## Mirković-Vilonen cycles and polytopes for a symmetric pair

HTML articles powered by AMS MathViewer

- by Jiuzu Hong
- Represent. Theory
**13**(2009), 19-32 - DOI: https://doi.org/10.1090/S1088-4165-09-00341-0
- Published electronically: February 13, 2009
- PDF | Request permission

## Abstract:

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.## References

- 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

## Bibliographic 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: hjzzjh@gmail.com
- 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: https://doi.org/10.1090/S1088-4165-09-00341-0
- MathSciNet review: 2480386