Miura opers and critical points of master functions Evgeny Mukhin Alexander Varchenko Article

Received: 02 March 2004 Accepted: 13 January 2005 DOI :
10.2478/BF02479193

Cite this article as: Mukhin, E. & Varchenko, A. centr.eur.j.math. (2005) 3: 155. doi:10.2478/BF02479193
Abstract Critical points of a master function associated to a simple Lie algebra\(\mathfrak{g}\) come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra\(^t \mathfrak{g}\) . 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.

Keywords Bethe Ansatz Miura opers flag varieties

MSC (2000) 82B23 17B67 14M15 Supported in part by NSF grant DMS-0140460

Supported in part by NSF grant DMS-0244579

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Authors and Affiliations Evgeny Mukhin Alexander Varchenko 1. Department of Mathematical Sciences Indiana University Purdue University Indianapolis Indianapolis USA 2. Department of Mathematics University of North Carolina at Chapel Hill Chapel Hill USA