TY - JOUR
T1 - Degenerate flag varieties of type A
T2 - Frobenius splitting and BW theorem
AU - Feigin, Evgeny
AU - Finkelberg, Michael
N1 - Funding Information:
We are grateful to Shrawan Kumar, Alexander Kuznetsov, and Peter Littelmann for useful discussions. We are also grateful to R. Bezrukavnikov and D. Kazhdan for organizing the 15th Midrasha Mathematicae “Derived Categories of Algebro-Geometric Origin and Integrable Systems” at IAS at the Hebrew University of Jerusalem where this work was conceived. This paper was written during the E. F. stay at the Hausdorff Research Institute for Mathematics. The hospitality and perfect working conditions of the Institute are gratefully acknowledged. The work of Evgeny Feigin was partially supported by the Russian President Grant MK-3312.2012.1, by the Dynasty Foundation, by the AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023, by the RFBR grants 12-01-00070, 12-01-00944 and by the Russian Ministry of Education and Science under the grant 2012-1.1-12-000-1011-016. M. F. was partially supported by the RFBR grant 12-01-00944, the National Research University Higher School of Economics’ Academic Fund award No.12-09-0062 and the AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023. This study was carried out within the National Research University Higher School of Economics Academic Fund Program in 2012-2013, research grant No. 11-01-0017. This study comprises research findings from the “Representation Theory in Geometry and in Mathematical Physics” carried out within The National Research University Higher School of Economics’ Academic Fund Program in 2012, grant No 12-05-0014.
PY - 2013/10
Y1 - 2013/10
N2 - Let Fλa be the PBW degeneration of the flag varieties of type An-1. These varieties are singular and are acted upon with the degenerate Lie group SLna. We prove that Fλa have rational singularities, are normal and locally complete intersections, and construct a desingularization Rλ of Fλa. The varieties Rλ can be viewed as towers of successive ℙ1-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties Rλ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel-Weil type theorem for Fλa. Using the Atiyah-Bott-Lefschetz formula for Rλ, we compute the q-characters of the highest weight sln-modules.
AB - Let Fλa be the PBW degeneration of the flag varieties of type An-1. These varieties are singular and are acted upon with the degenerate Lie group SLna. We prove that Fλa have rational singularities, are normal and locally complete intersections, and construct a desingularization Rλ of Fλa. The varieties Rλ can be viewed as towers of successive ℙ1-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties Rλ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel-Weil type theorem for Fλa. Using the Atiyah-Bott-Lefschetz formula for Rλ, we compute the q-characters of the highest weight sln-modules.
UR - http://www.scopus.com/inward/record.url?scp=84880385597&partnerID=8YFLogxK
U2 - 10.1007/s00209-012-1122-9
DO - 10.1007/s00209-012-1122-9
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:84880385597
SN - 0025-5874
VL - 275
SP - 55
EP - 77
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 1-2
ER -