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Register a new userGelfand-Kirillov dimension and mod p cohomology for GL2
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Let $p$ be a prime number, $F$ a totally real number field unramified at places above $p$ and $D$ a quaternion algebra of center $F$ split at places above $p$ and at no more than one infinite place. Let $v$ be a fixed place of $F$ above $p$ and $overline{r} : {rm Gal}(overline F/F)rightarrow mathrm{GL}_2(overline{mathbb{F}}_p)$ an irreducible modular continuous Galois representation which, at the place $v$, is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of $mathrm{GL}_2(F_v)$ over $overline{mathbb{F}}_p$ associated to $overline{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:mathbb{Q}]$, as well as several related results.
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Let $X$ be a topological space with Noetherian mod $p$ cohomology and let $C^*(X;mathbb{F}_p)$ be the commutative ring spectrum of $mathbb{F}_p$-valued cochains on $X$. The goal of this paper is to exhibit conditions under which the category of module spectra on $C^*(X;mathbb{F}_p)$ is stratified in the sense of Benson, Iyengar, Krause, providing a classification of all its localizing subcategories. We establish stratification in this sense for classifying spaces of a large class of topological groups including Kac--Moody groups as well as whenever $X$ admits an $H$-space structure. More generally, using Lannes theory we prove that stratification for $X$ is equivalent to a condition that generalizes Chouinards theorem for finite groups. In particular, this relates the generalized telescope conjecture in this setting to a question in unstable homotopy theory.
Let $F/F^+$ be a CM field and let $widetilde{v}$ be a finite unramified place of $F$ above the prime $p$. Let $overline{r}: mathrm{Gal}(overline{mathbb{Q}}/F)rightarrow mathrm{GL}_n(overline{mathbb{F}}_p)$ be a continuous representation which we assume to be modular for a unitary group over $F^+$ which is compact at all real places. We prove, under Taylor--Wiles hypotheses, that the smooth $mathrm{GL}_n(F_{widetilde{v}})$-action on the corresponding Hecke isotypical part of the mod-$p$ cohomology with infinite level above $widetilde{v}|_{F^+}$ determines $overline{r}|_{mathrm{Gal}(overline{mathbb{Q}}_p/F_{widetilde{v}})}$, when this latter restriction is Fontaine--Laffaille and has a suitably generic semisimplification.
Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $mathbb{F}_p$. We show that the supersingular irreducible $mathcal{H}^{(1)}$-modules of dimension 2 can be realized through the equivariant cohomology of the flag variety of the mod p Langlands dual group of $G$.
Let $F$ be a non-archimedean local field with residue field $mathbb{F}_q$ and let $G = GL_2/F$. Let $mathbf{q}$ be an indeterminate and let $H^{(1)}(mathbf{q})$ be Vigneras generic pro-p Iwahori-Hecke algebra of the p-adic group $G(F)$. Let $V_{widehat{G}}$ be the Vinberg monoid of the dual group of $G$. We establish a generic version for $H^{(1)}(mathbf{q})$ of the Kazhdan-Lusztig-Ginzburg antispherical representation, the Bernstein map and the Satake isomorphism. We define the flag variety for the monoid $V_{widehat{G}}$ and establish the characteristic map in its equivariant K-theory. These generic constructions recover the classical ones after the specialization $mathbf{q} = q in mathbb{C}$. At $mathbf{q} = q = 0 inoverline{mathbb{F}}_q$, the antispherical map provides a dual parametrization of all the irreducible $H^{(1)}_{overline{mathbb{F}}_q}(0)$-modules. This work supersedes our earlier work arXiv:1907.08808. We explain the relationship between the two articles in the introduction.
Recently, Anno, Bezrukavnikov and Mirkovic have introduced the notion of a real variation of stability conditions (which is related to Bridgelands stability conditions), and construct an example using categories of coherent sheaves on Springer fibers. Here we construct another example of representation theoretic significance, by studying certain sub-quotients of category O with a fixed Gelfand-Kirillov dimension. We use the braid group action on the derived category of category O, and certain leading coefficient polynomials coming from translation functors. Consequently, we use this to explicitly describe a sub-manifold in the space of Bridgeland stability conditions on these sub-quotient categories, which is a covering space of a hyperplane complement in the dual Cartan.
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