag varieties, K-theory 8 (1994), 541-585.; Panin, I., On the algebraic K-theory of twisted flag varieties, K-theory, 8, 541-585 (1994) · Zbl 0854.19002
[16] Richardson R., Röhrle G. and Steinberg R., Parabolic subgroups with Abelian unipotent radical, Invent. Math. 110 (1992), 649-671.; Richardson, R.; Röhrle, G.; Steinberg, R., Parabolic subgroups with Abelian unipotent radical, Invent. Math., 110, 649-671 (1992) · Zbl 0786.20029
[17] Seligman G. B., On Lie algebras of prime characteristic, Mem. Amer. Math. Soc. 19 (1965), 1-83.; Seligman, G. B., On Lie algebras of prime characteristic, Mem. Amer. Math. Soc., 19, 1-83 (1965) · Zbl 0071.02703
[18] Springer T., Linear algebraic groups, 2nd ed., Progr. Math. 9, Birkhäuser, Boston 1998.; Springer, T., Linear algebraic groups (1998) · Zbl 0927.20024
[19] Suslin A., Friedlander E. and Bendel C., Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc. 10 (1997), 693-728.; Suslin, A.; Friedlander, E.; Bendel, C., Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc., 10, 693-728 (1997) · Zbl 0960.14023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.