  In the context of Liouville conformal field theory, we construct the
highest-weight representations of the Virasoro algebra at the degenerate values
of the conformal weight (Kac table). We show that these modules are
irreducible, giving a complete characterization of the algebraic structure of
Liouville CFT. It also implies that all singular vectors vanish, which is one
of the main assumptions usually made in theoretical physics. Our proof uses
inputs from both probability theory and algebra, and gives new probabilistic
content to the Kac table.
  Combining the information that singular vectors vanish with the main
geometric properties of conformal blocks, we deduce that conformal blocks
involving degenerate primary fields satisfy null-vector equations. These
equations take the form of PDEs on the Teichm\"uller space of the underlying
surface and generalize previous works in several directions.
