##
**Affine Grassmannians and the geometric Satake in mixed characteristic.**
*(English)*
Zbl 1390.14072

The two main results of this paper are a construction of a mixed characteristic version of affine Grassmannians (as a “perfect” algebraic space, see below), and a proof of the geometric Satake isomorphism in this setting.

The affine Grassmannian (attached to a reductive group over a field \(k\), say), a variant of the classical Grassmann variety in the context of an affine root system, is an important object in algebraic geometry which plays a prominent role in the study of moduli spaces of vector bundles on a curve, in the Langlands program and in the arithmetic theory of Shimura varieties (and many other topics). See for instance, [A. Beauville and Y. Laszlo, Commun. Math. Phys. 164, No. 2, 385–419 (1994; Zbl 0815.14015)], [G. Faltings, J. Eur. Math. Soc. (JEMS) 5, No. 1, 41–68 (2003; Zbl 1020.14002)], [B. C. Ngô, Publ. Math., Inst. Hautes Étud. Sci. 111, 1–169 (2010; Zbl 1200.22011)], [G. Pappas et al., in: Handbook of moduli. Volume III. Somerville, MA: International Press; Beijing: Higher Education Press. 135–217 (2013; Zbl 1322.14014)] [G. Pappas and X. Zhu, Invent. Math. 194, No. 1, 147–254 (2013; Zbl 1294.14012)].

For the general linear group \(GL_n\), the affine Grassmannian can be viewed as the moduli space of all \(k[[t]]\)-lattices inside the vector space \(k((t))^n\) over the field of Laurent series over \(k\). From a slightly different point of view, the affine Grassmannian is the quotient (of fppf-sheaves, say) \(GL_n(k((t)))/GL_n(k[[t]])\) (where we understand \(GL_n(k((t)))\) as the functor \(R\mapsto GL_n(R((t)))\) on the category of \(k\)-algebras, and similarly for \(GL_n(k[[t]])\)). One shows that this functor is represented by an ind-scheme, more precisely by an inductive limit of projective \(k\)-schemes, where the transition maps are closed immersions.

It is an interesting question, whether this construction admits an analogue in mixed characteristic. Letting \(k\) denote a finite field or an algebraic closure of a finite field, we would look (starting with the group \(GL_n\), for simplicity) for a space which classifies lattices inside \(W(k)[1/p]^n\) over the ring \(W(k)\) of Witt vectors.

There have been various attempts at constructing such an object: A rather early one by G. Lusztig [Proc. Symp. Pure Math. 33, 171–175 (1979; Zbl 0421.22009)]; there the result is a pro-object rather than an ind-object, though. There is also work by W. J. Haboush [Tohoku Math. J. (2) 57, No. 1, 65–117 (2005; Zbl 1119.14004)] in the direction of constructing a mixed characteristic affine Grassmannian. Later, M. Kreidl has taken up the problem again in [M. Kreidl, Math. Z. 276, No. 3–4, 859–888 (2014; Zbl 1304.13010)]. He established basic properties of the quotient fpqc-sheaf \(GL_n(W(k)[1/p])/GL_n(W(k))\), but was unable to give a definition of the affine Grassmannian as a truly geometric object. As before, \(GL_n(W(k)[1/p])\) is meant as an abbreviation for the functor \(R\mapsto GL_n(W(R)[1/p])\). Since the ring of Witt vectors of an arbitrary ring is not usually a nice object, it might not be surprising that it is difficult to get one’s hands on these functors. There is one case where things work much better, though, namely when the ring \(R\) is perfect.

In the paper under review, the author presents a construction of the mixed characteristic affine Grassmannian as a “geometric object”. The key point of Zhu’s approach is to consider the above-mentioned functors as functors on the category of perfect \(k\)-algebras, rather than the category of all \(k\)-algebras. Restricting the functor attached to a scheme to this subcategory corresponds to passing to the perfection of the scheme, i.e., replacing the coordinate ring \(A\) of an affine scheme by its perfect hull \(A^{p^{-\infty}}\), the colimit over \(A\rightarrow A \rightarrow A \rightarrow\cdots\), where all maps are \(x\mapsto x^p\). In this “perfect” setting, the author is able to show that the affine Grassmann functor is representable by the perfect analogue of an algebraic space.

The construction proceeds by first handling the case of \(GL_n\). In this case, he shows that for fixed \(N\) and sufficiently large \(h\), the functor \[ \overline{\text{Gr}}_{N,h}(R) = \{ (\Lambda, \varepsilon);\;\Lambda \subseteq W(R)^n\;\text{a lattice}, \bigwedge^n\Lambda = p^N W(R), \varepsilon: W_h(R)^n \rightarrow \Lambda/p^h\Lambda\;\text{an isom.} \} \] is representable by a perfect scheme. Here \(W_h(R)\) denotes the ring of truncated Witt vectors. Passing to a suitable quotient, one can get rid of the “level structure” \(\varepsilon\) and obtains a perfect algebraic space \(\overline{\text{Gr}}_N\). Taking the inductive limit over all \(N\), we obtain the mixed characteristic affine Grassmannian for \(GL_n\). Perfect schemes, or algebraic spaces, of positive dimension are never of finite type over \(k\). There exists, for each \(N\), an algebraic space of finite presentation over \(k\) whose perfection is \(\overline{\text{Gr}}_N\), a so-called model or deperfection of \(\overline{\text{Gr}}_N\). (However it is not clear whether these can be put into a compatible family.)

By embedding a general reductive group \(G\) suitably into some \(GL_n\), one constructs a mixed characteristic affine Grassmannian \(Gr_G\) for \(G\).

While one does not obtain in this way an ind-scheme as in the classical case, this definition is powerful enough to allow for a proof of the geometric Satake isomorphism (i.e., an equivalence of categories between equivariant perverse sheaves on the affine Grassmannian and representations of the Langlands dual group) in this setting.

In fact, while certain geometric concepts (e.g., tangent spaces) are unavailable for such perfect spaces, passing to the perfection of a scheme does not change its étale site, and hence étale cohomology is still available. One can thus define \(\ell\)-adic sheaves, the \(\ell\)-adic derived category and the notion of (equivariant) perverse sheaf on a perfect scheme or perfect algebraic space which arises as the perfection of a proper scheme or proper algebraic space.

One obtains the hypercohomology functor \[ H^*(\text{Gr}_G, -):\text{Perv}_{L^+G}(\text{Gr}_G) \rightarrow \text{Vect}_{\overline{\mathbb Q}_\ell}, \] from the category of \(L^+G\)-equivariant perverse sheaves on \(Gr_G\) to the category of \(\overline{\mathbb Q}_\ell\)-vector spaces, which can be endowed with a canonical monoidal structure (even though the “classical” methods do not work directly in this setting). The author then proves:

Theorem. The monoidal functor \(H^*\) induces an equivalence between \(\text{Perv}_{L^+G}(\text{Gr}_G)\) and the category \(\text{Rep}_{\overline{\mathbb Q}_\ell}(\widehat{G})\) of finite-dimensional representations of the dual group \(\widehat{G}\) of \(G\) over \(\overline{\mathbb Q}_\ell\).

In particular, there exists a commutativity constraint on \(\text{Perv}_{L^+G}(\text{Gr}_G)\) such that \(H^*\) is a tensor functor. To construct it, the author uses a version of the classical “Gelfand trick”. To show that the result has the desired properties, he proves that those properties follow from a certain numerical result on the affine Hecke algebra. This numerical property is known by work of G. Lusztig and Z. Yun [J. Ramanujan Math. Soc. 28A, 311–340 (2013; Zbl 1295.20046)] based on the equal-characteristic version of the geometric Satake isomorphism. Since the affine Hecke algebra is the same in both cases, this yields the desired conclusion.

Finally, the author gives some applications of his theory to affine Deligne-Lusztig varieties and Rapoport-Zink spaces, certain moduli spaces of \(p\)-divisible groups which are closely related to the reductions of Shimura varieties. The construction of the mixed-characteristic affine Grassmannian provides a way to define an a priori geometric structure on affine Deligne-Lusztig varieties in this setting, an important piece of the theory which had been missing so far and which will have many applications. See for instance [L. Xiao and X. Zhu, “Cycles on Shimura varieties via geometric Satake”, Preprint, arXiv:1707.05700] for further work by L. Xiao and the author in this direction.

There are two appendices. Appendix A (Generalities on perfect schemes) contains foundational material on perfect schemes, perfect algebraic spaces, and \(\ell\)-adic sheaves on them.

In Appendix B, a few more topics concerning the mixed characteristic affine Grassmannian are discussed, notably the question whether there is a determinant line bundle as in the classical case. It is stated as a conjecture that such a line bundle exists, and that it induces embeddings of suitable finite presentation models of \(\overline{\text{Gr}}_N\) into projective space, so that \(\overline{\text{Gr}}_N\) is the perfection of a projective variety and in particular a perfect scheme, not merely a perfect algebraic space. These conjectures have in the meantime been proved by B. Bhatt and P. Scholze [Invent. Math. 209, No. 2, 329–423 (2017; Zbl 1397.14064)].

The affine Grassmannian (attached to a reductive group over a field \(k\), say), a variant of the classical Grassmann variety in the context of an affine root system, is an important object in algebraic geometry which plays a prominent role in the study of moduli spaces of vector bundles on a curve, in the Langlands program and in the arithmetic theory of Shimura varieties (and many other topics). See for instance, [A. Beauville and Y. Laszlo, Commun. Math. Phys. 164, No. 2, 385–419 (1994; Zbl 0815.14015)], [G. Faltings, J. Eur. Math. Soc. (JEMS) 5, No. 1, 41–68 (2003; Zbl 1020.14002)], [B. C. Ngô, Publ. Math., Inst. Hautes Étud. Sci. 111, 1–169 (2010; Zbl 1200.22011)], [G. Pappas et al., in: Handbook of moduli. Volume III. Somerville, MA: International Press; Beijing: Higher Education Press. 135–217 (2013; Zbl 1322.14014)] [G. Pappas and X. Zhu, Invent. Math. 194, No. 1, 147–254 (2013; Zbl 1294.14012)].

For the general linear group \(GL_n\), the affine Grassmannian can be viewed as the moduli space of all \(k[[t]]\)-lattices inside the vector space \(k((t))^n\) over the field of Laurent series over \(k\). From a slightly different point of view, the affine Grassmannian is the quotient (of fppf-sheaves, say) \(GL_n(k((t)))/GL_n(k[[t]])\) (where we understand \(GL_n(k((t)))\) as the functor \(R\mapsto GL_n(R((t)))\) on the category of \(k\)-algebras, and similarly for \(GL_n(k[[t]])\)). One shows that this functor is represented by an ind-scheme, more precisely by an inductive limit of projective \(k\)-schemes, where the transition maps are closed immersions.

It is an interesting question, whether this construction admits an analogue in mixed characteristic. Letting \(k\) denote a finite field or an algebraic closure of a finite field, we would look (starting with the group \(GL_n\), for simplicity) for a space which classifies lattices inside \(W(k)[1/p]^n\) over the ring \(W(k)\) of Witt vectors.

There have been various attempts at constructing such an object: A rather early one by G. Lusztig [Proc. Symp. Pure Math. 33, 171–175 (1979; Zbl 0421.22009)]; there the result is a pro-object rather than an ind-object, though. There is also work by W. J. Haboush [Tohoku Math. J. (2) 57, No. 1, 65–117 (2005; Zbl 1119.14004)] in the direction of constructing a mixed characteristic affine Grassmannian. Later, M. Kreidl has taken up the problem again in [M. Kreidl, Math. Z. 276, No. 3–4, 859–888 (2014; Zbl 1304.13010)]. He established basic properties of the quotient fpqc-sheaf \(GL_n(W(k)[1/p])/GL_n(W(k))\), but was unable to give a definition of the affine Grassmannian as a truly geometric object. As before, \(GL_n(W(k)[1/p])\) is meant as an abbreviation for the functor \(R\mapsto GL_n(W(R)[1/p])\). Since the ring of Witt vectors of an arbitrary ring is not usually a nice object, it might not be surprising that it is difficult to get one’s hands on these functors. There is one case where things work much better, though, namely when the ring \(R\) is perfect.

In the paper under review, the author presents a construction of the mixed characteristic affine Grassmannian as a “geometric object”. The key point of Zhu’s approach is to consider the above-mentioned functors as functors on the category of perfect \(k\)-algebras, rather than the category of all \(k\)-algebras. Restricting the functor attached to a scheme to this subcategory corresponds to passing to the perfection of the scheme, i.e., replacing the coordinate ring \(A\) of an affine scheme by its perfect hull \(A^{p^{-\infty}}\), the colimit over \(A\rightarrow A \rightarrow A \rightarrow\cdots\), where all maps are \(x\mapsto x^p\). In this “perfect” setting, the author is able to show that the affine Grassmann functor is representable by the perfect analogue of an algebraic space.

The construction proceeds by first handling the case of \(GL_n\). In this case, he shows that for fixed \(N\) and sufficiently large \(h\), the functor \[ \overline{\text{Gr}}_{N,h}(R) = \{ (\Lambda, \varepsilon);\;\Lambda \subseteq W(R)^n\;\text{a lattice}, \bigwedge^n\Lambda = p^N W(R), \varepsilon: W_h(R)^n \rightarrow \Lambda/p^h\Lambda\;\text{an isom.} \} \] is representable by a perfect scheme. Here \(W_h(R)\) denotes the ring of truncated Witt vectors. Passing to a suitable quotient, one can get rid of the “level structure” \(\varepsilon\) and obtains a perfect algebraic space \(\overline{\text{Gr}}_N\). Taking the inductive limit over all \(N\), we obtain the mixed characteristic affine Grassmannian for \(GL_n\). Perfect schemes, or algebraic spaces, of positive dimension are never of finite type over \(k\). There exists, for each \(N\), an algebraic space of finite presentation over \(k\) whose perfection is \(\overline{\text{Gr}}_N\), a so-called model or deperfection of \(\overline{\text{Gr}}_N\). (However it is not clear whether these can be put into a compatible family.)

By embedding a general reductive group \(G\) suitably into some \(GL_n\), one constructs a mixed characteristic affine Grassmannian \(Gr_G\) for \(G\).

While one does not obtain in this way an ind-scheme as in the classical case, this definition is powerful enough to allow for a proof of the geometric Satake isomorphism (i.e., an equivalence of categories between equivariant perverse sheaves on the affine Grassmannian and representations of the Langlands dual group) in this setting.

In fact, while certain geometric concepts (e.g., tangent spaces) are unavailable for such perfect spaces, passing to the perfection of a scheme does not change its étale site, and hence étale cohomology is still available. One can thus define \(\ell\)-adic sheaves, the \(\ell\)-adic derived category and the notion of (equivariant) perverse sheaf on a perfect scheme or perfect algebraic space which arises as the perfection of a proper scheme or proper algebraic space.

One obtains the hypercohomology functor \[ H^*(\text{Gr}_G, -):\text{Perv}_{L^+G}(\text{Gr}_G) \rightarrow \text{Vect}_{\overline{\mathbb Q}_\ell}, \] from the category of \(L^+G\)-equivariant perverse sheaves on \(Gr_G\) to the category of \(\overline{\mathbb Q}_\ell\)-vector spaces, which can be endowed with a canonical monoidal structure (even though the “classical” methods do not work directly in this setting). The author then proves:

Theorem. The monoidal functor \(H^*\) induces an equivalence between \(\text{Perv}_{L^+G}(\text{Gr}_G)\) and the category \(\text{Rep}_{\overline{\mathbb Q}_\ell}(\widehat{G})\) of finite-dimensional representations of the dual group \(\widehat{G}\) of \(G\) over \(\overline{\mathbb Q}_\ell\).

In particular, there exists a commutativity constraint on \(\text{Perv}_{L^+G}(\text{Gr}_G)\) such that \(H^*\) is a tensor functor. To construct it, the author uses a version of the classical “Gelfand trick”. To show that the result has the desired properties, he proves that those properties follow from a certain numerical result on the affine Hecke algebra. This numerical property is known by work of G. Lusztig and Z. Yun [J. Ramanujan Math. Soc. 28A, 311–340 (2013; Zbl 1295.20046)] based on the equal-characteristic version of the geometric Satake isomorphism. Since the affine Hecke algebra is the same in both cases, this yields the desired conclusion.

Finally, the author gives some applications of his theory to affine Deligne-Lusztig varieties and Rapoport-Zink spaces, certain moduli spaces of \(p\)-divisible groups which are closely related to the reductions of Shimura varieties. The construction of the mixed-characteristic affine Grassmannian provides a way to define an a priori geometric structure on affine Deligne-Lusztig varieties in this setting, an important piece of the theory which had been missing so far and which will have many applications. See for instance [L. Xiao and X. Zhu, “Cycles on Shimura varieties via geometric Satake”, Preprint, arXiv:1707.05700] for further work by L. Xiao and the author in this direction.

There are two appendices. Appendix A (Generalities on perfect schemes) contains foundational material on perfect schemes, perfect algebraic spaces, and \(\ell\)-adic sheaves on them.

In Appendix B, a few more topics concerning the mixed characteristic affine Grassmannian are discussed, notably the question whether there is a determinant line bundle as in the classical case. It is stated as a conjecture that such a line bundle exists, and that it induces embeddings of suitable finite presentation models of \(\overline{\text{Gr}}_N\) into projective space, so that \(\overline{\text{Gr}}_N\) is the perfection of a projective variety and in particular a perfect scheme, not merely a perfect algebraic space. These conjectures have in the meantime been proved by B. Bhatt and P. Scholze [Invent. Math. 209, No. 2, 329–423 (2017; Zbl 1397.14064)].

Reviewer: Ulrich Görtz (Essen)

### MSC:

14G35 | Modular and Shimura varieties |

11G18 | Arithmetic aspects of modular and Shimura varieties |

14M15 | Grassmannians, Schubert varieties, flag manifolds |