In this article, we pursue the study of the holomorphic dynamics of mapping class groups on two-dimensional character varieties, also called trace-map dynamics in the literature, as initiated in [44] (see also [20]). We show that the dynamics of pseudo-Anosov mapping classes resembles in many ways the dynamics of Hénon mappings, and then we apply this idea to answer open questions concerning

(1) the geometry of discrete and faithful representations of free groups into $\mathrm{SL}(2,C),$

(2) the dynamics of Painlevé sixth equations, and

(3) the spectrum of certain discrete Schrödinger operators

Every Fourier-Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms

In the first part of this article, we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field $F$ of characteristic zero. Our main tool is the Luna slice theorem.

In the second part, we apply this technique to symmetric pairs. In particular, we prove that the pairs $({\mathrm{GL}}_{n+k}(F),{\mathrm{GL}}_n(F)\times {\mathrm{GL}}_k(F))$ and $({\mathrm{GL}}_n(E),{\mathrm{GL}}_n(F))$ are Gelfand pairs for any local field $F$ and its quadratic extension $E$. In the non-Archimedean case, the first result was proved earlier by Jacquet and Rallis [JR] and the second result was proved by Flicker [F].

We also prove that any conjugation-invariant distribution on ${\mathrm{GL}}_n(F)$ is invariant with respect to transposition. For non-Archimedean $F$, the latter is a classical theorem of Gelfand and Kazhdan

Let $G$ be an unramified group over a $p$-adic field $F$, and let $E/F$ be a finite unramified extension field. Let $\theta$ denote a generator of $\mathrm{Gal}(E/F)$. This article concerns the matching, at all semisimple elements, of orbital integrals on $G(F)$ with $\theta$-twisted orbital integrals on $G(E)$. More precisely, suppose that $\phi$ belongs to the center of a parahoric Hecke algebra for $G(E)$. This article introduces a base change homomorphism $\phi \mapsto b\phi$ taking values in the center of the corresponding parahoric Hecke algebra for $G(F)$. It proves that the functions $\phi$ and $b\phi$ are associated in the sense that the stable orbital integrals (for semisimple elements) of $b\phi$ can be expressed in terms of the stable twisted orbital integrals of $\phi$. In the special case of spherical Hecke algebras (which are commutative), this result becomes precisely the base change fundamental lemma proved previously by Clozel [Cl4] and Labesse [L1]. As has been explained in [H1], the fundamental lemma proved in this article is a key ingredient for the study of Shimura varieties with parahoric level structure at the prime $p$