Exponential motives on the affine Grassmannian

TL;DR

This paper introduces exponential motives on prestacks with G_a-actions, establishes foundational results, and proves a motivic Casselman-Shalika equivalence linking geometric and algebraic structures in the Langlands program.

Contribution

It develops the theory of exponential motives, compares them with Whittaker motives, and proves a new motivic Casselman-Shalika equivalence for affine Grassmannians.

Findings

Established foundational results for exponential motives on affine flag varieties.

Proved a motivic Casselman-Shalika equivalence relating motives to ind-coherent sheaves.

Provided a new construction of the Whittaker module for the spherical Hecke algebra.

Abstract

We develop a notion of exponential motives on general prestacks equipped with a $\mathbf{G}_a$-action, and compare them with Whittaker motives via Gaitsgory's Kirillov model. We then establish foundational results for exponential motives on affine flag varieties concerning Tate motives and t-structures. We use this to prove a motivic Casselman-Shalika equivalence, relating exponential Tate motives on the affine Grassmannian to ind-coherent sheaves on the classifying stack of the Langlands dual group. The decategorification of this equivalence provides a new construction of the Whittaker module for the spherical Hecke algebra which works for arbitrary coefficients, including a generic version.