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A model for the E3 fusion-convolution product of constructible sheaves on the affine Grassmannian

Guglielmo Nocera

TL;DR

This work introduces a derived, higher-homotopical enhancement of the geometric Satake setting by constructing a left $\mathbb{E}_3$-monoidal structure on the spherical Hecke category on the affine Grassmannian. The authors build an intrinsically automorphic convolution over the Beilinson–Drinfeld Ran space, utilize factorization via topological and stratified-exodromy formalism, and pass to the analytic/topological realm to obtain an $\mathbb{E}_3$-algebra structure on the category of constructible sheaves with coefficients in a presentable stable ∞-category $\mathcal{E}$. The main result identifies ${\textup{Sph}}(G;\mathcal{E})^{\otimes}$ as an $\mathbb{E}_3$-algebra object with underlying category ${\textup{Cons}}_{G_{\mathcal{O}}^{an}}(\textup{Gr}^{an};\mathcal{E})$, and it yields a small, perverse-compatible version, along with left $t$-exactness when $\mathcal{E}=\mathrm{Mod}_R$. Specializing to a single point recovers a pointwise $\mathbb{E}_3$-structure, aligning with the fusion-convolution picture and supporting a refined automorphic Langlands perspective. Collectively, the results deepen the link between automorphic geometry and higher-algebraic structures, offering new avenues for the geometric Langlands program and the study of centers in derived representation theory. The framework also accommodates renormalization and D-module interpretations in suitable coefficient regimes, reflecting the broad applicability of the construction.

Abstract

Let $G$ be a complex reductive group. The spherical Hecke category of $G$ can be presented as the category of $G_{\mathcal O}$-equivariant constructible sheaves on the affine Grassmannian $\mathrm{Gr}_G$. This category admits a convolution product, extending the convolution product of equivariant perverse sheaves. In this paper, we upgrade the mentioned convolution product to a left t-exact $\mathbb E_3$-monoidal structure in $\infty$-categories. The construction is intrinsic to the automorphic side. Our main tools are the Beilinson--Drinfeld Grassmannian, Lurie's characterization of $\mathbb E_k$-algebras via the topological Ran space, the homotopy theory of stratified spaces, and the formalism of correspondences.

A model for the E3 fusion-convolution product of constructible sheaves on the affine Grassmannian

TL;DR

This work introduces a derived, higher-homotopical enhancement of the geometric Satake setting by constructing a left -monoidal structure on the spherical Hecke category on the affine Grassmannian. The authors build an intrinsically automorphic convolution over the Beilinson–Drinfeld Ran space, utilize factorization via topological and stratified-exodromy formalism, and pass to the analytic/topological realm to obtain an -algebra structure on the category of constructible sheaves with coefficients in a presentable stable ∞-category . The main result identifies as an -algebra object with underlying category , and it yields a small, perverse-compatible version, along with left -exactness when . Specializing to a single point recovers a pointwise -structure, aligning with the fusion-convolution picture and supporting a refined automorphic Langlands perspective. Collectively, the results deepen the link between automorphic geometry and higher-algebraic structures, offering new avenues for the geometric Langlands program and the study of centers in derived representation theory. The framework also accommodates renormalization and D-module interpretations in suitable coefficient regimes, reflecting the broad applicability of the construction.

Abstract

Let be a complex reductive group. The spherical Hecke category of can be presented as the category of -equivariant constructible sheaves on the affine Grassmannian . This category admits a convolution product, extending the convolution product of equivariant perverse sheaves. In this paper, we upgrade the mentioned convolution product to a left t-exact -monoidal structure in -categories. The construction is intrinsic to the automorphic side. Our main tools are the Beilinson--Drinfeld Grassmannian, Lurie's characterization of -algebras via the topological Ran space, the homotopy theory of stratified spaces, and the formalism of correspondences.

Paper Structure

This paper contains 27 sections, 69 theorems, 346 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Motivation: the Derived Satake Theorem
  4. Outline of the work
  5. Convolution over the Ran space
  6. The Beilinson--Drinfeld setting
  7. The Hecke stack over the Ran space
  8. The convolution Beilinson-Drinfeld Grassmannian
  9. The BD-convolution diagram as a 2-Segal object
  10. Topological factorization of the Hecke stack
  11. Consequences of analytification
  12. Setup for taking constructible sheaves
  13. The E3-structure
  14. Spherical Hecke category over the Ran space
  15. Specialization to a point
  16. ...and 12 more sections

Key Result

Theorem 1.4

Let $G$ be a complex reductive group and $\mathcal{E}$ a symmetric monoidal presentable stable $\infty$-category. Then there exists an object whose underlying $\infty$-category is the unbounded derived $\infty$-category of topological $G_\mathcal{O}^\textup{an}$-equivariant constructible sheaves over $\textup{Gr}^\textup{an}$, with coefficients in $\mathcal{E}$.

Theorems & Definitions (238)

  • Definition 1.1: Affine Grassmannian
  • Remark 1.2
  • Theorem 1.4: Main result, \ref{['final-theorem']}
  • Corollary 1.5: \ref{['corollary-small-subcategory']}, \ref{['t-exactness-of-convolution']}
  • Remark 1.6
  • Corollary 1.7: \ref{['corollary-Ind-completion']}
  • Theorem 1.8: Geometric Satake Equivalence, MV
  • Theorem 1.9: Derived Satake Theorem, Bezrukavnikov-Finkelberg
  • Remark 1.10
  • Proposition 1.11
  • ...and 228 more