Base change fundamental lemma for Bernstein centers of principal series blocks

TL;DR

This paper establishes a base change homomorphism for the Bernstein center of principal series blocks in unramified p-adic groups, supporting conjectures on twisted endoscopy and test functions for Shimura varieties.

Contribution

Introduces a new base change homomorphism for Bernstein centers and proves associated functions are related, advancing understanding of endoscopic transfer in p-adic groups.

Findings

Proves the base change homomorphism relates functions in Bernstein centers.

Provides evidence for conjectures on twisted endoscopic transfer.

Supports applications to test functions in Shimura varieties.

Abstract

Let $G$ be an unramified group over a $p$-adic field $F$. This article introduces a base change homomorphism for the Bernstein center of a principal series block, and proves that two functions related by this base change homomorphism are associated. This result provides new evidence for the conjecture on twisted endoscopic transfer of the stable Bernstein center proposed by T. Haines, which will be applied to a general conjecture on test functions for Shimura varieties due to R. Kottwitz and T. Haines.