We suggest a purely algebraic construction of the spin representation of an infinite-dimensional orthogonal Lie. al- gebra (sections 1 and 2) and a corresponding group (section 4). From this we deduce a construction of all level-one highest-weight representations of orthogonal affine Lie algebras in terms of cre- ation and annihilation operators on an infinite-dimensional Grass- mann algebra (section 3). We also give a similar construction of the level-one representations of the general linear affine. Lie al- gebra in an infinite-dimensional "wedge space." Along these lines we construct the corresponding representations of the universal central extension of. the group SL,(k(t,t-1)) in spaces of sections of line bundles over infinite-dimensional homogeneous spaces (sec- tion 5).

Spin and wedge representations of infinite-dimensional Lie algebras and groups (Clifford algebra/spin representation/affine Kac-Moody Lie algebra/highest weight representation/line bundle)

Wilson病(WD)是一种伴发原发性铜代谢障碍为特征的常染色体隐性遗传性疾病，病种稀少，对其早期发现和确定合适的治疗策略尤为重要，WD的发病机制与ATP7B基因突变及铜代谢障碍密切相关，临床表现具有差异性，适时选择肝移植对延长患者的生存时间和提高生存质量具有重要意义。

Wilson病与肝移植

For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the direction \"from automorphic to Galois\". Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.

Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale

We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.

/pdf/local-models-of-shimura-varieties-and-a-conjecture-of-2s0qzcjdce.pdf

Local models of Shimura varieties and a conjecture of Kottwitz

We prove that the Witt vector affine Grassmannian, which parametrizes W(k)-lattices in $$W(k)[\frac{1}{p}]^n$$
 for a perfect field k of characteristic p, is representable by an ind-(perfect scheme) over k. This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably h-descent results for vector bundles.

Projectivity of the Witt vector affine Grassmannian

We endow the set of lattices in Q_p^n with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake correspondence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.

Affine Grassmannians and the geometric Satake in mixed characteristic

We endow the set of lattices in Q^n_p with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake equivalence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.

/pdf/affine-grassmannians-and-the-geometric-satake-in-mixed-4bkuqpbx4y.pdf

We give a group theoretic definition of "local models" as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a $p$-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.

We construct a functor from the category of p-adic etale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection on its “base change to ${\mathrm {B}}_{{\text {dR}}}$”, which can be regarded as a first step towards the sought-after p-adic Riemann–Hilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some basic properties of the p-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties.

/pdf/rigidity-and-a-riemann-hilbert-correspondence-for-p-adic-2ixden6ytv.pdf

Xinwen Zhu

Papers

Local models of Shimura varieties and a conjecture of Kottwitz

Affine Grassmannians and the geometric Satake in mixed characteristic

Affine Grassmannians and the geometric Satake in mixed characteristic

Local models of Shimura varieties and a conjecture of Kottwitz

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems