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Register a new userEquivariant motives and geometric representation theory. (with an appendix by F. Hormann and M. Wendt)
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We consider categories of equivariant mixed Tate motives, where equivariant is understood in the sense of Borel. We give the two usual definitions of equivariant motives, via the simplicial Borel construction and via algebraic approximations of it. The definitions turn out to be equivalent and give rise to a full six-functor formalism. For rational etale motives over a finite field or the homotopical stable algebraic derivator arising from the semisimplified Hodge realization, the equivariant mixed Tate motives provide a graded version of the equivariant derived category. We show that, in sufficiently nice and clean cases, these categories admit weight structures; moreover, a tilting result holds which identifies the category of equivariant mixed Tate motives with the bounded homotopy category of the heart of its weight structure. This can be seen as a formality result for equivariant derived categories. We also discuss convolution functors on equivariant mixed Tate motives, and consequences for the categorification of the Hecke algebra and some of its modules.
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These myh lectures at the Park City conference in 1998.
We give an algebro-geometric interpretation of $C_2$-equivariant stable homotopy theory by means of the $b$-topology introduced by Claus Scheiderer in his study of $2$-torsion phenomena in etale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a $infty$-topos $mathscr{X}$ with $G$-action a presentable stable $infty$-category $mathrm{Sp}^G(mathscr{X})$, which recovers the $infty$-category $mathrm{Sp}^G$ of genuine $G$-spectra when $mathscr{X}$ is the terminal $G$-$infty$-topos. Given a scheme $X$ with $1/2 in mathcal{O}_X$, our construction then specializes to produce an $infty$-category $mathrm{Sp}^{C_2}_b(X)$ of $b$-sheaves with transfers as $b$-sheaves of spectra on the small etale site of $X$ equipped with certain transfers along the extension $X[i] rightarrow X$; if $X$ is the spectrum of a real closed field, then $mathrm{Sp}^{C_2}_b(X)$ recovers $mathrm{Sp}^{C_2}$. On a large class of schemes, we prove that, after $p$-completion, our construction assembles into a premotivic functor satisfying the full six functors formalism. We then introduce the $b$-variant $mathrm{SH}_b(X)$ of the $infty$-category $mathrm{SH}(X)$ of motivic spectra over $X$ (in the sense of Morel-Voevodsky), and produce a natural equivalence of $infty$-categories $mathrm{SH}_b(X)^{wedge}_p simeq mathrm{Sp}^{C_2}_b(X)^{wedge}_p$ through amalgamating the etale and real etale motivic rigidity theorems of Tom Bachmann. This involves a purely algebro-geometric construction of the $C_2$-Tate construction, which may be of independent interest. Finally, as applications, we deduce a $b$-rigidity theorem, use the Segal conjecture to show etale descent of the $2$-complete $b$-motivic sphere spectrum, and construct a parametrized version of the $C_2$-Betti realization functor of Heller-Ormsby.
The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts [64-66] that reflect a certain range of developments, but even in their totality they cannot be taken as a comprehensive survey. In the format of a 30-page contribution aimed at a general mathematical audience, I have decided to illustrate some of the basic ideas in one very interesting example - that of HilbpC2, nq, hoping to spark the curiosity of colleagues in those numerous fields of study where one should expect applications.
When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${mathsf{Perv}}_W(E_{mathbb C}, {mathcal{H}}_{mathbb C})$ of $W$-equivariant perverse sheaves on $E_{mathbb C}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtmans recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${mathsf{Perv}}_W(E_{mathbb C}, {mathcal{H}}_{mathbb C})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations.
The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant functions on a semi-simple group over a local non-Archimedian field, and Grothendieck group of equivariant coherent sheaves on Steinberg variety of the Langlands dual group; this isomorphism due to Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of an earlier joint work with S. Arkhipov, it relies on technical material developed in a paper with Z. Yun.
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