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Article

Symplectic Degenerate Flag Varieties

Canadian Journal of Mathematics. 2014. Vol. 66. No. 6. P. 1250-1286.
Feigin E., Michael Finkelberg, Littelmann P.

A simple finite dimensional module V_\lambda 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 is a module for the group G^a which can be roughly described as a semi-direct product of a Borel subgroup of G and a large commutative unipotent group. In analogy to the classical flag variety of G, we define the degenerate flag variety F^a_\lambda as the closure of the G^a-orbit through the highest weight line. 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 the symplectic group. The symplectic case is important for the conjecture because it is the first known case where, even for fundamental weights \omega, the varieties F^a_\omega differ from the classical partial flag varieties F_\omega. We give an explicit construction of the varieties F^a_\lambda and construct desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $F^a_\lambda$ 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 of Sp_{2n}-modules via the Atiyah-Bott-Lefschetz fixed point formula