Symplectic Degenerate Flag Varieties

Feigin, Evgeny ; Finkelberg, Michael ; Littelmann, Peter

Canadian journal of mathematics, Tome 66 (2014) no. 6, pp. 1250-1286

Résumé

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 $U\,(\text{Lie}\,G)$ . The associated graded space $\text{V}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ 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 $\mathbb{G}_{a}^{M}$ . In analogy to the flag variety ${{\mathcal{F}}_{\lambda }}\,=\,G.[{{v}_{\lambda }}]\,\,\subset \,\,\mathbb{P}({{V}_{\lambda }})$ , we call the closure $\overline{{{G}^{a}}\,.\,[{{v}_{\text{ }\!\!\lambda\!\!\text{ }}}]}\,\,\subset \,\,\mathbb{P}\,(V_{\text{ }\!\!\lambda\!\!\text{ }}^{a})$ of the ${{G}^{a}}$ -orbit through the highest weight line the degenerate flag variety $\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ . 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 $\mathcal{F}_{\text{ }\!\!\omega\!\!\text{ }}^{a}$ differ from ${{\mathcal{F}}_{\text{ }\!\!\omega\!\!\text{ }}}$ . We give an explicit construction of the varieties $\text{Sp}\,\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ and construct desingularizations, similar to the Bott–Samelson resolutions in the classical case. We prove that $\text{Sp}\,\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ 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 $\text{S}{{\text{p}}_{2\pi }}$ -modules via the Atiyah–Bott–Lefschetz fixed points formula.

Détail

DOI : 10.4153/CJM-2013-038-6

Mots-clés : 14M15, 22E46, Lie algebras, flag varieties, symplectic groups, representations