William Hardesty
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View article: Co-𝑡-structures on derived categories of coherent sheaves and the cohomology of tilting modules
Co-𝑡-structures on derived categories of coherent sheaves and the cohomology of tilting modules Open
We construct a co--structure on the derived category of coherent sheaves on the nilpotent cone of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employe…
View article: Integral exotic sheaves and the modular Lusztig–Vogan bijection
Integral exotic sheaves and the modular Lusztig–Vogan bijection Open
Let G be a reductive group over an algebraically closed field k of very good characteristic. The Lusztig-Vogan bijection is a bijection between the set of dominant weights for G and the set of irreducible G-equivariant vector bundles on ni…
View article: Nilpotent centralizers and good filtrations
Nilpotent centralizers and good filtrations Open
Let $G$ be a connected reductive group over an algebraically closed field $\Bbbk$. Under mild restrictions on the characteristic of $\Bbbk$, we show that any $G$-module with a good filtration also has a good filtration as a module for the …
View article: Silting complexes of coherent sheaves and the Humphreys conjecture
Silting complexes of coherent sheaves and the Humphreys conjecture Open
Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and …
View article: Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules
Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules Open
We construct a co-$t$-structure on the derived category of coherent sheaves on the nilpotent cone $\mathcal{N}$ of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These struct…
View article: Representation Theory of Disconnected Reductive Groups
Representation Theory of Disconnected Reductive Groups Open
We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dua…
View article: Calculations with graded perverse-coherent sheaves
Calculations with graded perverse-coherent sheaves Open
In this paper, we carry out several computations involving graded (or ${{\mathbb {G}}_{\textrm {m}}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper,…
View article: Conjectures on tilting modules and antispherical $p$-cells
Conjectures on tilting modules and antispherical $p$-cells Open
For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Ka…
View article: Representation Theory of Disconnected Reductive Groups
Representation Theory of Disconnected Reductive Groups Open
We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dua…
View article: On the centralizer of a balanced nilpotent section
On the centralizer of a balanced nilpotent section Open
Let $G$ be a split reductive algebraic group defined over a complete discrete valuation ring $\mathbb{O}$, with residue field $\mathbb{F}$ and fraction field $\mathbb{K}$, where the fiber $G_{\mathbb{F}}$ is geometrically standard. A balan…
View article: Explicit calculations in an infinitesimal singular block of $SL_N$
Explicit calculations in an infinitesimal singular block of $SL_N$ Open
Let $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$ there exists a singular block in the category of $G_1$-modules which contains precisely $n+1$ irreducible modules. We are intere…
View article: On the Humphreys conjecture on support varieties of tilting modules
On the Humphreys conjecture on support varieties of tilting modules Open
Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of th…