# Dynamical Weyl group

Plan:

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

• ${\displaystyle {\mathfrak {g}}}$ -----simple Lie algebra
• ${\displaystyle \lambda }$ -----------weight
• c -------dominant weight
• ${\displaystyle \lambda }$-----has singular vector
• ${\displaystyle \mu :(\lambda ,\mu )}$ and ${\displaystyle (\lambda +c,\mu +c)}$---- generic
• ${\displaystyle M_{\lambda }}$--------Verma module
• ${\displaystyle V}$ -------------finite-dimensional module
• ${\displaystyle \lambda }$-weight,
• ${\displaystyle V[\lambda ]}$ -weight subspace
• ${\displaystyle v\in V[\lambda -\mu ]}$
• ${\displaystyle \pi }$-------------dominant weight
• ${\displaystyle V_{\pi }}$----------finite-dimensional module
• ${\displaystyle Hom_{\mathfrak {g}}(M_{\lambda }\rightarrow M_{\mu }\otimes V)}$----space of g-invariant homomorphismata
• ${\displaystyle Hom_{\mathfrak {g}}(M_{\lambda +c}\rightarrow M_{\mu +c}\otimes V)}$----space of g-invariant homomorphisma
• ${\displaystyle Hom_{\mathfrak {g}}(M_{\lambda }\rightarrow M_{\mu }\otimes V)\rightarrow Hom_{\mathfrak {g}}(M_{\lambda +c}\rightarrow M_{\mu +c}\otimes V)}$
• For ${\displaystyle \lambda ,\mu }$ generic, ${\displaystyle Hom_{\mathfrak {g}}(M_{\lambda }\rightarrow M_{\mu }\otimes V)\cong V[\lambda -\mu ]}$
• For ${\displaystyle \lambda +c,\mu +c}$ generic, ${\displaystyle Hom_{\mathfrak {g}}(M_{\lambda +c}\rightarrow M_{\mu +c}\otimes V)\cong V[\lambda -\mu ]}$
• For ${\displaystyle \lambda }$ sufficiently large, ${\displaystyle s_{c}}$ is an isomorphism.
• ${\displaystyle w=s_{1}s_{2}}$
• ${\displaystyle A_{s_{1}}(\lambda )}$ -----------dynamical Weyl operator
• ${\displaystyle A_{s_{i}}}$----commutes with ${\displaystyle s_{c}}$

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