Horizontal Distribution Relations for Special Cycles on Unitary Shimura Varieties: Split Case

TL;DR

This paper investigates the local properties and distribution relations of special cycles on unitary Shimura varieties in the split case, extending previous work to higher-dimensional buildings and providing new formulas for their fields of definition.

Contribution

It establishes a local formula for the fields of definition of special cycles and proves a distribution relation between Galois and Hecke actions in the split case, advancing understanding of their arithmetic behavior.

Findings

Derived a local formula for fields of definition of special cycles

Proved a distribution relation linking Galois and Hecke actions

Extended combinatorial methods to higher-dimensional buildings

Abstract

We study the local behavior of special cycles on Shimura varieties for $\mathbf{U}(2, 1) \times \mathbf{U}(1, 1)$ in the setting of the Gan-Gross-Prasad conjectures at primes $\tau$ of the totally real field of definition of the unitary spaces which are split in the corresponding totally imaginary quadratic extension. We establish a local formula for their fields of definition, and prove a distribution relation between the Galois and Hecke actions on them. This complements work of \cite{jetchev:unitary} at inert primes, where the combinatorics of the formulas are reduced to calculations on the Bruhat--Tits trees, which in the split case must be replaced with higher-dimensional buildings.