Hilbert series of nearly holomorphic sections on generalized flag manifolds

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• 作者：Benjamin Schwarz ^{bschwarz@math.upb.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 32A50 ; secondary ; 32L10 ; 14M15 ; 22E46
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：26 February 2016
• 年：2016
• 卷：290
• 期：Complete
• 页码：293-313
• 全文大小：426 K

文摘

Let X=G/P$X=G/P$ be a complex flag manifold and E→X$E\to X$ be a G-homogeneous holomorphic vector bundle. Fix a U-invariant Kähler metric on X   with U⊆G$U\subseteq G$ maximal compact. We study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in E coincides with the space of U-finite smooth sections in E. The degree of nearly holomorphic sections defines a U-invariant filtration on this space. Using sheaf cohomology, we determine in suitable cases the corresponding Hilbert series. It turns out that this is given in terms of Lusztig's q-analog of Kostant's weight multiplicity formula, and hence gives a new representation theoretic interpretation of these formulas.