摘要翻译:
利用Mirkovic和Vilonen的几何Satake同构,还原群的分解数可以解释为Langlands对偶群的复仿射Grassmannian上的等变逆束的分解数。利用Malkin、Ostrik和Vybornov得到的仿射Grassmanian的最小退化的描述,我们可以从几何上恢复还原群的某些分解数。在另一个方向上,我们可以利用还原群的一些分解数来证明几何结果,如非光滑性结果的一个新的证明,以及某些奇点不等价的证明(Malkin,Ostrik和Vybornov的猜想)。我们还对Mirkovic和Vilonen的一个猜想给出了反例,即仿射Grassmanian上的整数上的标准反向束的茎是无扭转的,并提出了一个不含坏素数的修正猜想。
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英文标题:
《Modular representations of reductive groups and geometry of affine
Grassmannians》
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作者:
Daniel Juteau
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最新提交年份:
2008
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分类信息:
一级分类:Mathematics 数学
二级分类:Representation Theory 表象理论
分类描述:Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra
代数和群的线性表示,李理论,结合代数,多重线性代数
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一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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英文摘要:
By the geometric Satake isomorphism of Mirkovic and Vilonen, decomposition numbers for reductive groups can be interpreted as decomposition numbers for equivariant perverse sheaves on the complex affine Grassmannian of the Langlands dual group. Using a description of the minimal degenerations of the affine Grassmannian obtained by Malkin, Ostrik and Vybornov, we are able to recover geometrically some decomposition numbers for reductive groups. In the other direction, we can use some decomposition numbers for reductive groups to prove geometric results, such as a new proof of non-smoothness results, and a proof that some singularities are not equivalent (a conjecture of Malkin, Ostrik and Vybornov). We also give counterexamples to a conjecture of Mirkovic and Vilonen stating that the stalks of standard perverse sheaves over the integers on the affine Grassmannian are torsion-free, and propose a modified conjecture, excluding bad primes.
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PDF链接:
https://arxiv.org/pdf/0804.2041