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

• Verma modules
• BGG reciprocity
• BGG category
• Bott-Borel-Weil Theorem

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.