BGG resolution

==example of BGG resolution : sl_2

• \(W_{\lambda}\) : irreducible highest weight module
• \(V_{\lambda}\) : Verma modules
  
  • note that the Verma modules are free modules of rank 1 over \(\mathbb{C}[F]\)
• \(\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots\)
• \(W_{\lambda}=V_{\lambda}/V_{-\lambda-2}\)
• BGG resolution
  \(0\to V_{-\lambda-2}\to V_{\lambda}\to W\to 0\)
  
• 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 \(w(\lambda+\rho)=-\lambda-\rho\)
  

 

 

==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 in SL2
  
  • \(\lambda = \mu -2n\), \(n=0,1,2,\cdots\)
  • \((\lambda+1)^2 = (\mu+1)^2\)

 

 

==books

• James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008.