Co-rank $1$ Arithmetic Siegel--Weil II: Local Archimedean

TL;DR

This paper proves the key Archimedean local theorem for the arithmetic Siegel--Weil formula in co-rank 1, relating Green currents and derivatives of local Whittaker functions, advancing the understanding of special cycles on unitary Shimura varieties.

Contribution

It formulates and proves the Archimedean local theorem crucial for the arithmetic Siegel--Weil formula in the co-rank 1 case, introducing a new limiting method for Archimedean analysis.

Findings

Established the Archimedean local theorem for the arithmetic Siegel--Weil formula.

Proved an Archimedean local arithmetic Siegel--Weil formula relating Green currents and derivatives of Whittaker functions.

Developed a new limiting method parallel to non-Archimedean strategies.

Abstract

This is the second in a sequence of four papers, where we prove the arithmetic Siegel--Weil formula in co-rank $1$ for Kudla--Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. Our arithmetic Siegel--Weil formula implies that degrees of Kudla--Rapoport arithmetic special $1$-cycles are encoded in the first derivatives of unitary Eisenstein series Fourier coefficients. In this paper, we formulate and prove the key Archimedean local theorem. In the case of purely Archimedean intersection numbers, we also prove an Archimedean local arithmetic Siegel--Weil formula, relating Green currents of arbitrary degree and off-central derivatives of local Whittaker functions. The crucial input is a new limiting method, which is structurally parallel to our strategy at non-Archimedean places.