Dynamical Weyl group

Plan:

$A_{s_{1}s_{2}}(\lambda )=A_{s_{1}}(s_{2}\cdot \lambda )A_{s_{2}}(\lambda )$

• ${\mathfrak {g}}$ -----simple Lie algebra
• $\lambda$ -----------weight
  • c -------dominant weight
  • $\lambda$-----has singular vector
  • $\mu :(\lambda ,\mu )$ and $(\lambda +c,\mu +c)$---- generic
• $M_{\lambda }$--------Verma module
• $V$ -------------finite-dimensional module
  • $\lambda$-weight,
  • $V[\lambda ]$ -weight subspace
  • $v\in V[\lambda -\mu ]$
• $\pi$-------------dominant weight
• $V_{\pi }$----------finite-dimensional module
• $Hom_{\mathfrak {g}}(M_{\lambda }\rightarrow M_{\mu }\otimes V)$----space of g-invariant homomorphismata
• $Hom_{\mathfrak {g}}(M_{\lambda +c}\rightarrow M_{\mu +c}\otimes V)$----space of g-invariant homomorphisma
• $Hom_{\mathfrak {g}}(M_{\lambda }\rightarrow M_{\mu }\otimes V)\rightarrow Hom_{\mathfrak {g}}(M_{\lambda +c}\rightarrow M_{\mu +c}\otimes V)$

• For $\lambda ,\mu$ generic, $Hom_{\mathfrak {g}}(M_{\lambda }\rightarrow M_{\mu }\otimes V)\cong V[\lambda -\mu ]$
• For $\lambda +c,\mu +c$ generic, $Hom_{\mathfrak {g}}(M_{\lambda +c}\rightarrow M_{\mu +c}\otimes V)\cong V[\lambda -\mu ]$
• For $\lambda$ sufficiently large, $s_{c}$ is an isomorphism.
• $w=s_{1}s_{2}$
• $A_{s_{1}}(\lambda )$ -----------dynamical Weyl operator
• $A_{s_{i}}$----commutes with $s_{c}$

$A_{s_{1}s_{2}}(\lambda )=A_{s_{1}}(s_{2}\cdot \lambda )A_{s_{2}}(\lambda )$