Let P be a parabolic subgroup in G = SLn(k), for k an algebraically closed field. We show that there is a G-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural compactification of the cotangent bundle T*G/P. Restricting this identification to the conormal variety N*X(w) of a Schubert divisor X(w) in G/P, we show that there is a compactification of N*X(w) as an affine Schubert variety. It follows that N*X(w) is normal, Cohen–Macaulay, and Frobenius split.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
P. Achar, A. Henderson, Geometric Satake, Springer correspondence, and small representations, Selecta Math. 19 (2013), no. 4, 949–986.
B. Fu, Symplectic resolutions for nilpotent orbits, Invent. Math. 151 (2003), no. 1, 167–186.
G. Faltings, Algebraic loop groups and moduli spaces of bundles J. Eur. Math. Soc. 5 (2003), no. 1, 41–68.
W. Hesselink, Polarizations in the classical groups, Math. Z. 160 (1978), no. 3, 217–234.
V. Kac, Infinite Dimensional Lie Algebras, 3rd Ed., Cambridge University Press, Cambridge, 1990.
S. Kumar, Kac–Moody groups, their Flag Varieties and Representation Theory, Progress in Mathematics, Vol. 204, Birkhäuser Boston, Boston, MA, 2002.
V. Lakshmibai, Cotangent bundle to the Grassmann variety, Transform. Groups 21 (2016), no. 2, 519–530.
V. Lakshmibai, C. S. Seshadri, Geomtery of G/P–II, Proc. Ind. Acad. Sci. 87 (1978), no. 2, 1–54.
V. Lakshmibai, C. S. Seshadri, R. Singh, Cotangent bundle to the flag variety–I, Transform. Groups 24 (2019), no. 1, 127–147.
V. Lakshmibai, R. Singh, Conormal varieties on the cominuscule Grassmannian, arXiv:1712.06737v2 (2018).
G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169–178.
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498.
V. Mehta, A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. 122 (1985), no. 1, 27–40.
I. Mirković, M. Vybornov, Quiver varieties and Beilinson–Drinfeld Grassmannians of type A, arXiv:0712.4160v2 (2008).
B. Remy, Groupes de Kac–Moody Déployés et Presque Déployés, Astérisque 277 (2002).
R. Steinberg, Lecture on Chevalley Groups, University Lecture Series, Vol. 66, American Mathematical Society, RI, 2016.
E. Strickland, On the conormal bundle of the determinantal variety, J. Algebra 75 (1982), no. 2, 523–537.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
LAKSHMIBAI, V., SINGH, R. COTANGENT BUNDLES OF PARTIAL FLAG VARIETIES AND CONORMAL VARIETIES OF THEIR SCHUBERT DIVISORS. Transformation Groups 25, 127–148 (2020). https://doi.org/10.1007/s00031-019-09523-w