I am currently reading a text about the construction of a standard cyclic L-module of highest weight $ \lambda$ (L is a semisimple Lie algebra) and I am having trouble understanding the principle of the "induced module construction". I'd be very glad if someone could help me.
The construction goes as follows:
A standard cyclic module, viewed as a B-module ($\ B=B(\Delta)$, contains a one dimensional submodule spanned by the given maximal vector $\ v^+$. ( $ B(\Delta)=H+ \coprod_{\alpha \succ 0}L_\alpha $ ) is the Borel subalgebra.
Let $\ D_\lambda$ be a one dimensional vector space with $\ v^+$ as basis.
Define an action of B on $\ D_\lambda$ by the rule $\ (h+ \sum_{\alpha \succ 0}x\alpha).v^+=h.v^+=\lambda(h)v^+$, for fixed $\ \lambda \in H*$.
This makes $\ D_\lambda$ a B-module. Of course, $\ D_\lambda$ is equally well a $\ U(B)$-module, so it makes sense to form the tensor product $\ Z(\lambda)=U(L) \bigotimes_{U(B)}D_\lambda$ which becomes a U(L)-module under the natural (left) action of U(L).
We claim that $\ Z(\lambda)$ is standard cyclic for weight $\ \lambda$. $\ 1 \bigotimes v^+$ evidently generates $\ Z(\lambda)$. On the other hand, $\ 1 \bigotimes v^+$ is nonzero, because U(L) is a free U(B)-module with a basis consisting of 1 along with the various monomials $ y_{\beta_1}^{i_1}...y_{\beta_m}^{i_m} $.
Therefore $\ 1 \bigotimes v^+$ is a maximal vector of weight $\lambda$.
This construction also makes it clear that, if $N^-= \coprod_{\alpha \succ 0}L_\alpha$, then $Z(\lambda)$ viewed as $U(N^-)$-module is isomorphic to $U(N^-)$ itself.
To be precise, U(L) is isomorphic to $U(N^-) \bigotimes U(B)$, so that $Z(\lambda)$ is isomorphic to $U(N^-) \bigotimes F$ (as left U(N)-modules).