MirkovićVilonen cycles and polytopes for a symmetric pair
Author:
Jiuzu Hong
Journal:
Represent. Theory 13 (2009), 1932
MSC (2000):
Primary 20G05; Secondary 14M15
Published electronically:
February 13, 2009
MathSciNet review:
2480386
Fulltext PDF Free Access
Abstract 
References 
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Abstract: Let be a connected, simplyconnected, and almost simple algebraic group, and let be a Dynkin automorphism on . Then is a symmetric pair. In this paper, we get a bijection between the set of invariant MV cycles (polytopes) for and the set of MV cycles (polytopes) for , which is the fixed point subgroup of ; moreover, this bijection can be restricted to the set of MV cycles (polytopes) in irreducible representations. As an application, we obtain a new proof of the twining character formula.
 [A]
Jared
E. Anderson, A polytope calculus for semisimple groups, Duke
Math. J. 116 (2003), no. 3, 567–588. MR 1958098
(2004a:20047), http://dx.doi.org/10.1215/S0012709403116361
 [DM]
P. Deligne and J. Milne, Tannakian categories in ``Hodge cycles and motives'', Springer, Lecture Notes, 900 (1982), 101228.
 [G]
V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, preprint arXiv:alggeom/951107.
 [J]
Jens
Carsten Jantzen, Darstellungen halbeinfacher algebraischer Gruppen
und zugeordnete kontravariante Formen, Bonn. Math. Schr.
67 (1973), v+124. MR 0401935
(53 #5761)
 [K1]
J. Kamnitzer, MirkovićVilonen cycles and polytopes, preprint arXiv:math.AG/0501365.
 [KLP]
S. Kumar, G. Lusztig and D. Prasad, Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups, preprint arXiv: math.RT/0701615.
 [L1]
G.
Lusztig, Total positivity in reductive groups, Lie theory and
geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA,
1994, pp. 531–568. MR 1327548
(96m:20071)
 [MV]
I.
Mirković and K.
Vilonen, Geometric Langlands duality and representations of
algebraic groups over commutative rings, Ann. of Math. (2)
166 (2007), no. 1, 95–143. MR 2342692
(2008m:22027), http://dx.doi.org/10.4007/annals.2007.166.95
 [N]
Satoshi
Naito, Twining character formula of BorelWeilBott type, J.
Math. Sci. Univ. Tokyo 9 (2002), no. 4,
637–658. MR 1947485
(2003m:20061)
 [NS1]
S. Naito and D. Sagaki, Action of a diagram automorphism on MirkovićVilonen polytopes, a talk given at a conference in Karuizawa, June, 2007.
 [NS2]
Satoshi
Naito and Daisuke
Sagaki, A modification of the AndersonMirković conjecture
for MirkovićVilonen polytopes in types 𝐵 and
𝐶, J. Algebra 320 (2008), no. 1,
387–416. MR 2417995
(2009e:20106), http://dx.doi.org/10.1016/j.jalgebra.2008.02.009
 [ST]
Robert
Steinberg, Endomorphisms of linear algebraic groups, Memoirs
of the American Mathematical Society, No. 80, American Mathematical
Society, Providence, R.I., 1968. MR 0230728
(37 #6288)
 [V]
E.
Vasserot, On the action of the dual group on the cohomology of
perverse sheaves on the affine Grassmannian, Compositio Math.
131 (2002), no. 1, 51–60. MR 1895920
(2003b:20062), http://dx.doi.org/10.1023/A:1014743615104
 [A]
 J. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003), 567588. MR 1958098 (2004a:20047)
 [DM]
 P. Deligne and J. Milne, Tannakian categories in ``Hodge cycles and motives'', Springer, Lecture Notes, 900 (1982), 101228.
 [G]
 V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, preprint arXiv:alggeom/951107.
 [J]
 J. C. Jantzen, Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonner Math. Schriften 67 (1973). MR 0401935 (53:5761)
 [K1]
 J. Kamnitzer, MirkovićVilonen cycles and polytopes, preprint arXiv:math.AG/0501365.
 [KLP]
 S. Kumar, G. Lusztig and D. Prasad, Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups, preprint arXiv: math.RT/0701615.
 [L1]
 G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progress in Math. 123, Birkhäuser Boston, (1994) MR 1327548 (96m:20071)
 [MV]
 I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. 166 (2007), no. 1, 95143. MR 2342692 (2008m:22027)
 [N]
 S. Naito, Twining character formula of BorelWeilBott Type, J. Math. Sci. Univ. Tokyo, 9 (2002), 637658. MR 1947485 (2003m:20061)
 [NS1]
 S. Naito and D. Sagaki, Action of a diagram automorphism on MirkovićVilonen polytopes, a talk given at a conference in Karuizawa, June, 2007.
 [NS2]
 S. Naito and D. Sagaki, A modification of the AndersonMirković conjecture for MirkovićVilonen polytopes in types and , J. Algebra 320 (2008), no. 1, 387416. MR 2417995
 [ST]
 R. Steinberg, Endomorphism of linear algebraic groups. Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, RI. MR 0230728 (37:6288)
 [V]
 E. Vasserot, On the action of the dual group on the cohomology of perverse sheaves on the affine Grassmannian, Compositio Math. 131 (2002), no. 1, 5160. MR 1895920 (2003b:20062)
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Additional Information
Jiuzu Hong
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
Address at time of publication:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Email:
hjzzjh@gmail.com
DOI:
http://dx.doi.org/10.1090/S1088416509003410
PII:
S 10884165(09)003410
Received by editor(s):
May 13, 2008
Received by editor(s) in revised form:
November 15, 2008
Published electronically:
February 13, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
