Related papers: Inductive structure of Shalika germs and affine Sp…

Degenerating the action of the elliptic Hall algebra on the Fock space, we give a combinatorial formula for the Shalika germs of tamely ramified regular semisimple elements $\gamma$ of $GL_n$ over a nonarchimedean local field. As a…

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…

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…

The paper deals with two problems that seem to be of different nature: the local behavior of the weighted orbital integrals and the dependence of the affine Springer fibers on the root valuation datum of their defining elements. The…

This article constructs Shalika germs in the context of motivic integration, both for ordinary orbital integrals and kappa-orbital integrals. Based on transfer principles in motivic integration and on Waldspurger's endoscopic transfer of…

Let $F$ be a local field of characteristic $p$ and $G$ be a connected reductive group over $F$. Recall that Shalika's germ expansion of orbital integrals of regular semi-simple elements near the identity, when it exists, is a sum indexed by…

Let $\mathbf{G}$ be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $\gamma_0\in(\operatorname{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and…

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…

We study basic geometric properties of some group analogue of affine Springer fibers and compare with the classical Lie algebra affine Springer fibers. The main purpose is to formulate a conjecture that relates the number of irreducible…

Following Steinberg, we construct an adjoint quotient for the Vinberg semi-group and a section to this quotient. Then, after Ng\^o, we show the existence of a regular centralizer on it and use it to compute the affine Springer fibers for…

Let $G$ be a connected reductive group over an algebraically closed field $k$, and let $Fl$ be the affine flag variety of $G$. For every regular semisimple element $\gamma$ of $G(k((t)))$, the affine Springer fiber $Fl_{\gamma}$ can be…

Assuming a certain "purity" conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semi-simple element. We use this calculation to present a complex analog of…

We study the wave-front set of an element in a $p$-adic reductive Lie algebra (for $p\gg\operatorname{rank}$), namely the set of maximal nilpotent orbits appearing in its Shalika germ expansion. By adapting an algorithm of Waldspurger that…

Given a semisimple element in the loop Lie algebra of a reductive group, we construct a quasi-coherent sheaf on a partial resolution of the trigonometric commuting variety of the Langlands dual group. The construction uses affine Springer…

We prove a geometric local constancy theorem for affine Springer fibers in families of close local fields. Consequently, stable orbital integrals are locally constant in these families, and both the base change fundamental lemma and the…

We investigate the regularising properties of singular kernels at the level of germs, i.e. families of distributions indexed by points in $\mathbb{R}^d$. First we construct a suitable integration map which acts on general coherent germs.…

We explain an algorithm to calculate Arthur's weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of…

Assuming the purity conjecture for the affine Springer fibers which has been formulated by Goresky, Kottwitz and MacPherson, we prove a geometric analog of the fundamental lemma for unitary groups. Our approach is similar to the one of…

We establish dimension formulas for the Witt vector affine Springer fibers associated to a reductive group over a mixed characteristic local field, under the assumption that the group is essentially tamely ramified and the residue…

We determine a formula for the dimension of a family of affine Springer fibers associated to a symmetric space arising from the block diagonal embedding $\mathrm{GL}_n\times\mathrm{GL}_n\hookrightarrow\mathrm{GL}_{2n}$ . As an application,…