On the excursion algebra

The excursion algebra associated to a scheme X over a finite field and a reductive group G is the algebra of global functions on the stack of arithmetic G-local systems on X. When X is a curve, this algebra acts on the space of automorphic functions. In this paper we establish some basic properties of this algebra.

💡 Research Summary

The paper “On the Excursion Algebra” studies the algebra of global functions on the stack of arithmetic G‑local systems over a scheme X defined over a finite field 𝔽_q, denoted Exc(X,G). The authors work in a very general setting: X is any finite‑type scheme over 𝔽_q (not necessarily a curve) and G is a reductive group over the ℓ‑adic coefficient field e = ℚ_ℓ, with ℓ ≠ p = char 𝔽_q. The main goal is to understand the intrinsic algebraic properties of Exc(X,G) without reference to its action on automorphic functions, although the original motivation comes from V. Lafforgue’s construction of an action of Exc(W eil(X),G) on automorphic forms.

The paper is organized into five main parts.

1. Categories with an A¹‑action (Section 1).
   The authors develop a general categorical framework: a DG‑category C equipped with a G_m‑action gives rise to the fixed‑point category D = C^{G_m}. They prove (Theorem 1.1.7) that extending the G_m‑action to an action of the monoid A¹ (the affine line) is equivalent to providing a Z‑invariant filtration on D. This is a categorical analogue of the classical fact that a filtered vector space can be realized as the fiber at 1 of a quasi‑coherent sheaf on A¹/G_m. This machinery will later be applied to categories of sheaves on X.

2. Sheaf‑theoretic constructions (Section 2).
   The authors introduce several sheaf categories related to the Weil group of X. They define a pro‑algebraic group Z_{alg}^{wt} whose representations are finite‑dimensional vector spaces equipped with an automorphism whose eigenvalues are q‑Weil numbers. Its representation category Rep(Z_{alg}^{wt}) acts on the category of mixed Weil sheaves Shv_{Weil,wt}(X). By tensoring with the trivial vector space they obtain an equivalence \