Shadows in the Wild - Folded Galleries and Their Applications

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.

Tate motives on Witt vector affine flag varieties

Relying on recent advances in the theory of motives we develop a general formalism for derived categories of motives with $${\mathbf{Q}}$$ Q -coefficients on perfect $$\infty $$ ∞ -prestacks. We construct Grothendieck’s six functors for motives over perfect (ind-)schemes perfectly of finite presentation. Following ideas of Soergel–Wendt, this is used to study basic properties of stratified Tate motives on Witt vector partial affine flag varieties. As an application we give a motivic refinement of Zhu’s geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.

Riesz potentials and orthogonal radon transforms on affine Grassmannians

We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j,k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j-dimensional affine planes in ℝ n to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. The main results include sharp existence conditions of 𝓡 j,k f on L p -functions, Fuglede type formulas connecting 𝓡 j,k with Radon-John k-plane transforms and Riesz potentials, and explicit inversion formulas for 𝓡 j,k f under the assumption that f belongs to the range of the j-plane transform. The method extends to another class of Radon transforms defined on affine Grassmannians by inclusion.

Residues on affine Grassmannians

Given a linear group G over a field k, we define a notion of index and residue of an element g ∈ G ⁢ ( k ⁢ ( ( t ) ) ) {g\in G(k(\kern-1.707165pt(t)\kern-1.707165pt))} . The index r ⁢ ( g ) {r(g)} is a rational number and the residue a group homomorphism res ( g ) : 𝔾 a ⁢ or ⁢ 𝔾 m → G {\mathop{\rm res}\nolimits(g):\mathbb{G}_{a}\text{ or }\mathbb{G}_{m}\to G} . This provides an alternative proof of Gabber’s theorem stating that G has no subgroups isomorphic to 𝔾 a {\mathbb{G}_{a}} or 𝔾 m {\mathbb{G}_{m}} iff G ⁢ ( k ⁢ [ [ t ] ] ) = G ⁢ ( k ⁢ ( ( t ) ) ) {G(k[\kern-1.13811pt[t]\kern-1.13811pt])=G(k(\kern-1.707165pt(t)\kern-1.707165% pt))} . In the case of a reductive group, we offer an explicit connection with the theory of affine Grassmannians.

Beilinson–Drinfeld Schubert varieties and global Demazure modules

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson–Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra.

On a conjecture of Pappas and Rapoport about the standard local model for GL_d

In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

The test function conjecture for local models of Weil-restricted groups

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $p\geqslant 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.