A Borel--Weil--Bott theorem for Quot schemes on ${\mathbb{P}}^1$

We study the cohomology groups of tautological bundles on Quot schemes over the projective line, which parametrize rank $r$ quotients of a vector bundle $V$ on ${\mathbb{P}}^1$. Our main result is an analogue of the Borel–Weil–Bott theorem for Quot schemes. As a corollary, we prove recent conjectures of Marian, Oprea, and Sam on the exterior and symmetric powers of tautological bundles.