Algebraic geometry
Parabolic subgroups and automorphism groups of Schubert varieties
Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 542-549.

Let G be a simple algebraic group of adjoint type over the field C of complex numbers, B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and X(w) be the Schubert variety in G/B corresponding to w. In this article we show that given any parabolic subgroup P of G containing B properly, there is an element wW such that P is the connected component, containing the identity element of the group of all algebraic automorphisms of X(w).

Soit G un groupe algébrique du type adjoint sur le corps des nombres complexes C et B un sous-groupe de Borel de G contenant un tore maximal T. Soit w un élément du groupe de Weil W et X(w) la variété de Schubert dans G/B correspondant à w. Dans cet article, nous montrons que, pour tout sous-groupe parabolique P de G contenant B, il existe un élément w dans W tel que P est la composante connexe contenant l'élément identité du groupe des automorphismes algébriques de X(w).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.001
Kannan, Subramaniam Senthamarai 1; Saha, Pinakinath 1

1 Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Siruseri, Kelambakkam, 603103, India
@article{CRMATH_2018__356_5_542_0,
     author = {Kannan, Subramaniam Senthamarai and Saha, Pinakinath},
     title = {Parabolic subgroups and automorphism groups of {Schubert} varieties},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {542--549},
     publisher = {Elsevier},
     volume = {356},
     number = {5},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.001},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.04.001/}
}
TY  - JOUR
AU  - Kannan, Subramaniam Senthamarai
AU  - Saha, Pinakinath
TI  - Parabolic subgroups and automorphism groups of Schubert varieties
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 542
EP  - 549
VL  - 356
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.04.001/
DO  - 10.1016/j.crma.2018.04.001
LA  - en
ID  - CRMATH_2018__356_5_542_0
ER  - 
%0 Journal Article
%A Kannan, Subramaniam Senthamarai
%A Saha, Pinakinath
%T Parabolic subgroups and automorphism groups of Schubert varieties
%J Comptes Rendus. Mathématique
%D 2018
%P 542-549
%V 356
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.04.001/
%R 10.1016/j.crma.2018.04.001
%G en
%F CRMATH_2018__356_5_542_0
Kannan, Subramaniam Senthamarai; Saha, Pinakinath. Parabolic subgroups and automorphism groups of Schubert varieties. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 542-549. doi : 10.1016/j.crma.2018.04.001. http://www.numdam.org/articles/10.1016/j.crma.2018.04.001/

[1] Akhiezer, D.N. Lie Group Actions in Complex Analysis, Aspects of Mathematics, vol. E 27, Vieweg, Braunschweig/Wiesbaden, Germany, 1995

[2] Balaji, V.; Kannan, S.S.; Subrahmanyam, K.V. Cohomology of line bundles on Schubert varieties—I, Transform. Groups, Volume 9 (2004) no. 2, pp. 105-131

[3] Bourbaki, N. Lie Groups and Lie Algebras, Chapters 4–6, Springer-Verlag, Berlin, Heidelberg, New York, 2002

[4] Brion, M. On automorphism groups of fiber bundles, Publ. Mat. Urug., Volume 12 (2011), pp. 39-66

[5] Brion, M.; Kumar, S. Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics, vol. 231, Birkhäuser, Boston, Inc., Boston, MA, USA, 2005

[6] Demazure, M. A very simple proof of Bott's theorem, Invent. Math., Volume 33 (1976), pp. 271-272

[7] Humphreys, J.E. Introduction to Lie Algebras and Representation Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1972

[8] Humphreys, J.E. Linear Algebraic Groups, Springer-Verlag, Berlin, Heidelberg, New York, 1975

[9] Jantzen, J.C. Representations of Algebraic Groups, Mathematical Surveys and Monographs, vol. 107, 2003

[10] Kannan, S.S. On the automorphism group of a smooth Schubert variety, Algebr. Represent. Theory, Volume 19 (2016) no. 4, pp. 761-782

[11] Lakshmibai, V.; Sandhya, B. Criterion for smoothness of Schubert varieties in Sl(n)/B, Proc. Indian Acad. Sci. Math. Sci., Volume 100 (1990) no. 1, pp. 45-52

[12] Matsumura, H.; Oort, F. Representability of group functors, and automorphisms of algebraic schemes, Invent. Math., Volume 4 (1967), pp. 1-25

Cited by Sources: