Miura Opers and Critical Points of Master Functions

Critical points of a master function associated to a simple Lie algebra g come in families called the populations [MV1]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra g . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY = 0 can be written explicitly in terms of critical points composing the population. ∗ Department of Mathematical Sciences, Indiana University Purdue University Indianapolis, 402 North Blackford St., Indianapolis, IN 46202-3216, USA Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA December, 2003