Mathematische Zeitschrift

, Volume 275, Issue 1, pp 55–77

Degenerate flag varieties of type A: Frobenius splitting and BW theorem

Article

DOI: 10.1007/s00209-012-1122-9

Cite this article as:
Feigin, E. & Finkelberg, M. Math. Z. (2013) 275: 55. doi:10.1007/s00209-012-1122-9

Abstract

Let \(\mathcal F ^a_\lambda \) be the PBW degeneration of the flag varieties of type \(A_{n-1}\). These varieties are singular and are acted upon with the degenerate Lie group \(SL_n^a\). We prove that \(\mathcal F ^a_\lambda \) have rational singularities, are normal and locally complete intersections, and construct a desingularization \(R_\lambda \) of \(\mathcal F ^a_\lambda \). The varieties \(R_\lambda \) can be viewed as towers of successive \(\mathbb{P }^1\)-fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties \(R_\lambda \) are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for \(\mathcal F ^a_\lambda \). Using the Atiyah–Bott–Lefschetz formula for \(R_\lambda \), we compute the \(q\)-characters of the highest weight \(\mathfrak sl _n\)-modules.

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsNational Research University Higher School of EconomicsMoscowRussia
  2. 2.Tamm Theory DivisionLebedev Physics InstituteMoscowRussia
  3. 3.IMU, IITP, and National Research University Higher School of EconomicsMoscowRussia

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