arXiv subjects

Peter Dillery

Publications and source records attributed to Peter Dillery.

10 recordsLinked to original sources

arXiv2026

The extended Fargues--Scholze spectral action

Let $G$ be a connected reductive group over a non-archimedean local field. The main result of Fargues and Scholze [FS21] for the geometrization of the local Langlands correspondence is the construction of a ``spectral action'' on the category of $\ell$-adic sheaves on $\text{Bun}_{G}$, the stack of $G$-torsors on the Fargues-Fontaine curve. The goal of this paper is to prove a conjecture of Fargues which says that one can extend this construction to the larger stack $\text{Bun}_G^e$ of $G$-torsors on the Kaletha gerbe over the curve, as introduced by Fargues [Far22]. This ``extended spectral action'' allows for a version of the categorical local Langlands conjecture for an arbitrary connected reductive group $G$, and is the first such statement for those $G$ which are not extended pure inner forms of a quasi-split group, such as non-trivial inner forms of $\mathrm{SL}_{n}$. Finally, we prove this conjecture for tori, following the original argument of [Zou24] and, under the same assumptions as [Zou26] (including connected center), we reduce the ``extended'' version of the categorical conjecture to the one in Fargues--Scholze.

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arXiv2026

Moduli of $G$-bundles on rigid gerbes over affine curves

We geometrize the basic cohomology set $H^{1}(\text{Kal}_{F}, G)_{\text{basic}}$ for a global function field $F$. We do this by constructing a v-stack $\text{Bun}_{G,F}^{e}$ which has localization maps to Fargues' analogous stack $\text{Bun}_{G,F_{v}}^{e}$ for all places $v$ of $F$ and whose semistable locus is the disjoint union of $\text{Bun}_{G_{b},F}$ for all $b \in H^{1}(\text{Kott}_{F} \times_{F} \text{Kal}_{F},G)_{\text{basic}}$. We also prove a version of Tate-Nakayama duality for $H^{1}(\text{Kott}_{F} \times_{F} \text{Kal}_{F},G)_{\text{basic}}$, which lets us state a conjectural multiplicity formula for discrete automorphic representations of $G(\mathbb{A}_{F})$ adapted to this new cohomology set.

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arXiv2026

A Tannakian description of the local Kaletha gerbe

We construct, for a $p$-adic field $F$, an explicit semisimple Tannakian category $\text{RigIsoc}_{F}$ whose category of fiber functors recovers Kaletha's Galois gerbe $\mathcal{E}_{\text{Kal}}$. We then classify and write down the simple objects in $\text{RigIsoc}_{F}$, all of which come from elliptic twisted Levi subgroups of $\mathrm{GL}_{n}$.

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arXiv2024

Non-basic rigid packets for discrete $L$-parameters

This article initiates the study of non-basic rigid inner forms over $p$-adic local fields, extending the basic theory developed by Kaletha. Motivated by the recent work of Bertoloni Meli--Oi on the $B(G)$-parametrization of the local Langlands conjectures, our main application is to extend the basic rigid refined local Langlands conjectures for a discrete $L$-parameter $\phi$ of a quasi-split connected reductive group $G$. The packets of our extended construction are Weyl orbits of representations of inner forms of twisted Levi subgroups $N$ of $G$ for which $\phi$ factors through a member of the canonical $L$-embeddings $^{L}N_{\pm} \to {^{L}G}$ constructed by Kaletha.

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arXiv2023

Isocrystals and limits of rigid local Langlands correspondences

We show that, over a nonarchimedean local field, the rigid refined local Langlands correspondence and associated endoscopic character identities for connected reductive $G$ follow if one only has them for all such $G$ with connected center. The strategy is to construct a projective system of central extensions and then take limits of the Langlands correspondences (and endoscopic data) of each group in the system. As an application, we prove the equivalence of the rigid refined local Langlands correspondence and its analogue for isocrystals, generalizing the work of [Kal18] in the $p$-adic case.

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arXiv2023

A stacky generalized Springer correspondence and rigid enhancements of L-parameters

Motivated by applications to the Langlands program, Aubert-Moussaoui-Solleveld extended Lusztig's generalized Springer correspondence to disconnected reductive groups. We use stacks to give a more geometric account of their theory, in particular, formulating a truly geometric version of the (relevant analogue of the) Bernstein-Zelevinsky Geometrical Lemma and explaining how to compare the correspondence on the group and the Lie algebra using quasi-logarithms. As an application, we study Kaletha's rigid enhancements of L-parameters and draw the same conclusions as Aubert-Moussaoui-Solleveld for this enhancement: there exists a cuspidal support map and its fibers are parameterized by irreducible representations of twisted group algebras.

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arXiv2021

Rigid inner forms over global function fields

We construct an fpqc gerbe $\mathcal{E}_{\dot{V}}$ over a global function field $F$ such that for a connected reductive group $G$ over $F$ with finite central subgroup $Z$, the set of $G_{\mathcal{E}_{\dot{V}}}$-torsors contains a subset $H^{1}(\mathcal{E}_{\dot{V}}, Z \to G)$ which allows one to define a global notion of ($Z$-)rigid inner forms. There is a localization map $H^{1}(\mathcal{E}_{\dot{V}}, Z \to G) \to H^{1}(\mathcal{E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [Kal16, Dil23]) which allows us to organize local rigid inner forms across all places $v$ into coherent families. Doing so enables a construction of (conjectural) global $L$-packets and a conjectural formula for the multiplicity of an automorphic representation $\pi$ in the discrete spectrum of $G$ in terms of these $L$-packets. We also show that, for a connected reductive group $G$ over a global function field $F$, the adelic transfer factor $\Delta_{\mathbb{A}}$ for the ring of adeles $\mathbb{A}$ of $F$ serving an endoscopic datum for $G$ decomposes as the product of the normalized local transfer factors from [Dil20].

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arXiv2020

Rigid inner forms over local function fields

We generalize the concept of rigid inner forms, defined by Kaletha in [Kal16], to the setting of a local function field $F$ in order state the local Langlands conjectures for arbitrary connected reductive groups over $F$. To do this, we define for a connected reductive group $G$ over $F$ a new cohomology set $H^{1}(\mathcal{E}, Z \to G) \subset H_{\text{fpqc}}^{1}(\mathcal{E}, G)$ for a gerbe $\mathcal{E}$ attached to a class in $H_{\text{fppf}}^{2}(F, u)$ for a certain canonically-defined profinite commutative group scheme $u$, building up to an analogue of the classical Tate-Nakayama duality theorem. We define a relative transfer factor for an endoscopic datum serving a connected reductive group $G$ over $F$, and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and $\dot{s}$-stable virtual characters for a semisimple $\dot{s}$ associated to a tempered Langlands parameter.

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arXiv2017

The canonical join complex for biclosed sets

The canonical join complex of a semidistributive lattice is a simplicial complex whose faces are canonical join representations of elements of the semidistributive lattice. We give a combinatorial classification of the faces of the canonical join complex of the lattice of biclosed sets of segments supported by a tree, as introduced by the third author and McConville. We also use our classification to describe the elements of the shard intersection order of the lattice of biclosed sets. As a consequence, we prove that this shard intersection order is a lattice.

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arXiv2015

Minimal Length Maximal Green Sequences and Triangulations of Polygons

We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type $\mathbb{A}$. We prove that such sequences have length $n+t$, where $n$ is the number of vertices and $t$ is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.

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