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Clifton Cunningham

Publications and source records attributed to Clifton Cunningham.

At least 19 recordsLinked to original sources
arXiv2025

Representations of $p$-adic groups and orbits with smooth closure in a variety of Langlands parameters

Let $G$ be a reductive $p$-adic group for which the local Langlands correspondence is known, and $\lambda$ an infinitesimal parameter of $G$. In this paper, we prove that if the $p$-adic Kazhdan-Lusztig hypothesis holds for $\lambda$, then for a Langlands parameter $\phi$ with infinitesimal parameter $\lambda$, if $\Pi_\phi(G)$ contains a generic representation, then $L(s, \phi, \Ad)$ is regular at $s=1$. We then prove an analogous statement for ABV-packets, which together with Vogan's conjecture on ABV-packets implies that if Arthur's conjectures for $G$ are known, then one direction of Shahidi's enhanced genericity conjecture holds: If an Arthur packet $\Pi_\psi(G)$ contains a generic representation, then $\phi_\psi$ is tempered. We also offer some speculation about the relationship between Arthur type representations and singularities in varieties of Langlands parameters defined by Vogan.

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arXiv2024

Whittaker normalization of $p$-adic ABV-packets and Vogan's conjecture for tempered representations

We show that ABV-packets for $p$-adic groups do not depend on the choice of a Whittaker datum, but the function from the ABV-packet to representations of the appropriate microlocal equivariant fundamental group does, and we find this dependence exactly. We study the relation between open parameters and tempered parameters and Arthur parameters and generic representations. We state a genericity conjecture for ABV-packets and prove this conjecture for quasi-split classical groups and their pure inner forms. Motivated by this we study ABV-packets for open parameters and prove that they are L-packets, and further that the function from the packet to the fundamental group given by normalized vanishing cycles coincides with the one given by the Langlands correspondence. From this conclude Vogan's conjecture on A-packets for tempered representations: ABV-packets for tempered parameters are Arthur packets and the function from the packet to the fundamental group given by normalized vanishing cycles coincides with the one given by Arthur.

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arXiv2024

Koszul duality for generalised Steinberg representations of $p$-adic groups

Let $G$ be a semisimple group, split over a non-Archimedean field $F$. We prove that the category of modules over the extension algebra of generalised Steinberg representations of $G(F)$ is equivalent to a full subcategory of equivariant perverse sheaves on the variety of Langlands parameters for these representations. Specifically, we establish an equivalence \[ \textbf{Mod}(\text{Ext}_G^\bullet(\Sigma_\lambda, \Sigma_\lambda)) \simeq \textbf{Per}_{\widehat{G}}^\circ(X_\lambda), \] where $\Sigma_\lambda$ is the direct sum of generalised Steinberg representations and $\textbf{Per}_{\widehat{G}}^\circ(X_\lambda)$ is the subcategory of perverse sheaves on the variety of Langlands parameters $X_\lambda$ corresponding to these representations under Vogan's geometrisation of the Langlands correspondence. Furthermore, we demonstrate that this equivalence is a true Koszul duality by showing that the extension algebra of generalised Steinberg representations is Koszul dual to the endomorphism algebra of the direct sum of corresponding equivariant perverse sheaves, taken in the equivariant derived category $D_{\widehat{G}}^b(X_\lambda)$.

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arXiv2024

Generic representations, open parameters and ABV-packets for $p$-adic groups

If $\pi$ is a representation of a $p$-adic group $G(F)$, and $\phi$ is its Langlands parameter, can we use the moduli space of Langlands parameters to find a geometric property of $\phi$ that will detect when $\pi$ is generic? In this paper we show that if $G$ is classical or if we assume the Kazhdan-Lusztig hypothesis for $G$, then the answer is yes, and the property is that the orbit of $\phi$ is open. We also propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter $\phi$ for a $p$-adic group $G(F)$, the ABV-packet $\Pi^{\mathrm{ABV}}_\phi(G(F))$ contains a generic representation if and only if the local adjoint L-function $L(s,\phi,\mathop{\text{Ad}})$ is regular at $s=1$, and show that this condition is equivalent to the "open parameter" condition above. We show that this genericity conjecture for ABV-packets follows from other standard conjectures and we verify its validity with the same conditions on $G$. We show that, in this case, the ABV-packet for $\phi$ coincides with its $L$-packet. Finally, we prove Vogan's conjecture on $A$-packets for tempered parameters.

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arXiv2022

Proof of Vogan's conjecture on Arthur packets: irreducible parameters of p-adic general linear groups

In this paper we prove Vogan's conjecture on local Arthur packets, for Arthur parameters of $p$-adic general linear groups that are irreducible as representations of $W_F \times SL_2(\mathbb{C}) \times SL_2(\mathbb{C})$ - we refer to such parameters as irreducible Arthur parameters. This result shows that these Arthur packets may be characterized by properties of simple perverse sheaves on a moduli space of Langlands parameters.

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arXiv2021

Appearance of the Kashiwara-Saito singularity in the representation theory of $p$-adic $\mathop{GL}_{16}$

In 1993 David Vogan proposed a basis for the vector space of stable distributions on $p$-adic groups using the microlocal geometry of moduli spaces of Langlands parameters. In the case of general linear groups, distribution characters of irreducible admissible representations, taken up to equivalence, form a basis for the vector space of stable distributions. In this paper we show that these two bases, one putative, cannot be equal. Specifically, we use the Kashiwara-Saito singularity to find a non-Arthur type irreducible admissible representation of $p$-adic $\mathop{GL}_{16}$ whose ABV-packet, as defined in earlier work, contains exactly one other representation; remarkably, this other admissible representation is of Arthur type. In the course of this study we strengthen the main result concerning the Kashiwara-Saito singularity. The irreducible admissible representations in this paper illustrate a fact we found surprising: for general linear groups, while all A-packets are singletons, some ABV-packets are not.

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arXiv2021

Toward the endoscopic classification of unipotent representations of $p$-adic $G_2$

We begin this paper by reviewing the Langlands correspondence for unipotent representations of the exceptional group of type $G_2$ over a $p$-adic field $F$ and present it in an explicit form. Then we compute all ABV-packets, as defined in [CFM+21] following ideas from Vogan's 1993 paper The local Langlands Conjecture, and prove that these packets satisfy properties derived from the expectation that they are generalized A-packets. We attach distributions to ABV-packets for $G_2$ and its endoscopic groups and study a geometric endoscopic transfer of these distributions. This paper builds on earlier work by the same authors.

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arXiv2020

Arthur packets for $G_2$ and perverse sheaves on cubics

This paper begins the project of defining Arthur packets of all unipotent representations for the $p$-adic exceptional group $G_2$. Here we treat the most interesting case by defining and computing Arthur packets with component group $S_3$. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. In forthcoming work we will treat the remaining unipotent representations and their endoscopic classification and strengthen our result on stability.

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arXiv2019

Commutative character sheaves and geometric types for supercuspidal representations

We show that some types for supercuspidal representations of tamely ramified $p$-adic groups that appear in Jiu-Kang Yu's work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. We then use these commutative character sheaves to geometrize the linear characters that appear in the types introduced by Jiu-Kang Yu, assuming that the character vanishes on a certain derived subgroup. To define geometric types, we combine commutative character sheaves with Gurevich and Hadani's geometrization of the Weil representation and Lusztig's character sheaves.

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arXiv2017

Arthur packets for $p$-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples

In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

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arXiv2015

From the function-sheaf dictionary to quasicharacters of $p$-adic tori

We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$ and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$ and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$-adic tori.

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arXiv2014

A note on L-packets and abelian varieties over local fields

A polarized abelian variety (X,λ) of dimension g over a local field K determines an admissible representation of GSpin_{2g+1}(K). We show that the restriction of this representation to Spin_{2g+1}(K) is reducible if and only if X is isogenous to its twist by the quadratic unramified extension of K. When g=1 and K = Q_p, we recover the well-known fact that the admissible GL_2(K) representation attached to an elliptic curve E is reducible upon restriction to SL_2(K) if and only if E has supersingular reduction.

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arXiv2011

Geometrization of continuous characters of $\mathbb{Z}_p^\times$

We define the $p$-adic trace of certain rank-one local systems on the multiplicative group over $p$-adic numbers, using Sekiguchi and Suwa's unification of Kummer and Artin-Schrier-Witt theories. Our main observation is that, for every non-negative integer $n$, the $p$-adic trace defines an isomorphism of abelian groups between local systems whose order divides $(p-1)p^n$ and $\ell$-adic characters of the multiplicative group of $p$-adic integers of depth less than or equal to $n$.

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arXiv2010

On the computability of some positive-depth supercuspidal characters near the identity

This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of $p$-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which Harish-Chandra local character expansion holds). We construct a parameter space $B$ (that depends on the group and a real number $r>0$) for the set of equivalence classes of the representations of minimal depth $r$ satisfying some additional assumptions. This parameter space is essentially a geometric object defined over $\Q$. Given a non-Archimedean local field $\K$ with sufficiently large residual characteristic, the part of the character table near the identity element for $G(\K)$ that comes from our class of representations is parameterized by the residue-field points of $B$. The character values themselves can be recovered by specialization from a constructible motivic exponential function. The values of such functions are algorithmically computable. It is in this sense that we show that a large part of the character table of the group $G(\K)$ is computable.

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arXiv2008

Character Sheaves of Algebraic Groups Defined over Non-Archimedean Local Fields

This paper concerns character sheaves of connected reductive algebraic groups defined over non-Archimedean local fields and their relation with characters of smooth representations. Although character sheaves were devised with characters of representations of finite groups of Lie type in mind, character sheaves are perfectly well defined for reductive algebraic groups over any algebraically closed field. Nevertheless, the relation between character sheaves of an algebraic group $G$ over an algebraic closure of a field $K$ and characters of representations of $G(K)$ is well understood only when $K$ is a finite field and when $K$ is the field of complex numbers. In this paper we consider the case when $K$ is a non-Archimedean local field and explain how to match certain character sheaves of a connected reductive algebraic group $G$ with virtual representations of $G(K)$. In the final section of the paper we produce examples of character sheaves of general linear groups and matching admissible virtual representations.

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