Abstract view
Symplectic Degenerate Flag Varieties


Published:20131204
Printed: Dec 2014
Evgeny Feigin,
National Research University Higher School of Economics, Department of Mathematics, Vavilova str. 7, 117312, Moscow, Russia
Michael Finkelberg,
IMU, IITP, and National Research University Higher School of Economics, Department of Mathematics, Vavilova str. 7, 117312, Moscow, Russia
Peter Littelmann,
Mathematisches Institut, Universität zu Köln, Weyertal 8690, D50931 Köln, Germany
Abstract
A simple finite dimensional module $V_\lambda$ of a simple complex
algebraic group $G$ is naturally endowed with a filtration induced by the PBWfiltration
of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module
for the group $G^a$, which can be roughly described as a semidirect product of a
Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy
to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$,
we call the closure
$\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$
of the $G^a$orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$.
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
$\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit
construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations,
similar to the BottSamelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$
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 BorelWeil theorem and obtain a $q$character formula
for the characters of irreducible $Sp_{2n}$modules via the AtiyahBottLefschetz fixed
points formula.