A formula for certain Shalika germs of ramified unitary groups

TL;DR

This paper derives explicit formulas for certain Shalika germs in ramified unitary groups, linking them to hyperelliptic curves and Frobenius eigenvalues, with implications for stability, transfer, and representation theory.

Contribution

It provides the first closed-form expressions for these Shalika germs in ramified cases, extending previous work and connecting them to algebraic geometry and representation theory.

Findings

Formulas expressed via Frobenius eigenvalues on hyperelliptic curves.

Implications for stability and endoscopic transfer of orbital integrals.

Connections to local character expansions and Whittaker models.

Abstract

In this article, for nilpotent orbits of ramified quasi-split unitary groups with two Jordan blocks, we give closed formulas for their Shalika germs at certain equi-valued elements with half-integral depth previously studied by Hales. These elements are parametrized by hyperelliptic curves defined over the residue field, and the numbers we obtain can be expressed in terms of Frobenius eigenvalues on the $\ell$-adic $H^1$ of the curves, generalizing previous result of Hales on stable subregular Shalika germs. These Shalika germ formulas imply new results on stability and endoscopic transfer of nilpotent orbital integrals of ramified unitary groups. We mention also how the same numbers appear in the local character expansion of specific supercuspidal representations and consequently dimensions of degenerate Whittaker models.