Functorial transfer for reductive groups and central complexes

TL;DR

This paper establishes functorial transfer properties for certain $ ext{Weyl group}$-equivariant $ ext{l}$-adic complexes on tori and extends these results to categories on reductive groups, providing new geometric proofs for transfer formulas.

Contribution

It introduces functorial monoidal transfer for central complexes and applies this to categories on reductive groups, connecting to the Braverman-Kazhdan conjecture.

Findings

Functorial monoidal transfer for central complexes established.

Transfer properties for bi-Whittaker and vanishing complexes on reductive groups proven.

New geometric proof of Laumon-Letellier's transfer formula provided.

Abstract

A class of Weyl group equivariant $\ell$-adic complexes on a torus, called the central complexes, was introduced and studied in our previous work on Braverman-Kazhdan conjecture. In this note we show that the category of central complexes admits functorial monoidal transfers with respect to morphisms between the dual groups. Combining with the work of Bezrukavnikov-Deshpande, we show that the $\ell$-adic bi-Whittaker categories (resp. the category of vanishing complexes, the category of stable complexes) on reductive groups admit functorial transfers. As an application, we give a new geometric proof of Laumon-Letellier's fromula for transfer maps of stable functions on finite reductive groups.