Skip to Main Content

Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Mirković-Vilonen cycles and polytopes for a symmetric pair
HTML articles powered by AMS MathViewer

by Jiuzu Hong PDF
Represent. Theory 13 (2009), 19-32 Request permission


Let $G$ be a connected, simply-connected, and almost simple algebraic group, and let $\sigma$ be a Dynkin automorphism on $G$. Then $(G,G^\sigma )$ is a symmetric pair. In this paper, we get a bijection between the set of $\sigma$-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for $G^\sigma$, which is the fixed point subgroup of $G$; 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.
  • Jared E. Anderson, A polytope calculus for semisimple groups, Duke Math. J. 116 (2003), no. 3, 567–588. MR 1958098, DOI 10.1215/S0012-7094-03-11636-1
  • P. Deligne and J. Milne, Tannakian categories in “Hodge cycles and motives”, Springer, Lecture Notes, 900 (1982), 101-228.
  • V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, preprint arXiv:alg-geom/951107.
  • Jens Carsten Jantzen, Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonn. Math. Schr. 67 (1973), v+124. MR 401935
  • J. Kamnitzer, Mirković-Vilonen cycles and polytopes, preprint arXiv:math.AG/0501365.
  • S. Kumar, G. Lusztig and D. Prasad, Characters of simply-laced nonconnected groups versus characters of nonsimply-laced connected groups, preprint arXiv: math.RT/0701615.
  • 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, DOI 10.1007/978-1-4612-0261-5_{2}0
  • 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, DOI 10.4007/annals.2007.166.95
  • Satoshi Naito, Twining character formula of Borel-Weil-Bott type, J. Math. Sci. Univ. Tokyo 9 (2002), no. 4, 637–658. MR 1947485
  • S. Naito and D. Sagaki, Action of a diagram automorphism on Mirković-Vilonen polytopes, a talk given at a conference in Karuizawa, June, 2007.
  • Satoshi Naito and Daisuke Sagaki, A modification of the Anderson-Mirković conjecture for Mirković-Vilonen polytopes in types $B$ and $C$, J. Algebra 320 (2008), no. 1, 387–416. MR 2417995, DOI 10.1016/j.jalgebra.2008.02.009
  • Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
  • 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, DOI 10.1023/A:1014743615104
Similar Articles
  • Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 20G05, 14M15
  • Retrieve articles in all journals with MSC (2000): 20G05, 14M15
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
  • MR Author ID: 862719
  • Email:
  • Received by editor(s): May 13, 2008
  • Received by editor(s) in revised form: November 15, 2008
  • Published electronically: February 13, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 13 (2009), 19-32
  • MSC (2000): Primary 20G05; Secondary 14M15
  • DOI:
  • MathSciNet review: 2480386