Excursion algebra - Knowlpedia

Fix a global function field $F$ , a reductive group $G$ , a level, and coefficients in an $\ell$ -adic field. The excursion algebra $\mathcal B$ is the subalgebra of the endomorphism algebra of the cuspidal automorphic space generated by all excursion operators

S_{I,f,(\gamma_i)}

as $I$ , $f$ , and the Galois tuple vary.

Commutativity

The creation, coalescence, partial-Frobenius, and annihilation identities imply that the generators commute. Thus the cuspidal space has a canonical decomposition into generalized eigenspaces for characters $\nu:\mathcal B\to\overline{\mathbb Q}_\ell$ .

The use of generalized eigenspaces matters: the excursion algebra is finite dimensional on a fixed finite-level cuspidal space but is not known in this construction to be reduced.

Characters and parameters

Lafforgue's reconstruction theorem associates to every character $\nu$ a unique conjugacy class of continuous semisimple parameter from the absolute Galois group

\sigma_\nu: \operatorname{Gal}(\overline F/F) \longrightarrow \widehat G(\overline{\mathbb Q}_\ell)

such that

\nu(S_{I,f,(\gamma_i)}) = f((\sigma_\nu(\gamma_i))_{i\in I}).

For a nonsplit group, the parameter is valued in the appropriate $L$ -group.

Relation to Hecke operators

At an unramified place, a spherical Hecke operator is recovered as a particular excursion operator. Therefore the excursion decomposition refines the simultaneous Hecke eigenspace decomposition and determines the unramified Satake classes.

What it does not provide

The algebra labels canonical summands but does not by itself compute their dimensions or the full Arthur multiplicity formula. Different automorphic representations can lie in the same parameter summand.

References

Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” Proposition 11.7 and Theorem 11.11. arXiv.