Related papers: Reducedness of affine Grassmannian slices in type …
We study spherical Schubert varieties in the affine Grassmannian. These Schubert varieties have a natural conjectural modular description due to Finkelberg-Mirkovi\'c. This modular description is easily seen to be set-theoretically correct,…
Given a simply-connected simple algebraic group $G$, we determine the tangent space of any Finkelberg-Mirkovi\'c Schubert scheme at the base point of the affine Grassmannian of $G$. As a consequence, we exhibit non-reduced…
We give a simple and uniform proof of a conjecture of Haines-Richarz characterizing the smooth locus of Schubert varieties in twisted affine Grassmannians. Our method is elementary and avoids any representation theoretic techniques, instead…
We consider a certain class of Schubert varieties of the affine Grassmannian of type A. By embedding a Schubert variety into a finite-dimensional Grassmannian, we construct an explicit basis of sections of the basic line bundle by…
We classify all normal Schubert varieties in the affine Grassmannian of a semisimple group over an arbitrary field with special attention to small positive characteristic. The proof is elementary and relies on tangent space calculations for…
We prove a conjecture of Frenkel, Gaitsgory, Kazhdan and Vilonen, related to Fourier coefficients of spherical perverse sheaves on the affine Grassmannian associated to a a split reductive group. Our proof is an extension of the proof given…
This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the…
The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Pl\"ucker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part…
We show that every smooth Schubert variety of affine type $\tilde{A}$ is an iterated fibre bundle of Grassmannians, extending an analogous result by Ryan and Wolper for Schubert varieties of finite type $A$. As a consequence, we finish a…
Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical…
We give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the…
We realise the Bott-Samelson resolutions of type A Schubert varieties as quiver Grassmannians. In order to explicitly describe this isomorphism, we introduce the notion of a \textit{geometrically compatible} decomposition for any…
We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians --- these are subalgebras of the Yangian we introduce which…
In this paper, a description of the set-theoretical defining equations of symplectic (type C) Grassmannian/flag/Schubert varieties in corresponding (type A) algebraic varieties is given as linear polynomials in Pl$\ddot{u}$cker coordinates,…
This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also…
In this paper, we introduce Schubert decompositions for quiver Grassmannians and investigate example classes of quiver Grassmannians with a Schubert decomposition into affine spaces. The main theorem puts the cells of a Schubert…
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…
The Pappas-Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to the line bundles on spherical Schubert varieties in the…
A general framework for the reduction of the equations defining classes of spherical varieties to (maybe infinite dimensional) grassmannians is proposed. This is applied to model varieties of type A, B and C; in particular a standard…
Let $L$ be a Levi subgroup of $GL_N$ which acts by left multiplication on a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. We say that $X(w)$ is a spherical Schubert variety if $X(w)$ is a spherical variety for the action of $L$. In…