Dynamical Weyl group

Plan:

$A_{s_1{s}_2}(\lambda )=A_{s_1}(s_2\cdot \lambda )A_{s_2}(\lambda )$

• $𝔤$ -----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
• $Ho{m}_{𝔤}(M_{\lambda }\to M_{\mu }\otimes V)$----space of g-invariant homomorphismata
• $Ho{m}_{𝔤}(M_{\lambda +c}\to M_{\mu +c}\otimes V)$----space of g-invariant homomorphisma
• $Ho{m}_{𝔤}(M_{\lambda }\to M_{\mu }\otimes V)\to Ho{m}_{𝔤}(M_{\lambda +c}\to M_{\mu +c}\otimes V)$

• For $\lambda ,\mu$ generic, $Ho{m}_{𝔤}(M_{\lambda }\to M_{\mu }\otimes V)\cong V[\lambda -\mu ]$
• For $\lambda +c,\mu +c$ generic, $Ho{m}_{𝔤}(M_{\lambda +c}\to M_{\mu +c}\otimes V)\cong V[\lambda -\mu ]$
• 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 $s_c$

$A_{s_1{s}_2}(\lambda )=A_{s_1}(s_2\cdot \lambda )A_{s_2}(\lambda )$