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The affine Grassmannian as a presheaf quotient

Kestutis Cesnavicius

TL;DR

The paper investigates when the affine Grassmannian $\mathrm{Gr}_G$ can be defined without resorting to fpqc or étale sheafifications of the presheaf quotient $LG/L^+G$. By leveraging the geometry of torsors on $\mathbb{P}^1_A$ and the moduli stack $\mathrm{Bun}_G$, it proves that for many reductive group schemes $\mathcal{G}$ (notably when descended from a reductive group over $A$) the Zariski sheafification suffices, giving $\mathrm{Gr}_{\mathcal{G}} \cong (L\mathcal{G}/L^+\mathcal{G})_{\mathrm{Zar}}$. In the totally isotropic case, the presheaf quotient already captures $\mathrm{Gr}_G$ with no sheafification, and even allows realization via polynomial loops $L_{\mathrm{poly}}G/L^+_{\mathrm{poly}}G$. The key technical engine is a suite of descent and patching results for $G$-torsors on $\mathbb{P}^1_A$ and their interaction with $\mathrm{Bun}_G$, extending the understanding of loop groups, torsors, and moduli in the relative setting. These results simplify the definition of the affine Grassmannian in many cases and illuminate the precise role of different topologies in its construction, with potential impact on geometric Langlands and related moduli problems.

Abstract

For a reductive group $G$ over a ring $A$, its affine Grassmannian $\mathrm{Gr}_G$ plays important roles in a wide range of subjects and is typically defined as the étale sheafification of the presheaf quotient $LG/L^+G$ of the loop group $LG$ by its positive loop subgroup $L^+G$. We show that the Zariski sheafification gives the same result. Moreover, for totally isotropic $G$ (for instance, for quasi-split $G$), we show that no sheafification is needed at all: $\mathrm{Gr}_G$ is already the presheaf quotient $LG/L^+G$, which seems new already in the classical case of $G$ over $\mathbb{C}$. For totally isotropic $G$, we also show that the affine Grassmannian may be formed using polynomial loops. We deduce all of these results from the study of $G$-torsors on $\mathbb{P}^1_A$ that is ultimately built on the geometry of $\mathrm{Bun}_G$.

The affine Grassmannian as a presheaf quotient

TL;DR

The paper investigates when the affine Grassmannian can be defined without resorting to fpqc or étale sheafifications of the presheaf quotient . By leveraging the geometry of torsors on and the moduli stack , it proves that for many reductive group schemes (notably when descended from a reductive group over ) the Zariski sheafification suffices, giving . In the totally isotropic case, the presheaf quotient already captures with no sheafification, and even allows realization via polynomial loops . The key technical engine is a suite of descent and patching results for -torsors on and their interaction with , extending the understanding of loop groups, torsors, and moduli in the relative setting. These results simplify the definition of the affine Grassmannian in many cases and illuminate the precise role of different topologies in its construction, with potential impact on geometric Langlands and related moduli problems.

Abstract

For a reductive group over a ring , its affine Grassmannian plays important roles in a wide range of subjects and is typically defined as the étale sheafification of the presheaf quotient of the loop group by its positive loop subgroup . We show that the Zariski sheafification gives the same result. Moreover, for totally isotropic (for instance, for quasi-split ), we show that no sheafification is needed at all: is already the presheaf quotient , which seems new already in the classical case of over . For totally isotropic , we also show that the affine Grassmannian may be formed using polynomial loops. We deduce all of these results from the study of -torsors on that is ultimately built on the geometry of .
Paper Structure (3 sections, 11 theorems, 27 equations)

This paper contains 3 sections, 11 theorems, 27 equations.

Table of Contents

  1. The affine Grassmannian and its modular description
  2. The affine Grassmannian as a Zariski quotient
  3. The affine Grassmannian as a presheaf quotient

Key Result

Proposition 3

For a scheme $\mathcal{X}$ over $A(\!( t )\!)$, or over $A\{t\}[\frac{1}{t}]$, or over $(A[t]_{1 + tA[t]})[\frac{1}{t}]$, or over $A[t, t^{-1}]$ (resp., over $A\llbracket t \rrbracket$, or over $A\{t\}$, or over $A[t]_{1 + tA[t]}$, or over $A[t]$) as in §pp:loops such that every quasi-compact open o

Theorems & Definitions (26)

  • Proposition 3
  • proof
  • Remark 4
  • Corollary 5
  • proof
  • Proposition 7
  • proof
  • Theorem 8: totally-isotropic*Theorem 3.6
  • Proposition 9
  • proof
  • ...and 16 more