## Abelian ideals and cohomology of symplectic type

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**137**(2009), 479-485 Request permission

## Abstract:

Let $\mathfrak {b}$ be a Borel subalgebra of the symplectic Lie algebra $\mathfrak {sp}(2n,\mathbb {C})$ and let $\mathfrak {n}$ be the corresponding maximal nilpotent subalgebra. We find a connection between the abelian ideals of $\mathfrak {b}$ and the cohomology of $\mathfrak {n}$ with trivial coefficients. Using this connection, we are able to enumerate the number of abelian ideals of $\mathfrak {b}$ with given dimension via the Poincaré polynomials of Weyl groups of types $A_{n-1}$ and $C_n$.## References

- Raoul Bott,
*Homogeneous vector bundles*, Ann. of Math. (2)**66**(1957), 203–248. MR**89473**, DOI 10.2307/1969996 - Paola Cellini and Paolo Papi,
*Abelian ideals of Borel subalgebras and affine Weyl groups*, Adv. Math.**187**(2004), no. 2, 320–361. MR**2078340**, DOI 10.1016/j.aim.2003.08.011 - James E. Humphreys,
*Reflection groups and Coxeter groups*, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR**1066460**, DOI 10.1017/CBO9780511623646 - Bertram Kostant,
*Lie algebra cohomology and the generalized Borel-Weil theorem*, Ann. of Math. (2)**74**(1961), 329–387. MR**142696**, DOI 10.2307/1970237 - Bertram Kostant,
*The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations*, Internat. Math. Res. Notices**5**(1998), 225–252. MR**1616913**, DOI 10.1155/S107379289800018X - Bertram Kostant,
*Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra*, Invent. Math.**158**(2004), no. 1, 181–226. MR**2090363**, DOI 10.1007/s00222-004-0370-7 - Dmitri I. Panyushev,
*Abelian ideals of a Borel subalgebra and long positive roots*, Int. Math. Res. Not.**35**(2003), 1889–1913. MR**1995141**, DOI 10.1155/S1073792803211053 - Dmitri Panyushev and Gerhard Röhrle,
*Spherical orbits and abelian ideals*, Adv. Math.**159**(2001), no. 2, 229–246. MR**1825058**, DOI 10.1006/aima.2000.1959 - I. Schur,
*Zur Theorie der vertauschbaren Matrizen*, J. Reine Angew. Math.**130**(1905), 66–76. - Ruedi Suter,
*Abelian ideals in a Borel subalgebra of a complex simple Lie algebra*, Invent. Math.**156**(2004), no. 1, 175–221. MR**2047661**, DOI 10.1007/s00222-003-0337-0

## Additional Information

**Li Luo**- Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
- Email: luoli@amss.ac.cn
- Received by editor(s): January 24, 2008
- Published electronically: September 29, 2008
- Communicated by: Gail R. Letzter
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 479-485 - MSC (2000): Primary 17B05, 17B56; Secondary 17B20, 17B30
- DOI: https://doi.org/10.1090/S0002-9939-08-09685-8
- MathSciNet review: 2448567