Extending torsors on the punctured Spec(A inf)

We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine’s ring A inf {A_{\mathrm{inf}}} , extend to the whole spectrum. Using descent we can extend a similar result for the ring 𝔖 {\mathfrak{S}} of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the B dR + {B^{+}_{\mathrm{dR}}} -affine Grassmannian to the Witt vector affine flag variety.

Steenrod operators, the Coulomb branch and the Frobenius twist

We observe a fundamental relationship between Steenrod operations and the Artin–Schreier morphism. We use Steenrod's construction, together with some new geometry related to the affine Grassmannian, to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the $K$ -theoretic version of the quantum Coulomb branch. At the end of the paper, we investigate what our ideas produce on the categorical level. We find that they yield, after a little fiddling, a construction which corresponds, under the geometric Satake equivalence, to the Frobenius twist functor for representations of the Langlands dual group. We also describe the unfiddled answer, conditional on a conjectural ‘modular derived Satake’, and, though it is more complicated to state, it is in our opinion just as neat and even more compelling.

Mirabolic Satake equivalence and supergroups

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of $\operatorname{GL}(N-1,{\mathbb {C}}[\![t]\!])$ -equivariant perverse sheaves on the affine Grassmannian of $\operatorname{GL}_N$ . We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

Parity sheaves and Smith theory

Let p be a prime number and let X be a complex algebraic variety with an action of ℤ / p ⁢ ℤ {\mathbb{Z}/p\mathbb{Z}} . We develop the theory of parity complexes in a certain 2-periodic localization of the equivariant constructible derived category D ℤ / p ⁢ ℤ b ⁢ ( X , ℤ p ) {D^{b}_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_{p})} . Under certain assumptions, we use this to define a functor from the category of parity sheaves on X to the category of parity sheaves on the fixed-point locus X ℤ / p ⁢ ℤ {X^{\mathbb{Z}/p\mathbb{Z}}} . This may be thought of as a categorification of Smith theory. When X is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.

Tempered D-modules and Borel–Moore homology vanishing

We characterize the tempered part of the automorphic Langlands category $$\mathfrak {D}({\text {Bun}}_G)$$ D ( Bun G ) using the geometry of the big cell in the affine Grassmannian. We deduce that, for G non-abelian, tempered D-modules have no de Rham cohomology with compact support. The latter fact boils down to a concrete statement, which we prove using the Ran space and some explicit t-structure estimates: for G non-abelian and $$\Sigma $$ Σ a smooth affine curve, the Borel–Moore homology of the indscheme $${\text {Maps}}(\Sigma ,G)$$ Maps ( Σ , G ) vanishes.

The Equations Defining Affine Grassmannians in Type A and a Conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called shuffles, and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps; first, we reinterpret the shuffle equations in terms of Frobenius twists of symmetric functions. Using this, we reduce to a finite-dimensional problem, which we solve. For the 2nd step, we introduce a finite-dimensional analogue of the affine Grassmannian of $SL_n$, which we conjecture to be precisely the reduced subscheme of a finite-dimensional Grassmannian consisting of subspaces invariant under a nilpotent operator.

Families of affine Grassmannians

This chapter studies families of affine Grassmannians. In the geometric case, if X is a smooth curve over a field k, Beilinson-Drinfeld defined a family of affine Grassmannians whose fiber parametrizes G-torsors on X. If one fixes a coordinate at x, this gets identified with the affine Grassmannian considered previously. Over fibers with distinct points xi, one gets a product of n copies of the affine Grassmannian, while over fibers with all points xi = x equal, one gets just one copy of the affine Grassmannian: This is possible as the affine Grassmannian is infinite-dimensional. However, sometimes it is useful to remember more information when the points collide. The chapter then discusses the convolution affine Grassmannian in the setting of the previous lecture.