Category $\mathcal{O}$ for the Lie algebra of vector fields on the line

Let $\mathfrak{W}$ be the Lie algebra of vector fields on the line. Via computing extensions between all simple modules in the category $\mathcal{O}$, we give the block decomposition of $\mathcal{O}$, and show that the representation type of each block of $\mathcal{O}$ is wild using the Ext-quiver. Each block of $\mathcal{O}$ has infinite simple objects. This result is very different from that of $\mathcal{O}$ for complex semisimple Lie algebras. To find a connection between $\mathcal{O}$ and the module category over some associative algebra, we define a subalgebra $H_1$ of $U(\mathfrak{b})$. We give an exact functor from $\mathcal{O}$ to the category $\Omega$ of finite dimensional modules over $H_1$. We also construct new simple $\mathfrak{W}$-modules from Weyl modules and modules over the Borel subalgebra $\mathfrak{b}$ of $\mathfrak{W}$.