PBW basis and grades of Virasoro Verma modules

The bracket [F2] shows that ${\mathfrak{n}}_{-}$, $\mathfrak{h}$, and ${\mathfrak{n}}_{+}$ from [F1] are subalgebras: same-sign nonzero indices add without a central term, while $L_0,C$ preserve each block. Their direct sum is the full Virasoro vector space. Order a basis with every negative mode first, then $L_0,C$, then every positive mode. PBW [F3] gives the unique ordered factorization $U({\mathfrak{n}}_{-})U(\mathfrak{h})U({\mathfrak{n}}_{+})$.

Multiplication therefore identifies $U(\mathrm{V}\mathrm{i}\mathrm{r})$, as a right $U(\mathfrak{b})$-module, with the direct sum of one copy of $U(\mathfrak{b})$ for each ordered negative-mode monomial. Tensoring this free right module with the one-dimensional module ${\mathbb{C}}_{c,h}$ kills positive modes and evaluates $L_0,C$ at $h,c$, but imposes no relation among the negative monomials. Hence they give a basis of $M(c,h)$.

The relation $[L_0,L_{-n}]=n{L}_{-n}$ shows by induction on the number of factors that a monomial of total index $N$ has eigenvalue $h+N$. Distinct total indices have distinct eigenvalues; the ordered monomials at fixed total index are indexed exactly by partitions of $N$. This proves the grade and dimension assertions. At $N=0$, the only monomial is the empty one and it is the nonzero vector from step 2.1.