Plan:
-----simple Lie algebra
-----------weight
- c -------dominant weight
-----has singular vector
and
---- generic
--------Verma module
-------------finite-dimensional module
-weight,
-weight subspace
![{\displaystyle v\in V[\lambda -\mu ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df17848573191548c295d5840e1ad10be16777d7)
-------------dominant weight
----------finite-dimensional module
----space of g-invariant homomorphismata
----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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8be4991b919bf82ed1c35740cd4fc9611a6df2)
- For
generic, ![{\displaystyle Hom_{\mathfrak {g}}(M_{\lambda }\rightarrow M_{\mu }\otimes V)\cong V[\lambda -\mu ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b01353ebaed4d7ee85c8a9e0023206ba3c76d833)
- For
generic, ![{\displaystyle Hom_{\mathfrak {g}}(M_{\lambda +c}\rightarrow M_{\mu +c}\otimes V)\cong V[\lambda -\mu ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60a7874d90211dcf051109c4463cef9e352f9e63)
- For
sufficiently large,
is an isomorphism.
![{\displaystyle w=s_{1}s_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f59de4aeaf0b52359372c7bea2c85f51f76ab9a2)
-----------dynamical Weyl operator
----commutes with ![{\displaystyle s_{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91c4d1b073949e8f032e50ff913afbf43bb5acb7)