This paper studies the analytic continuation of Liouville eigenstates and shows that they assemble into irreducible highest-weight representations of the Virasoro algebra, for all values of the conformal weights. This builds on previous results from the first author and Guillarmou, Kupiainen, Rhodes & Vargas, where such representations were constructed except for the conformal weight on the Kac table. In order to extend these results to the degenerate weights, we find explicit analytic expressions for the Virasoro descendants and uncover the probabilistic meaning of the Kac table. In the algebraic approach to conformal field theory, the irreducibility is a crucial property that must be satisfied by the representations in the spectrum, and is usually taken as an axiom. Computationally, it leads to the celebrated null-vector (or BPZ) equations for correlation functions and conformal blocks, which are the cornerstone of the integrability of the theory.