TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS

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 the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

LA VARIANTE INFINITÉSIMALE DE LA FORMULE DES TRACES DE JACQUET-RALLIS POUR LES GROUPES LINÉAIRES

We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated à la Arthur multiplied by the absolute value of the determinant to the power $s\in \mathbb{C}$. It has a geometric side which is a sum of distributions $I_{\mathfrak{o}}(s,\cdot )$ indexed by the invariants of the adjoint action of $\text{GL}_{n}(\text{F})$ on $\mathfrak{gl}_{n+1}(\text{F})$ as well as a «spectral side» consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $I_{\mathfrak{o}}(s,\cdot )$ are invariant and depend only on the choice of the Haar measure on $\text{GL}_{n}(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $I_{\mathfrak{o}}(s,\cdot )$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $I_{\mathfrak{o}}(s,\cdot )$ in terms of relative orbital integrals regularised by means of zeta functions.

On the arithmetic fundamental lemma in the minuscule case

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.

Elliptic and hypoelliptic orbital integrals

This chapter constructs semisimple orbital integrals associated with the heat kernel for the hypoelliptic Laplacian ℒbX. By making b → 0, the chapter shows that the corresponding supertrace coincides with the orbital integral associated with the standard elliptic heat kernel. Throughout this chapter, the same assumptions as in chapters 2 and 3 will be made, and this chapter uses corresponding notation. Also if V = V₊ ⊕ V₋ is a finite dimensional Z₂-graded vector space, if τ‎ = ±1 is the involution of V that defines the Z₂-grading, if A ∈ End(V), the chapter defines its supertrace Tr″[A] by Tr″[A] = Tr[τ‎A].

Elementary Proof of the Fundamental Lemma For a Unitary Group

The fundamental lemma in the theory of automorphic forms is proven for the (quasi-split) unitary group U(3) in three variables associatedwith a quadratic extension of p-adic fields, and its endoscopic group U(2), by means of a new, elementary technique. This lemma is a prerequisite for an application of the trace formula to classify the automorphic and admissible representations of U(3) in terms of those of U(2) and base change to GL(3). It compares the (unstable) orbital integral of the characteristic function of the standard maximal compact subgroup K of U(3) at a regular element (whose centralizer T is a torus), with an analogous (stable) orbital integral on the endoscopic group U(2). The technique is based on computing the sum over the double coset space T\G/K which describes the integral, by means of an intermediate double coset space H\G/K for a subgroup H of G= U(3) containing T. Such an argument originates from Weissauer's work on the symplectic group. The lemma is proven for both ramified and unramified regular elements, for which endoscopy occurs (the stable conjugacy class is not a single orbit).

Unipotent Orbital Integrals of Hecke Functions for GL(n)

Let G = GL(n, F) where F is a p-adic field, and let 𝓗(G) denote the Hecke algebra of spherical functions on G. Let u1,..., up denote a complete set of representatives for the unipotent conjugacy classes in G. For each 1 ≤ i ≤ p, let μi be the linear functional on such that μi(f) is the orbital integral of f over the orbit of ui. Waldspurger proved that the μi, 1 ≤ i ≤ p, are linearly independent. In this paper we give an elementary proof of Waldspurger's theorem which provides concrete information about the Hecke functions needed to separate orbits. We also prove a twisted version of Waldspurger's theorem and discuss the consequences for SL(n, F).

Relative Kloosterman Integrals for GL(3): II

Let G′ be a quasi–split reductive group over a local field F, ƒ′ the characteristic function of a maximal compact subgroup K′ of G′, N′ a maximal unipotent subgroup of G′. We consider the orbits of maximal dimension for the action of N′ × N′ on G′ and the weighted orbital integral of f′ on such an orbit, the weight being a generic character. The resulting integral, we call a Kloosterman integral. A relative version of this construction is to consider a symmetric space S associated to a quasi-split group G, a maximal unipotent subgroup N of G, a maximal compact K of G and the orbits of maximal dimension for the action of N on S. The weighted orbital integral of the characteristic function f of K ∩ S on such an orbit is what we call a relative Kloosterman integral; the weight is an appropriate character of N. We conjecture that a relative Kloosterman integral is actually a Kloosterman integral for an appropriate group G′. We prove the conjecture in a simple case: E is an unramified quadratic extension of F,G is GL(3, E), S is the set of 3 × 3 matrices s such that the group G′ is then the quasi-split unitary group in three variables.

On Twisted Orbital Integral Identities for PGL(3) Over A p-adic Field

The object of this paper is to prove certain p-adic orbital integral identities needed in order to accomplish the symmetric square transfer via the twisted Arthur trace formula. Only §5 of this article contains original material, the rest of it is due to R. Langlands. Very briefly, we reduce the problem of proving certain orbital integral identities for “matching” functions in the respective Hecke algebras to two counting problems on the buildings. We give Langlands’ solution of one of these problems in the case of the unit elements of the respective Hecke algebras and §5 provides the solution to the other one, again, in the unit element case. The main results assume p ≠ 2.

Stable base change for spherical functions

Let E/F be an unramified cyclic extension of local non-archimedean fields, G a connected reductive group over F, K(F) (resp. K(E)) a hyper-special maximal compact subgroup of G(F) (resp. G(E)), and H(F) (resp. H(E)) the Hecke convolution algebra of compactly-supported complex-valued K(F) (resp. G(E))-biinvariant functions on G(F) (resp. G(E)). Then the theory of the Satake transform defines (see § 2) a natural homomorphism H(E) → H(F), θ→f. There is a norm map N from the set of stable twisted conjugacy classes in G(E) to the set of stable conjugacy classes in G(F); it is an injection (see [Ko]). Let Ω‱(x, f) denote the stable orbital integral of f in H(F) at the class x, and Ω‱(y, θ) the stable twisted orbital integral of θ in H(E) at the class y.