Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Standard parabolic subsets of highest weight modules

Author: Apoorva Khare
Journal: Trans. Amer. Math. Soc. 369 (2017), 2363-2394
MSC (2010): Primary 17B10; Secondary 17B20, 52B15, 52B20
Published electronically: June 20, 2016
Erratum: Trans. Amer. Math. Soc. 369 (2017), 3015-3015.
MathSciNet review: 3592514
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study certain fundamental and distinguished subsets of weights of an arbitrary highest weight module over a complex semisimple Lie algebra. These sets $ {\rm wt}_J \mathbb{V}^\lambda $ are defined for each highest weight module $ \mathbb{V}^\lambda $ and each subset $ J$ of simple roots; we term them "standard parabolic subsets of weights''. It is shown that for any highest weight module, the sets of simple roots whose corresponding standard parabolic subsets of weights are equal form intervals in the poset of subsets of the set of simple roots under containment. Moreover, we provide closed-form expressions for the maximum and minimum elements of the aforementioned intervals for all highest weight modules $ \mathbb{V}^\lambda $ over semisimple Lie algebras $ \mathfrak{g}$. Surprisingly, these formulas only require the Dynkin diagram of $ \mathfrak{g}$ and the integrability data of $ \mathbb{V}^\lambda $. As a consequence, we extend classical work by Satake, Borel-Tits, Vinberg, and Casselman, as well as recent variants by Cellini-Marietti to all highest weight modules.

We further compute the dimension, stabilizer, and vertex set of standard parabolic faces of highest weight modules and show that they are completely determined by the aforementioned closed-form expressions. We also compute the $ f$-polynomial and a minimal half-space representation of the convex hull of the set of weights. These results were recently shown for the adjoint representation of a simple Lie algebra, but analogues remain unknown for any other finite- or infinite-dimensional highest weight module. Our analysis is uniform and type-free, across all semisimple Lie algebras and for arbitrary highest weight modules.

References [Enhancements On Off] (What's this?)

  • [ABH] Federico Ardila, Matthias Beck, Serkan Hoşten, Julian Pfeifle, and Kim Seashore, Root polytopes and growth series of root lattices, SIAM J. Discrete Math. 25 (2011), no. 1, 360-378. MR 2801233 (2012e:52043),
  • [Bou] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4-6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629 (2003a:17001)
  • [CM] Paola Cellini and Mario Marietti, Root polytopes and Borel subalgebras, Int. Math. Res. Not. IMRN 12 (2015), 4392-4420. MR 3356759,
  • [CP] Paola Cellini and Paolo Papi, Abelian ideals of Borel subalgebras and affine Weyl groups, Adv. Math. 187 (2004), no. 2, 320-361. MR 2078340 (2005e:17012),
  • [CDR] Vyjayanthi Chari, R. J. Dolbin, and Tim Ridenour, Ideals in parabolic subalgebras of simple Lie algebras, Symmetry in mathematics and physics, Contemp. Math., vol. 490, Amer. Math. Soc., Providence, RI, 2009, pp. 47-60. MR 2555969 (2010i:17017),
  • [CG] Vyjayanthi Chari and Jacob Greenstein, A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras, Adv. Math. 220 (2009), no. 4, 1193-1221. MR 2483719 (2010h:16067),
  • [CKR] Vyjayanthi Chari, Apoorva Khare, and Tim Ridenour, Faces of polytopes and Koszul algebras, J. Pure Appl. Algebra 216 (2012), no. 7, 1611-1625. MR 2899824,
  • [Dix] Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR 1393197 (97c:17010)
  • [Fe] Suren L. Fernando, Lie algebra modules with finite-dimensional weight spaces. I, Trans. Amer. Math. Soc. 322 (1990), no. 2, 757-781. MR 1013330 (91c:17006),
  • [HK] Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971 (2002m:17012)
  • [Hu] James E. Humphreys, Representations of semisimple Lie algebras in the BGG category $ \mathcal {O}$, Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008. MR 2428237 (2009f:17013)
  • [Ka1] Joel Kamnitzer, Mirković-Vilonen cycles and polytopes, Ann. of Math. (2) 171 (2010), no. 1, 245-294. MR 2630039 (2011g:20070),
  • [Ka2] Joel Kamnitzer, Categorification of Lie algebras, Séminaire Bourbaki 1072 (2013), 1-22.
  • [Kh] Apoorva Khare, Faces and maximizer subsets of highest weight modules, J. Algebra 455 (2016), 32-76. MR 3478853,
  • [KR] Apoorva Khare and Tim Ridenour, Faces of weight polytopes and a generalization of a theorem of Vinberg, Algebr. Represent. Theory 15 (2012), no. 3, 593-611. MR 2912474,
  • [LCL] Zhuo Li, You'an Cao, and Zhenheng Li, Orbit structures of weight polytopes, preprint, math.RT/1411.6140.
  • [Me] Karola Mészáros, Root polytopes, triangulations, and the subdivision algebra, II, Trans. Amer. Math. Soc. 363 (2011), no. 11, 6111-6141. MR 2817421 (2012g:52021),
  • [PR] Mohan S. Putcha and Lex E. Renner, The system of idempotents and the lattice of $ {\mathcal {J}}$-classes of reductive algebraic monoids, J. Algebra 116 (1988), no. 2, 385-399. MR 953159 (89k:20098),
  • [Vi] Ernest B. Vinberg, Some commutative subalgebras of a universal enveloping algebra, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 3-25, 221 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 1-22. MR 1044045 (91b:17015)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 17B10, 17B20, 52B15, 52B20

Retrieve articles in all journals with MSC (2010): 17B10, 17B20, 52B15, 52B20

Additional Information

Apoorva Khare
Affiliation: Departments of Mathematics and Statistics, Stanford University, Stanford, California 94305

Keywords: Highest weight module, parabolic Verma module, Weyl polytope, standard parabolic subset, face map, inclusion relations, $f$-polynomial
Received by editor(s): September 30, 2014
Received by editor(s) in revised form: March 10, 2015, March 11, 2015, and April 1, 2015
Published electronically: June 20, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society