Symplectic degenerate flag varieties

Evgeny Feigin, Michael Finkelberg, Peter Littelmann

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A simple finite dimensional module Vλ of a simple complex algebraic group G is naturally endowed with a filtration induced by the PBW-filtration of U(Lie G). The associated graded space Vaλ is a module for the group Ga, which can be roughly described as a semi-direct product of a Borel subgroup of G and a large commutative unipotent group GMa. In analogy to the flag variety fλl = G-[νλ] C P(Vλ), we call the closure Ga.[νλ] C P(Vaλ) of the Ga-orbit through the highest weight line the degenerate flag variety faa7lambda;. In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of G being die symplectic group. The symplectic case is important for the conjecture because it is the first known case where, even for fundamental weights ω, the varieties faω differ from fω. We give an explicit construction of the varieties Sp faλ and construct desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that Sp faλ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil theorem and obtain a q-character formula for the characters of irreducible Sp2n-modules via die Atiyah-Bott-Lefschetz fixed points formula.

Original languageEnglish
Pages (from-to)1250-1286
Number of pages37
JournalCanadian Journal of Mathematics
Issue number6
StatePublished - 1 Dec 2014
Externally publishedYes


  • Flag varieties
  • Lie algebras
  • Representations
  • Symplectic groups


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