BGG resolution

introduction

• Weyl character formula. For $\lambda\in \Lambda^{+}$,

$$ \operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda) $$

• realize this formula as an Euler characteristic
• The BGG resolution (cf. [5, 31]) resolves a finite-dimensional (simple) g-module $L(\lambda)$ by direct sums of Verma modules indexed by weights "of the same length" in the twisted Weyl orbit $W\cdot \lambda$
• This is used to compute the cohomologies of $\mathfrak{n}^+$.

thm (Bernstein-Gelfand-Gelfand Resolution).

Fix $\lambda\in \Lambda^{+}$. 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.

overview

• property of character map on short exact sequences
• Euler-Poincare mapping
• Weyl character formula
• principal block : filtering through central characters
  • is a block a $U(\mathfrak{g})$-submodule?
  • how to check that it preserves the exactness : any homomorphism between modules belonging to different blocks will be zero
• combinatorial results
  • consider the set of sum of k distinct roots. Which elements are linked to $0$?
  • let $\beta$ be a sum of $k$-distinct negative root. Can we find $w\in W$ such that $w\cdot 0=\beta$?
  • yes, if and only if $\beta$ is a sum of $w \Pi^+ \cap \Pi^-$

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 \(w(\lambda+\rho)=-\lambda-\rho\)

$\mathfrak{sl}_3$

• http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf

Euler-Poincare characteristic

• $A$ : abelian category
• Euler-Poincare map $\varphi$ on an object in $A$
• $\varphi$ turns a short exact sequence into an alternating sum
• we can define Euler-Poincare characteristic $\chi_{\varphi}$ of a complex as the alternating sum of Euler-Poincare map on the homology
• the main result is that the Euler-Poincre characteristic can be computed from a different resolution and it is independent of the choice of it

Verma modules

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 $w\in W$.

action of center on Verma modules

• check

maximal submodule of Verma modules

• Maximal Submodule of $M(\lambda), \lambda \in \Lambda+$ (see 2.6)

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.

thm (Weak BGG resolution)

There is an exact sequence $$0 = D_m^{\lambda} \to \subset D_{m-1}^{\lambda} \to \cdots \to D_2^{\lambda} \to D_1^{\lambda} \to L(0) \to 0$$ where $D_{k}^{\lambda}$ has a standard filtration involving exactly once each of the Verma modules $M(w\cdot \lambda)$ with $\ell(w)=k$

• we prove this for $\lambda=0$ and apply the translation function to extend it

strategy of the proof

• The sequence of modules $D_k:=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\Lambda^{k}(\mathfrak{g}/\mathfrak{b})$ is a relative version of the standard resolution of the trivial module in Lie algebra cohomology

standard resolution of trivial module

• free $U(\mathfrak{g})$-modules $U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{k}(\mathfrak{g})$
• standard resolution of trivial module

$$\cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{p}(\mathfrak{g})\to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{p-1}(\mathfrak{g})\to \cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{0}(\mathfrak{g})\to L(0)$$

• $D_k$ are free only over $U(\mathfrak{n}^{−})$

extensions of Verma modules

• $\mu, \lambda\in \mathfrak{h}^{*}$
• $\mu \uparrow \lambda$ if $\mu = \lambda$ or there is a root $\alpha$ such that $\mu=s_{\alpha}\cdot \lambda < \lambda $

thm

Let $\lambda\in \mathfrak{h}^{*}$.

(a) If $\operatorname{Ext}_{\mathcal{O}}(M(\mu),M(\lambda))\neq 0$ for $\mu\in \mathfrak{h}^{*}$, then $\mu \uparrow \lambda$ but $\mu \neq \lambda$

(b) Let $\lambda\in \Lambda^{+}$ and $w,w'\in W$. If $\operatorname{Ext}_{\mathcal{O}}(M(\mu),M(\lambda))\neq 0$, then $w<w'$ in the Bruhat ordering. In particular, $\ell(w)<\ell(w')$.

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]).

related items

• 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.
• Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html
• [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.