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 $$ 0 \to M_{w_0\cdot \lambda}\to \cdots \bigoplus_{w\in W, \ell(w)=k}M_{w\cdot \lambda}\to \cdots M_{\lambda}\to L_{\lambda}\to 0 $$ 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
$\mathfrak{sl}_2$
- \(L_{\lambda}\) : irreducible highest weight module
- \(M_{\lambda}\) : Verma modules
- note that the Verma modules are free modules of rank 1 over \(\mathbb{C}[F]\) where $F$ is the annihilation operator of $\mathfrak{sl}_2$
- \(\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots\)
- \(L_{\lambda}=M_{\lambda}/M_{-\lambda-2}\)
- BGG resolution
\[0\to M_{-\lambda-2}\to M_{\lambda}\to L_{\lambda}\to 0\]
- number of modules = 2 (=order of Weyl group in general)
- character of $L_{\lambda}$ = alternating sum of characters of Verma modules
\[\chi_{L_{\lambda}}=\chi_{M_{\lambda}}-\chi_{M_{-\lambda-2}}=\frac{q^{\lambda}}{1-q^{-2}}-\frac{q^{-\lambda-2}}{1-q^{-2}}\]
- comparison with Weyl-Kac character formula
\[ch(L_{\lambda})=\frac{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho})}{e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}=\frac{q^{\lambda+1}-q^{-\lambda-1}}{q^{1}(1-q^{-2})}\] where I used \(\rho=1,\alpha=2\) and \(w(\lambda+\rho)=-\lambda-\rho\)
$\mathfrak{sl}_3$
maps between Verma modules
- 2 conditions to have non-zero homomorphisms \(V_{\lambda}\to V_{\mu}\) between two Verma modules
- \(\lambda+\rho, \mu+\rho\) are in the same orbit of Weyl group
- \(V_{\lambda}\leq V_{\mu}\), i.e. \(\lambda = \mu -\sum \alpha\), where the sum is over some positive roots.
example
- SL2
- \(\lambda = \mu -2n\), \(n=0,1,2,\cdots\)
- \((\lambda+1)^2 = (\mu+1)^2\)
composition series of Verma modules
- thm
The Verma module $M(\lambda)$ has a finite composition series $$ M(\lambda)=N_0\supset N_1\supset N_2\supset \cdots N_{r}=O $$ where each $N_i$ is a submodule of $M(\lambda)$ and $N_{i+1}$ is a maximal submodule of $N_i$. Moreover, $N_i/N_{i+1}$ is isomorphic to $L(w\cdot \lambda)$ for some $w\in W$.
weak BGG resolution
standard filtration
- We say that $M \in O$ has a standard filtration (also sometimes called a Verma flag) if there is a sequence of submodules
$0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_n = M$ for which each $M^i := M_i/M_{i−1}\, (1 \le i \le n)$ is isomorphic to a Verma module.
standard resolution of trivial module
- The sequence of modules $D_k$ is a relative version of the standard resolution of the trivial module in Lie algebra cohomology
- free $U(\mathfrak{g})$-modules $U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{k}(\mathfrak{g})$
- standard resolution of trivial module
$$\cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{p}(\mathfrak{g})\to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{p-1}(\mathfrak{g})\to \cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{0}(\mathfrak{g})\to \mathbb{C}$$
- $D_k$ are free only over $U(\mathfrak{n}^{−})$
generalization
- There exist generalizations to symmetrizable Kac-Moody algebras, cf. [34].
- Kempf obtained a resolution of finite-dimensional L(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]).
- Verma modules
- BGG reciprocity
- BGG category
- Kostant theorem on Lie algebra cohomology of nilpotent radical
- 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.