arXiv subjects

Sean Cotner

Publications and source records attributed to Sean Cotner.

8 recordsLinked to original sources

arXiv2026

Central isogenies and conjugacy classes in reductive groups

Steinberg described the group of components of the centralizer of a semisimple element of a connected semisimple algebraic group $G$ as a subgroup of the fundamental group of $G$. We show that this description can be generalized to explain the fact that centralizers of unipotent elements can fail to be reduced when the universal cover of $G$ is not \'etale. As applications, we compute generic multiplicities in the special fibers of moduli spaces of L-parameters and universal deformation rings, and we show there is no Springer isomorphism for $\mathrm{PGL}_p$ in characteristic $p$.

math.RT↗

arXiv2024

Connected components of the moduli space of L-parameters

Recently, in order to formulate a categorical version of the local Langlands correspondence, several authors have constructed moduli spaces of $\mathbf{Z}[1/p]$-valued L-parameters for $p$-adic groups. The connected components of these spaces over various $\mathbf{Z}[1/p]$-algebras $R$ are conjecturally related to blocks in categories of $R$-representations of $p$-adic groups. Dat-Helm-Kurinczuk-Moss described the components when $R$ is an algebraically closed field and gave a conjectural description when $R = \overline{\mathbf{Z}}[1/p]$. In this paper, we prove a strong form of this conjecture applicable to any integral domain $R$ over $\overline{\mathbf{Z}}[1/p]$.

math.NT↗

arXiv2023

Hom schemes for algebraic groups

In SGA3, Demazure and Grothendieck showed that if $G$ and $H$ are smooth affine group schemes over a scheme $S$ and $G$ is reductive, then the functor of $S$-homomorphism $G \to H$ is representable. In this paper we extend this result to cover cases in which $G$ is not reductive, with much simpler proofs. Our results apply in particular to parabolics over any base, and they are essentially optimal over a field. We also relate the closed orbits in Hom schemes to Serre's theory of complete reducibility, answer a question of Furter--Kraft, and provide many examples.

math.AG↗

arXiv2023

Morphisms of character varieties

Let $k$ be a field, let $H \subset G$ be (possibly disconnected) reductive groups over $k$, and let $\Gamma$ be a finitely generated group. Vinberg and Martin have shown that the induced morphism of character varieties \[ \underline{\mathrm{Hom}}_{k\textrm{-gp}}(\Gamma, H)//H \to \underline{\mathrm{Hom}}_{k\textrm{-gp}}(\Gamma, G)//G \] is finite. In this note, we generalize this result (with a significantly different proof) by replacing $k$ with an arbitrary locally noetherian scheme, answering a question of Dat. Along the way, we use Bruhat-Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings.

math.RT↗

arXiv2022

Springer isomorphisms over a general base scheme

We establish the existence of Springer isomorphisms for reductive group schemes over general base schemes. For this, we first study centralizers of fiberwise regular sections of reductive group schemes, and we establish their flatness in many cases. At the end, we give several arguments to show that the hypotheses in our results are essentially optimal. Our results clarify some aspects of Springer isomorphisms even over a field, and the arguments simplify considerably in this case.

math.AG↗

arXiv2022

Lifting $G$-Valued Galois Representations when $\ell \neq p$

In this paper we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$. In particular, under certain technical conditions applicable to any root datum we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\textrm{GL}_n$-valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm{G}_2$-valued representations.

math.NT↗

arXiv2022

Lefschetz theorems in flat cohomology and applications

We prove a version of the Lefschetz hyperplane theorem for fppf cohomology with coefficients in any finite commutative group scheme over the ground field. As consequences, we establish new Lefschetz results for the Picard scheme.

math.AG↗

arXiv2022

Centralizers of sections of a reductive group scheme

This paper proves a number of flatness results for centralizers of sections of a reductive group scheme over a general base scheme. To this end, we establish relative versions of the Jordan decomposition. Using our results, we obtain a canonical flattening stratification for the universal centralizer of a simply connected semisimple group scheme over a base of good characteristic. We also investigate the structure of centralizers and conjugacy classes of unipotent and nilpotent sections over general bases.

math.AG↗