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In ‘Quantization of Hitchin's Integrable System and Hecke Eigensheaves’, Beilinson and Drinfeld introduced the ‘very good’ property for a smooth complex equidimensional stack. They prove that for a semisimple group   $G$   over   $\mathbb {C}$  , the moduli stack   $\text {Bun}_G(X)$   of G-bundles over a smooth complex projective curve   $X$   is ‘very good’, as long as   $X$   has genus   $g &gt; 1$  . In the case of the projective line, when   $g = 0$  , this is not the case. However, the result can sometimes be extended to the projective line by introducing additional parabolic structure at a collection of marked points and slightly modifying the definition of a ‘very good’ stack. We provide a sufficient condition for the moduli stack of parabolic vector bundles over   $\mathbb {P}^1$   to be very good. We then use this property to study the space of solutions to the Deligne–Simpson problem.

PARABOLIC BUNDLES OVER THE PROJECTIVE LINE AND THE DELIGNE–SIMPSON PROBLEMS