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
 * $$Hom_\mathfrak{g} (M_\lambda \rightarrow M_\mu \otimes V) \rightarrow Hom_\mathfrak{g}(M_{\lambda +c}\rightarrow M_{\mu+c}\otimes V) $$

$$A_{s_1 s_2}(\lambda) = A_{s_1}(s_2 \cdot \lambda ) A_{s_2}(\lambda)$$
 * 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 $$s_c$$