Notes on Motivic Periods

The second part of a set of notes based on lectures given at the IHES in 2015 on Feynman amplitudes and motivic periods

Nilpotence and descent in equivariant stable homotopy theory

Let $G$ be a finite group and let $\mathscr{F}$ be a family of subgroups of $G$. We introduce a class of $G$-equivariant spectra that we call $\mathscr{F}$-nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable $\infty$-category, with which we begin. We then develop some of the basic properties of $\mathscr{F}$-nilpotent $G$-spectra, which are explored further in the sequel to this paper. In the rest of the paper, we prove several general structure theorems for $\infty$-categories of module spectra over objects such as equivariant real and complex $K$-theory and Borel-equivariant $MU$. Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex $K$-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property.Comment: 63 pages. Revised version, to appear in Advances in Mathematic

A unified approach on Springer fibers in the hook, two-row and two-column cases

We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections.Comment: 42 page

The Galois group of a stable homotopy theory

To a "stable homotopy theory" (a presentable, symmetric monoidal stable $\infty$-category), we naturally associate a category of finite \'etale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call the Galois group. We then calculate the Galois groups in several examples. For instance, we show that the Galois group of the periodic $\mathbf{E}_\infty$-algebra of topological modular forms is trivial and that the Galois group of $K(n)$-local stable homotopy theory is an extended version of the Morava stabilizer group. We also describe the Galois group of the stable module category of a finite group. A fundamental idea throughout is the purely categorical notion of a "descendable" algebra object and an associated analog of faithfully flat descent in this context.Comment: 93 pages. To appear in Advances in Mathematic

Calogero-Moser spaces vs unipotent representations

Lusztig's classification of unipotent representations of finite reductive groups depends only on the associated Weyl group $W$ (endowed with its Frobenius automorphism). All the structural questions (families, Harish-Chandra series, partition into blocks...) have an answer in a combinatorics that can be entirely built directly from $W$. Over the years, we have noticed that the same combinatorics seems to be encoded in the Poisson geometry of a Calogero-Moser space associated with $W$ (roughly speaking, families correspond to ${\mathbb{C}}^\times$-fixed points, Harish-Chandra series correspond to symplectic leaves, blocks correspond to symplectic leaves in the fixed point subvariety under the action of a root of unity). The aim of this survey is to gather all these observations, state precise conjectures and provide general facts and examples supporting these conjectures.Comment: 53 page