Exotic symmetric space over a finite field, I
Exotic symmetric space over a finite field, I
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DOI：
10.1007/s00031-013-9237-6
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发表时间：
2012-07
影响因子：
0.7
通讯作者：
T. Shoji;K. Sorlin
中科院分区：
文献类型：
--
作者：
T. Shoji;K. Sorlin
设V是一个代数闭域k上的2n维向量空间，chk≠ 2。设G = GL（V），H = Sp 2n为G上对合θ的辛群H =Gθ.我们还用θ表示= LieG上的诱导对合。考虑变量G/H×Von，它自然地起作用。设为θin的-1本征空间中的幂零元集。在我们的设置中，G的幂幺簇的作用由×V扮演，这与Kato的奇异幂零锥相一致。Kato应用仿射Hecke代数的Ginzburg理论，在k = C的情形下，建立了C型Weyl群的不可约表示集与× V中的H-轨道集之间的Springer对应。本文发展了G/H×V上的特征层理论，并基于特征层理论给出了Kato关于Springer对应的结果的另一种证明.
LetVbe a 2n-dimensional vector space over an algebraically closed field k with chk≠ 2. Let G = GL(V) andH= Sp2nbe the symplectic group obtained asH=Gθfor an involutionθonG. We also denote byθthe induced involution on= LieG. Consider the varietyG/H×Von whichHacts naturally. Letbe the set of nilpotent elements in the -1 eigenspace ofθin. The role of the unipotent variety forGin our setup is played by×V, which coincides with Kato’s exotic nilpotent cone. Kato established, in the case where k = C, the Springer correspondence between the set of irreducible representations of the Weyl group of typeCnand the set ofH-orbits in×Vby applying Ginzburg theory for affine Hecke algebras. In this paper we develop a theory of character sheaves onG/H×V, and give an alternate proof for Kato’s result on the Springer correspondence based on the theory of character sheaves.