Affine Grassmannians and the geometric Satake in mixed characteristic

By Xinwen Zhu. 2014

177 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/18b534545dc25c39e97767e2748d53b420479658

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In Geometry and Topology

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Citations in the corpus

Rapoport–Zink spaces for spinor groups (2015, methodology): “Indeed, then one can even use the Witt vector affine Grassmannian as in [Zhu14] and [BS15] to obtain a more straightforward and general construction of (at least) the reduced locus of Rapoport-Zink schemes but only up to perfection.”
Geometrization of the local Langlands correspondence: an overview (2016, background): “If G is unramified there is an object sitting over Spa(OE) whose special fiber over Fq is the Witt vectors affine Grassmanian as defined by Zhu ([32], [29]) and whose generic fiber over E is the BdR-affine Grassmanian.”
p-ADIC GEOMETRY (2017, methodology): “The special fibre of GrG,SpdZp is given by the Witt vector affine Grassmannian GrWG constructed in [Zhu17] and [BS17] that can be written as an increasing union of perfections of projective varieties along closed immersions.”
Cycles on Shimura varieties via geometric Satake (2017, methodology): “This construction induces an isomorphism (3.1.9) Grµ1,...,µt ∼= Grµ1,...,µs×̃Grµs+1,...,µt := Gr(∞)µ1,...,µs × L+G Grµs+1,...,µt , where Gr (∞) µ1,...,µs is the L +G-torsor over Grµ1,...,µs classifying a point (Es 99K · · · 99K E0 = E0) of Grµ1,...,µs together with an isomorphism Es ' E0.”
Projectivity of the Witt vector affine Grassmannian (2015, methodology): “One can also determine the connected components of GrG in case G is parahoric, using Kottwitz’ map κ : LG(k̄)→ π1(G)GalK , defined for any algebraically closed field k̄ containing k. Proposition 9.7 ([Zhu14, Proposition 1.21]).”
Stack (2020, methodology): “Recall the following deperfection result from [Ans+22] that relies on [Zhu17, Proposition A.15]: Proposition 3.31 ([Ans+22, Proposition 3.3]).”