Convergence and semisimplicity of big quantum cohomology of flag varieties

We prove the folklore conjecture that the big quantum cohomology of a flag variety $G\mathrm{/}P$ of general Lie type is convergent near the large-radius limit and is generically semisimple. The convergence relies on a slightly modified version of Iritani’s reconstruction theorem, together with Cauchy’s formula that avoids directly taking the non-equivariant limit at the origin. Generic semisimplicity follows from a surprising application of the big quantum $D$-module mirror symmetry for $G\mathrm{/}P$ established by the first, third and fourth authors jointly with Hinault and Zhang. Moreover, we prove the regularity conjecture on the big quantum cohomology of $G\mathrm{/}P$ formulated by Perrin and Smirnov, which implies generic semisimplicity.