  Topological photonics provides a fundamental framework for robust
manipulation of light, including directional transport and localization with
built-in immunity to disorder. Combined with an optical gain, active
topological cavities hold special promise for a design of light-emitting
devices. Most studies to date have focused on lasing at topological edges of
finite systems or domain walls. Recently discovered higher-order topological
phases enable strong high-quality confinement of light at the corners. Here we
demonstrate lasing action of corner states in a nanophotonic topological
cavity. We identify four multipole corner modes with distinct emission profiles
via hyperspectral imaging and discern signatures of non-Hermitian radiative
coupling of leaky topological states. In addition, depending on the pump
position in a large-size cavity, we selectively generate lasing from either
edge or corner states within the topological bandgap. Our findings introduce
pathways to engineer collective resonances and tailor generation of light in
active topological circuits.
