arXiv subjects

Pablo Boixeda Alvarez

Publications and source records attributed to Pablo Boixeda Alvarez.

8 recordsLinked to original sources

arXiv2026

Affine Springer fiber and the small quantum group

We find a new geometric incarnation for the principal block in the category of modules over a quantum group at a root of unity, realizing it as a full subcategory of microsheaves on a certain affine Springer fiber. We also prove a related geometric Langlands type equivalence with wild ramification, identifying the latter category with a category of coherent sheaves on the Springer resolution for the dual group. This can also be viewed as a version of homological mirror symmetry for the Springer resolution.

math.AG↗

arXiv2025

Weight modules and gluing of sheaves on the flag variety

We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric realization of this category as unipotently monodromic sheaves on the flag variety satisfying a singular support condition. We then explain that a derived version $\mathcal{D}$ of this category is Koszul dual to the Kazhdan-Laumon Category O, a different enlargement of the BGG Category O obtained from a gluing construction for sheaves on the flag variety. This characterizes $\mathcal{D}$ as the category of algebras over a natural monad on a direct sum of copies of the derived Category O. These results give new interpretations of classical algebraic constructions of weight modules over semisimple Lie algebras due to Fernando and Mathieu. We also conjecture that our results fit naturally into a proposed Koszul duality relating the small quantum group and the semi-infinite flag variety to the geometry of affine Springer fibers.

math.RT↗

arXiv2023

A geometric realization of the center of the small quantum group

We propose a new geometric model for the center of the small quantum group using the cohomology of certain affine Springer fibers. More precisely, we establish an isomorphism between the equivariant cohomology of affine Spaltenstein fibers for a split element and the center of the deformed graded modules for the small quantum group. We also obtain an embedding from the invariant part of the nonequivariant cohomology under the action of the extended affine Weyl group to the invariant part of the center of the small quantum group under Langlands dual group action, which we conjecture to be an isomorphism. Finally, we give a dimension formula for the invariants on the cohomology side, thus providing a lower bound for the dimension of the center.

math.RT↗

arXiv2022

Non-abelian Hodge moduli spaces and homogeneous affine Springer fibers

Starting from a homogeneous affine Springer fiber $Fl_ψ$, we construct three moduli spaces that correspond to the Dolbeault, de Rham and Betti aspects of a hypothetical Simpson correspondence with wild ramifications. We show that $Fl_ψ$ is homeomorphic to the central Lagrangian fiber in the Dolbeault space, prove that the Dolbeaut and de Rham spaces both have the same cohomology as $Fl_ψ$, and construct a map from the de Rham space to the Betti space which we conjecture to be an analytic isomorphism.

math.AG↗

arXiv2021

Affine Springer Fibers, Procesi bundles, and Cherednik algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{t}$ its Cartan subalgebra and $W$ the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homology of certain affine Springer fibers for $\mathfrak{g}$ and the global sections of a bundle related to a Procesi bundle on the smooth locus of a partial resolution of $(\mathfrak{t}\oplus \mathfrak{t}^*)/W$. We deduce some applications of our isomorphism including a conditional application to the center of the small quantum group. Our main method is to compare certain bimodules over rational and trigonometric Cherednik algebras.

math.RT↗

arXiv2020

Affine Springer Fibers and the Affine Matrix Ball Construction for Rectangular Type Nilpotents

In this paper, we study the affine Springer fiber $\mathcal{F} l_N$ in type $A$ for rectangular type semisimple nil-element $N$ and calculate the relative position between irreducible components. In particular, we use the affine matrix ball construction to show the relative position map is compatible with the Kazhdan-Lusztig cell structure, generalizing the work of Steinberg and van Leeuwen.

math.RT↗

arXiv2019

Toward partial Verma functors of $\mathcal{U}^{[r]}(\mathfrak{g})$ and related results

This note extends the Steinberg Tensor product theorem from the Frobenius kernel $G_{(r)}$ to the deformation $\mathcal{U}^{[r]}(\mathfrak{g})$ of its distribution algebra. As a Corollary we proof some conjectures from \cite{Wes}. Further it describes the graded representation theory of $\mathcal{U}^{[r]}(\mathfrak{g})$ at a generic semisimple p-central character as a twist of the category of graded $G_{(r-1)}$-modules. We conclude by explaining this relation for $SL_2$ explicitely as well as computing the center in this case.

math.RT↗

arXiv2019

Fixed points and components of equivalued affine Springer fibers

For G a semisimple algebraic group, we revisit the description of the components of the affne Springer fiber given by ts, with s a regular semisimple element. We then compute the fixed points of each component of a particular affne Springer fiber for Type A

math.AG↗