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