[Paper Review] Stacks in Representation Theory. What is a continuous representation of an algebraic group ?

Joseph Bernstein|arXiv (Cornell University)|Oct 2, 2014

Advanced Algebra and Geometry3 references3 citations

TL;DR

This paper proposes a new framework in representation theory by interpreting representations of algebraic groups as sheaves on algebraic stacks, particularly the stack $B\mathcal{G} = \text{pt}/\mathcal{G}$, thereby extending the standard category of continuous representations to a larger category $\mathcal{M}(\mathcal{G}, F)$ that unifies irreducible representations across all pure inner forms of $\mathcal{G}$. The key result is that the irreducible objects in $\mathcal{M}(\mathcal{G}, F)$ are naturally in bijection with the disjoint union of irreducible representations of all pure inner forms $G_i$ of $G = \mathcal{G}(F)$.

ABSTRACT

In this note I introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group $G$ as a sheaf on some geometric object (a stack). This point of view necessarily leads to a conclusion that the standard approach to (continuous) representations of algebraic groups is ideologically inconsistent. I propose a way to modify this approach using sheaves on stacks. In new version I corrected some misprints and added an explanation how stack's approach is related to Vogan's picture of representations.

Motivation & Objective

• To address the philosophical inconsistency in the standard approach to continuous representations of algebraic groups over local or finite fields.
• To propose a geometric framework using algebraic stacks to better capture the intuitive structure of representations.
• To extend the category $Rep(G)$ of continuous representations to a larger category $\mathcal{M}(\mathcal{G}, F)$ that incorporates representations from all pure inner forms of $G$.
• To establish a natural equivalence between sheaves on the stack $B\mathcal{G}$ and equivariant sheaves on various $G_i$-spaces, thereby unifying the classification of irreducible representations.

Proposed method

• Model representations as sheaves on algebraic stacks, particularly $B\mathcal{G} = \text{pt}/\mathcal{G}$, using the site of schemes over a field $F$ with the étale or smooth topology.
• Define the category $Sh(B\mathcal{G})$ of sheaves on the stack $B\mathcal{G}$ as the category of functors from the groupoid $B\mathcal{G}(F)$ to the category of complex vector spaces.
• Use the descent property of stacks to relate $F$-points of a stack $\mathcal{X}$ to fixed points under Galois actions on $L$-points for finite Galois extensions $L/F$.
• Realize the category $Sh(\mathcal{X})$ for a quotient stack $\mathcal{X} = \mathcal{Z}/\mathcal{G}$ as a product $\prod_i Sh_{G_i}(Z_i)$, where $G_i = \mathcal{G}_i(F)$ and $\mathcal{G}_i$ are pure inner forms of $\mathcal{G}$.
• Leverage Hilbert's Theorem 90 to reduce the study of $\mathcal{M}(\mathcal{G}, F)$ to a single group $P \cong GL(n)$, allowing reduction to $P$-equivariant sheaves on a suitable space $W$.

Experimental results

Research questions

• RQ1 Why is the standard category $Rep(G)$ of continuous representations of $G = \mathcal{G}(F)$ philosophically inconsistent for algebraic groups?
• RQ2 How can representations of an algebraic group $\mathcal{G}$ over a local or finite field $F$ be naturally unified across all pure inner forms of $G$?
• RQ3 What is the correct geometric object (stack) on which to define representations so that irreducible representations arise naturally from a disjoint union over inner forms?
• RQ4 How can sheaves on an algebraic stack $\mathcal{X} = \mathcal{Z}/\mathcal{G}$ be described in terms of equivariant sheaves on $F$-points of torsors?
• RQ5 What role does Galois descent play in relating $F$-points of a stack to its $L$-points for Galois extensions $L/F$?

Key findings

• The category $\mathcal{M}(\mathcal{G}, F)$ of stacky $G$-modules is naturally equivalent to the category $Sh(B\mathcal{G})$ of sheaves on the stack $B\mathcal{G}$, providing a geometric interpretation of representations.
• The category $Sh(B\mathcal{G})$ is equivalent to the product $\prod_i Sh_{G_i}(Z_i)$, where $G_i$ are the $F$-points of pure inner forms $\mathcal{G}_i$ of $\mathcal{G}$, and $Z_i$ are associated $G_i$-spaces.
• The set of irreducible objects in $\mathcal{M}(\mathcal{G}, F)$ is naturally isomorphic to the disjoint union $\coprod_i \text{Irr}(G_i)$, resolving the need to treat representations of different inner forms separately.
• For the case $\mathcal{Z} = \text{pt}$, the irreducible stacky $G$-modules correspond exactly to the disjoint union of irreducible representations of all pure inner forms of $G$.
• The construction allows reduction to $GL(n)$-equivariant sheaves via embedding $\mathcal{G} \hookrightarrow \mathcal{P} \cong GL(n)$, simplifying computations via Hilbert's Theorem 90.