LS-图库、路径模型和MV-圈
LS-Galleries, the path model and MV-cycles
摘要：
本文利用复半单代数群$G$的仿射Grassmannian几何，给出了复半单代数群的表示\cite{Lit1}的路径模型的一种解释.在这种情况下，路径被与$G$的Weyl群相关联的仿射Coxeter复形中的LS-gallery所取代。为了解释与几何的联系，考虑一个Demazure-Hansen-Bott-Samelson去奇异化$\hatΣ(\lam)$，它是仿射格拉斯曼轨道$G(\bc[[t]]).\lam$的闭包。$\hatΣ(\lam)$的同源性有一个由Biaidynicki-Birula细胞给出的基，它由$\hatΣ(\lam)$中的$T$-不动点索引。现在，$\hatΣ(\lam)$的点可以用与$G$相关联的仿射Tits建筑中的固定类型的图库来标识，并且$T$ --不动点在这种语言中对应于仿射Coxeter复形中的固定类型的组合图库。我们确定这些画廊，使相关的细胞有一个非空的交集与$G(\bc[[t]]).\lam$（确定与一个开放的子集$\hatΣ(\lam)$），我们表明，关闭的层相关的LS画廊正是MV-周期\cite{MV}，这形成了一个基础上的代表$V(\lam)$的朗兰的双重组$G^\vee$。
英文摘要：
We give an interpretation of the path model of a representation \cite{Lit1} of a complex semisimple algebraic group $G$ in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by LS--galleries in the affine Coxeter complex associated to the Weyl group of $G$. To explain the connection with geometry, consider a Demazure--Hansen--Bott--Samelson desingularization $\hatΣ(\lam)$ of the closure of an orbit $G(\bc[[t]]).\lam$ in the affine Grassmannian. The homology of $\hatΣ(\lam)$ has a basis given by Białynicki--Birula cell's, which are indexed by the $T$--fixed points in $\hatΣ(\lam)$. Now the points of $\hatΣ(\lam)$ can be identified with galleries of a fixed type in the affine Tits building associated to $G$, and the $T$--fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a non-empty intersection with $G(\bc[[t]]).\lam$ (identified with an open subset of $\hatΣ(\lam)$), and we show that the closures of the strata associated to LS-galleries are exactly the MV--cycles \cite{MV}, which form a basis of the representation $V(\lam)$ for the Langland's dual group $G^\vee$.
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