BGG resolution

introduction

The BGG resolution (cf. [5, 31]) resolves a finite-dimensional (simple) g-module (V(λ)) by direct sums of Verma modules indexed by weights “of the same length”in the twisted Weyl orbit (W•λ).
This is used to compute the cohomologies of n+.

thm (Bernstein-Gelfand-Gelfand Resolution).

There is an exact sequence of Verma modules $$ 0 \to V_{w_0\cdot \lambda}\to \cdots \bigoplus_{w\in W, \ell(w)=k}V_{w\cdot \lambda}\to \cdots V_{\lambda}\to W_{\lambda}\to 0 $$ where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element of maximal length. Here $\rho$ is half the sum of the positive roots.

example of BGG resolution : $\mathfrak{sl}_2$

$W_{\lambda}$ : irreducible highest weight module
$V_{\lambda}$ : Verma modules
- note that the Verma modules are free modules of rank 1 over $\mathbb{C}[F]$ where $F$ is the annihilation operator of $\mathfrak{sl}_2$
$\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots$
$W_{\lambda}=V_{\lambda}/V_{-\lambda-2}$
BGG resolution

\[0\to V_{-\lambda-2}\to V_{\lambda}\to W_{\lambda}\to 0\]

number of modules = 2 (=order of Weyl group in general)
character of W = alternating sum of characters of Verma modules

\[\chi_{W_{\lambda}}=\chi_{V_{\lambda}}-\chi_{V_{-\lambda-2}}=\frac{q^{\lambda}}{1-q^{-2}}-\frac{q^{-\lambda-2}}{1-q^{-2}}\]

comparison with Weyl-Kac character formula

\[ch(W_{\lambda})=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho})}{e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}=\frac{q^{\lambda+1}-q^{-\lambda-1}}{q^{1}(1-q^{-2})}\] where I used $\rho=1,\alpha=2$ and $w(\lambda+\rho)=-\lambda-\rho$

generalization

There exist generalizations to symmetrizable Kac-Moody algebras, cf. [34].
Kempf obtained a resolution of finite-dimensional V(λ) in terms of the Grothendieck-Cousin complex in [26], which is dual to the BGG resolution.
This was extended by Kumar to arbitrary Kac-Moody algebras; he thus obtained the BGG resolution here, and computed the Weyl-Kac character formula and the cohomologies of n+ (cf. [30,§9.3]).

related items

books

[30] Shrawan Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Birkhauser, Progress in Math. 204, Boston, 2002
James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008.

expositions

http://rvirk.com/notes/student/catObasics.pdf
BGG resolution http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf
Wang, Jing Ping. “Representations of sl(2,C) in the BGG Category O and Master Symmetries.” arXiv:1408.3437 [nlin], August 14, 2014. http://arxiv.org/abs/1408.3437.
http://stanford.edu/~khare/EoM-BGG-O.pdf

articles

Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748.
[34] A. Rocha-Caridi, Splitting Criteria for $\mathfrak{g}$-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible $\mathfrak{g}$-module, Trans. Amer. Math. Soc.262 (1980), no. 2, 335–366
[26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396
[31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511.
J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92
[5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of $\mathfrak{g}$-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996.
Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9.