Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras

Shalika germs for sl(n) and sp(2n) are motivic

We prove that Shalika germs on the Lie algebras sl(n) and sp(2n) belong to the class of so-called `motivic functions' defined by means of a first-order language of logic. We also prove, for these Lie algebras, a uniform bound of the form q^a (where q is the cardinality of the residue field) for the normalized Shalika germs. Our proof of the bound uses the theorem of Harish-Chandra that normalized Shalika germs are bounded, and a model-theoretic statement for uniform bounds of motivic functions from Appendix B to [arXiv:1208.1945].

A formula for certain Shalika germs of ramified unitary groups

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 …

Computations of orbital integrals and Shalika germs

For a reductive group $G$ over a non-archimedean local field, with some assumptions on (residue) characteristic we give an method to compute certain orbital integrals using a method close to that of Goresky-Kottiwitz-MacPherson but in a different language. These orbital integrals allow us to compute the Shalika germs at some ``very elliptic'' elements in terms of number of rational points on some quasi-finite covers of the Hessenberg varieties of GKM, which are subvarieties of (partial) flag varieties. Such values of Shalika germs determine the Harish-Chandra local character expansions of the …

Inductive structure of Shalika germs and affine Springer fibers

This article has two parallel perspectives: to demonstrate an inductive structure of Shalika germs, and to show an analogous inductive structure for affine Springer fibers. More precisely, we give an algorithm to compute arbitrary Shalika germs (resp. affine Springer fibers up to stratification) in terms of three ingredients: Shalika germs (resp. affine Springer fibers) for twisted Levi subgroups, a finite list of combinatorial objects, and the numbers of rational points on varieties over the residue field (resp. varieties themselves) among an explicit finite list of such. We also discuss some…

任意特征下 SL(2) 的芽展开

Germ expansion for SL(2) in arbitrary characteristics

本文研究任意特征局部域上 SL(2) 的轨道积分芽展开，指出 p=2 时 Shalika 芽展开失效，并利用内窥展开构造了新的芽展开，进而提出任意群的猜想。

On Shalika germs

Let $G$ be a reductive group over a local field $F$ satisfying the assumptions of \cite{Deb1}, $G_{reg}\subset G$ the subset of regular elements. Let $T\subset G$ be a maximal torus. We write $T_{reg}=T\cap G_{reg}$. Let $dg ,dt$ be Haar measures on $G$ and $T$. They define an invariant measure $dg/dt$ on $G/T$. Let $\mathcal{H}$ be the space of complex valued locally constant functions on $G$ with compact support. For any $f\in \mathcal{H} ,t\in T_{reg}$ we define $I_t(f)=\int_{G/T}f(\bar gt\bar g^{-1})dg/dt$. Let $P$ be the set of conjugacy classes of unipotent elements in $G$. For any $Ω\in…