Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Base change fundamental lemma for Bernstein centers of principal series blocks

Shenghao Li

TL;DR

This work establishes a base change homomorphism $b_r$ between the Bernstein centers of the $r$-th unramified extension $G_r$ and $G$ for principal blocks, proving that any $ au\, ext{in}\, ext{Z}(G_r, ho_r)$ is associated with $b_r( au)$. It constructs explicit principal-block types by merging Roche’s approach with Kim–Yu data, and develops descent formulas and Labesse’s elementary functions to reduce the comparison to strongly regular elliptic elements. The results connect the base change on the (stable) Bernstein centers and verify compatibility with the stable center base change $b_r'$, providing evidence for Haines’s twisted endoscopic transfer conjecture and informing test-function conjectures in Shimura varieties. The methods introduce a concrete, depth-agnostic construction of principal-block types for unramified groups and a robust descent mechanism to handle deeper levels beyond depth-zero or parahoric cases.

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.

Base change fundamental lemma for Bernstein centers of principal series blocks

TL;DR

This work establishes a base change homomorphism between the Bernstein centers of the -th unramified extension and for principal blocks, proving that any is associated with . It constructs explicit principal-block types by merging Roche’s approach with Kim–Yu data, and develops descent formulas and Labesse’s elementary functions to reduce the comparison to strongly regular elliptic elements. The results connect the base change on the (stable) Bernstein centers and verify compatibility with the stable center base change , providing evidence for Haines’s twisted endoscopic transfer conjecture and informing test-function conjectures in Shimura varieties. The methods introduce a concrete, depth-agnostic construction of principal-block types for unramified groups and a robust descent mechanism to handle deeper levels beyond depth-zero or parahoric cases.

Abstract

Let be an unramified group over a -adic field . 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.
Paper Structure (38 sections, 46 theorems, 151 equations)

This paper contains 38 sections, 46 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Basic notations
  3. Bernstein blocks and types
  4. The construction of types for principal blocks
  5. Main result
  6. Construction of types following Kim-Yu
  7. Construction of G-datum
  8. The twisted Levi sequence $(\vec{G},M^d)$
  9. The character sequence
  10. The depth zero datum and the generic embeddings of buildings
  11. The concrete description of the type
  12. Bernstein center, stable Bernstein center and base change homomorphism
  13. The variety structure on a principal block
  14. Bernstein center
  15. Stable Bernstein center
  16. ...and 23 more sections

Key Result

Theorem 1.2

If $\phi\in \mathcal{H}(G_r,K_r)$, then $\phi$ and $b\phi$ are associated.

Theorems & Definitions (91)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1.3 : Haines
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 3.1
  • Theorem 3.2 : roche1998types
  • Definition 3.3
  • Remark 3.4
  • ...and 81 more