A Formula for Some Shalika Germs

Abstract

In this article, for nilpotent orbits in (the Lie algebras of) ramified quasi-split unitary groups with two Jordan blocks, we give the values of 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 values we obtain can be expressed in terms of Frobenius eigenvalues on the l-adic H^1 of the curves, generalizing previous result of Hales on stable subregular Shalika germs. Using the Shalika germ formulas, we obtain some new results on stability and endoscopic transfer of nilpotent orbital integrals.