Kac moody verma module

For $\lambda \in {\mathfrak{h}}^{\ast }$ let $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ and let ${\mathbb{C}}_{\lambda }$ be the one-dimensional $\mathfrak{b}$-module with $hv=\lambda (h)v$ and ${\mathfrak{n}}^{+}v=0$. Define $M_A(\lambda )=U(\mathfrak{g}){\otimes }_{U(\mathfrak{b})}{\mathbb{C}}_{\lambda }$. Define $\tilde{M}(\lambda )$ in the same way for $\tilde{\mathfrak{g}}$ and its positive Borel.

The tensor quotient is The universal enveloping algebra as a tensor quotient. Order a homogeneous negative basis, then a Cartan basis, then a positive basis. PBW for countably presented Kac Moody Lie algebras makes multiplication $U({\mathfrak{n}}^{-})\otimes U(\mathfrak{b})\to U(\mathfrak{g})$ a vector-space isomorphism and a right $U(\mathfrak{b})$-module isomorphism. Tensoring gives $M_A(\lambda )\cong U({\mathfrak{n}}^{-})$, with $1\otimes 1$ corresponding to 1. The same reasoning gives $\tilde{M}(\lambda )\cong U({\tilde{\mathfrak{n}}}^{-})$. Thus the top space has dimension one, and all other weights are $\lambda -\beta$ for $\beta \in Q^{+}\setminus \{0\}$. At a fixed height there are finitely many monomials: only finitely many root degrees and basis elements of height at most that height can occur, with bounded exponents. Hence $M_A(\lambda )$ belongs to Kac moody category o. The module $\tilde{M}(\lambda )$ has the same finite-weight-space and downward-cone properties as a $\tilde{\mathfrak{g}}$-module; no factorization of its action through $\mathfrak{g}(A)$ is asserted. Mapping $u\otimes 1\mapsto uv$ gives the unique module map to any module with a specified highest vector $v$ of weight $\lambda$, because the tensor relations hold for that vector.

Source comparison: Kleshchev, §9.1, pp.116–117; local countable PBW verification.