We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on .

We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.

We correct an error in our paper ‘Connected components of affine Deligne–Lusztig varieties in mixed characteristic’.

An important result of Arkhipov–Bezrukavnikov–Ginzburg relates constructible sheaves on the affine Grassmannian to coherent sheaves on the dual Springer resolution. In this paper, we prove a positive-characteristic analogue of this statement, using the framework of ‘mixed modular sheaves’ recently developed by the first author and Riche. As an application, we deduce a relationship between parity sheaves on the affine Grassmannian and Bezrukavnikov’s ‘exotic t-structure’ on the Springer resolution.

For $T$ a compact torus and $E_{T}^{*}$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_{T}^{*}$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $H_{T}^{*}\left( X \right)$ as an $H_{T}^{*}\left( pt \right)$-module when $X$ is a direct limit of smooth complex projective ${{T}_{\mathbb{C}}}$-varieties. We perform this computation on the affine Grassmannian of a complex semisimple group.

We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

For an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$, we consider the equivariant homology $H^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ and K-theory $K^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$. They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of $\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of $G(\mathbb{C}[[{\rm t}]])$-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of GrG is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin–Loktev fusion product of $G(\mathbb{C}[[{\rm t}]])$-modules.