On $$p$$p-adic lattices and Grassmannians
On $$p$$p-adic lattices and Grassmannians
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关于 $$p$$p-adic 格子和格拉斯曼矩阵
DOI：
10.1007/s00209-013-1225-y
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发表时间：
2010
影响因子：
0.8
通讯作者：
M. Kreidl
中科院分区：
文献类型：
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作者：
M. Kreidl
众所周知，对于域$$k$$ k上的约化群$$G$$ G，其协集空间$$G(k((z)))/G(k[[z]])$$ G(k((z)))/G（k[[z]]）具有射影$$k$$ k格式的归纳极限的几何结构。这种$$k$$ k-ind格式被称为$$G$$ G的仿射Grassmannian格式。从数论的角度来看，获得类似于$$\mathcal {G}(\mathbf {W}(k)[1/p])/\mathcal {G}(\mathbf {W}(k))$$ G(W(k)[1/p])/G（W(k)）形式的商的几何解释将是有趣的，其中$$p$$ p是有理素数，$$\mathbf {W}$$ W表示$$p$$ p-典型Witt向量的环格式，$$k$$ k是特征$$p$$ p的完美域，$$\mathcal {G}$$ G是$$\mathbf {W}(k)$$ W(k)上的约简群方案。本文试图描述哪些构造从函数场情况延续到$$p$$ p进情况，更确切地说，延续到特殊线性群$$\mathcal {G}=\mathbf {SL}_{n}$$ G=SLn的$$p$$ p进仿射Grassmannian的情况。我们首先描述$$\mathbf {SL}_{n}$$ SLn的$$p$$ p进仿射Grassmannian的$$R$$ R值点在$$\mathbf {W}(R)$$ W(R)上的格，其中$$R$$ R是一个完美的$$k$$ k代数。为了得到与几何的联系，我们进一步构造了多阶Hilbert格式的射影$$k$$ k-子变体，它们等价地映射到$$p$$ p进仿射Grassmannian。这些模态的意象在$$p$$ p-adic背景下起着舒伯特变体的作用。此外，对于任何约简$$k$$ k代数$$R$$ R，这些态射在多重Hilbert格式中各自开放轨道的$$R$$ R值点集与$$\mathbf {SL}_{n}$$ SLn的$$p$$ p进仿射Grassmannian的相应Schubert单元之间诱导出双射映射。
It is well-known that the coset spaces $$G(k((z)))/G(k[[z]])$$G(k((z)))/G(k[[z]]), for a reductive group $$G$$G over a field $$k$$k, carry the geometric structure of an inductive limit of projective $$k$$k-schemes. This $$k$$k-ind-scheme is known as the affine Grassmannian for $$G$$G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form $$\mathcal {G}(\mathbf {W}(k)[1/p])/\mathcal {G}(\mathbf {W}(k))$$G(W(k)[1/p])/G(W(k)), where $$p$$p is a rational prime, $$\mathbf {W}$$W denotes the ring scheme of $$p$$p-typical Witt vectors, $$k$$k is a perfect field of characteristic $$p$$p and $$\mathcal {G}$$G is a reductive group scheme over $$\mathbf {W}(k)$$W(k). The present paper is an attempt to describe which constructions carry over from the function field case to the $$p$$p-adic case, more precisely to the situation of the $$p$$p-adic affine Grassmannian for the special linear group $$\mathcal {G}=\mathbf {SL}_{n}$$G=SLn. We start with a description of the $$R$$R-valued points of the $$p$$p-adic affine Grassmannian for $$\mathbf {SL}_{n}$$SLn in terms of lattices over $$\mathbf {W}(R)$$W(R), where $$R$$R is a perfect $$k$$k-algebra. In order to obtain a link with geometry we further construct projective $$k$$k-subvarieties of the multigraded Hilbert scheme which map equivariantly to the $$p$$p-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the $$p$$p-adic setting. Further, for any reduced $$k$$k-algebra $$R$$R these morphisms induce bijective maps between the sets of $$R$$R-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the $$p$$p-adic affine Grassmannian for $$\mathbf {SL}_{n}$$SLn.