arXiv subjects

Jeffrey Adams

Publications and source records attributed to Jeffrey Adams.

At least 19 recordsLinked to original sources

arXiv2026

The Unitarity of Arthur Packets for Real Reductive Groups

Let $G$ be a connected reductive algebraic group defined over $\mathbb{R}$. In the 1980s, Arthur conjectured the existence of certain packets of irreducible admissible representations of $G(\mathbb{R})$ satisfying various remarkable properties. These packets were given a precise definition in the book of Adams, Barbasch, and Vogan in terms of microlocal geometry on a space of Langlands parameters. A longstanding conjecture, originally due to Arthur, is that all Arthur packets consist of $\textit{unitary}$ representations. In this paper, we prove this conjecture in general. The main new idea is a `Jordan decomposition' for Arthur packets: a canonical two-step process for realizing an arbitrary Arthur packet via real parabolic and cohomological induction from a unipotent Arthur packet for a certain Levi subgroup. This process is analogous to the decomposition of an element of a complex algebraic group as a (unique) commuting product of elliptic, hyperbolic, and unipotent parts. Using our Jordan decomposition, we reduce the question of unitarity to the case of unipotent Arthur packets, where the answer is already known (by work of Adams-Arancibia-Mezo, Adams-van Leeuwen-Miller-Vogan, Arthur, Barbasch, Barbasch-Ma-Sun-Zhu, and Davis-Mason-Brown). As an application of the same methods, we also give a proof of Jiang's conjecture for real reductive groups, which gives an upper bound on the wavefront sets of the members of an Arthur packet in terms of the Barbasch-Vogan dual of the Arthur $SL_2(\mathbb{C})$.

math.RT↗

arXiv2024

Nilpotent Invariants for Generic Discrete Series of Real Groups

Let $G(\mathbb{R})$ be a real reductive group. Suppose $\pi$ is an irreducible representation of $G(\mathbb{R})$ having a Whittaker model, and consider three invariants of $\pi$ related to nilpotents elements of the Lie algebra of $G$ (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $\pi$ has a Whittaker model. If $\pi$ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related results, including an elementary proof that passage from $\pi$ to the three invariants defines natural bijections between the generic discrete series in an $L$-packet, the possible Whittaker data for $G(\mathbb{R})$, and the appropriate sets of nilpotent orbits. Given one of the three invariants, we also explain how to reconstruct the other two. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant-Sekiguchi correspondence.

math.RT↗

arXiv2024

Discrete series L-packets for real reductive groups

We give a modern exposition of the construction, parameterization, and character relations for discrete series L-packets of real reductive groups, which are fundamental results due to Langlands and Shelstad. This exposition incorporates recent developments not present in the original sources, such as normalized geometric transfer factors and the canonical double covers of tori and endoscopic groups, allowing for simpler statements and proofs. We also prove some new results, such as a simple criterion for detecting generic representations for a prescribed Whittaker datum, and an explicit formula for the factor $\Delta_I$ in terms of covers of tori.

math.RT↗

arXiv2024

Substitute adjustment via recovery of latent variables

The deconfounder was proposed as a method for estimating causal parameters in a context with multiple causes and unobserved confounding. It is based on recovery of a latent variable from the observed causes. We disentangle the causal interpretation from the statistical estimation problem and show that the deconfounder in general estimates adjusted regression target parameters. It does so by outcome regression adjusted for the recovered latent variable termed the substitute. We refer to the general algorithm, stripped of causal assumptions, as substitute adjustment. We give theoretical results to support that substitute adjustment estimates adjusted regression parameters when the regressors are conditionally independent given the latent variable. We also introduce a variant of our substitute adjustment algorithm that estimates an assumption-lean target parameter with minimal model assumptions. We then give finite sample bounds and asymptotic results supporting substitute adjustment estimation in the case where the latent variable takes values in a finite set. A simulation study illustrates finite sample properties of substitute adjustment. Our results support that when the latent variable model of the regressors hold, substitute adjustment is a viable method for adjusted regression.

math.ST↗

arXiv2024

Lowest $K$-types in the local Langlands correspondence

Consider the irreducible representations of a real reductive group $G(\mathbb{R})$, and their parametrization by the local Langlands correspondence. We ask: does the parametrization give easily accessible information on the restriction of representations to a maximal compact subgroup $K(\mathbb{R})$ of $G(\mathbb{R})$? We find a natural connection between the set of lowest $K$-types of a representation and its Langlands parameters. For our results, it is crucial to use the refined version of the local Langlands correspondence, involving (coverings of) component groups attached to $L$-homomorphisms. The first part of the paper is a simplified description of this refined parametrization.

math.RT↗

arXiv2021

Equivalent definitions of Arthur packets for real classical groups

Arthur has conjectured the existence of what are now known as Arthur packets of representations of reductive algebraic groups over local and global fields. In the case of classical groups he subsequently gave a definition of these packets, using local and global methods. For general real groups, an alternative approach to the definition of Arthur packets has been given by Adams-Barbasch-Vogan. This construction is purely local and uses geometric methods. Our main result is that these two definitions agree in the case of real classical groups.

math.RT↗

arXiv2021

Associated varieties for real reductive groups

We give an algorithm to compute the associated variety of a Harish- Chandra module for a real reductive group $G({\mathbb R})$. The algorithm is implemented in the atlas software package.

math.RT↗

arXiv2020

Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry.

math.RT↗

arXiv2019

From conjugacy classes in the Weyl group to semisimple conjugacy classes

Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple conjugacy classes of $G$. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case.

math.RT↗

arXiv2017

Unitary representations of real reductive groups

We present a finite algorithm for computing the set of irreducible unitary representations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant Hermitian form as a deformation of one of the unitary representations in Harish-Chandra's Plancherel formula. The behavior of these deformations was determined to a first approximation in the Kazhdan-Lusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. The basic idea of our algorithm is to follow the behavior of the signature of the Hermitian form through this deformation, counting changes through singularities of the form at reducibility points. An important technical tool is replacing the classical invariant form (in which the real form of the Lie algebra acts by skew-adjoint operators) by forms in which the compact form of the Lie algebra acts by skew-adjoint operators.

math.RT↗

arXiv2017

Computing twisted KLV polynomials

In order to compute Hermitian forms on representations of real reductive groups, in the unequal rank case, it is necessary to compute twisted Kazhdan-Lusztig-Vogan polynomials. These were defined by Lusztig and Vogan (Quasisplit Hecke algebras and Symmetric Spaces, Duke, 2014) and discussed further by Adams and Vogan (Parameters for twisted representations, 2015). These notes contain the details necessary to go from what is in those papers to an explicit algorithm. This algorithm has been implemented in the Atlas of Lie Groups and Representations software.

math.RT↗

arXiv2017

Deforming Representations of SL(2,R)

The spherical principal series representations $π(ν)$ of SL(2,$\mathbb R$) is a family of infinite dimensional representations parametrized by $ν\in\mathbb C$. The representation $π(ν)$ is irreducible unless $ν$ is an odd integer, in which case it is indecomposable. We find a new continuous family of representations $Π(ν)$ such that $π(ν)$ and $Π(ν)$ have the same composition factors, and $Π(ν)$ is completely reducible, for all $ν$. We also describe a connection between this construction and families of invariant Hermitian forms on the representations.

math.RT↗

arXiv2016

Galois and Cartan Cohomology of Real Groups

Real forms of a complex reductive group are classified by Galois cohomology H^1(Gamma,G_ad) where G_ad is the adjoint group. Cartan's classification of real forms in terms of maximal compact subgroups can be stated in terms of H^(Z/2Z,G_ad) where the action is by a (holomorphic) Cartan involution. The main result is that for any complex reductive group, possibly disconnected, there is a canonical isomorphism between H^1(Gamma,G) and H^1(Z/2Z,G). As applications we give short proofs of some well known results, including the Sekiguchi correspondence, Matsuki duality, results on Cartan subgroups, the rational Weyl group, and strong real forms. We also compute H^1(Gamma,G) for all simple, simply connected real groups.

math.GR↗

arXiv2016

Lifting of elements of Weyl groups

Suppose $G$ is a reductive algebraic group, $T$ is a Cartan subgroup, $N=\text{Norm}(T)$, and $W=N/T$ is the Weyl group. If $w\in W$ has order $d$, it is natural to ask about the orders lifts of $w$ to $N$. It is straightforward to see that the minimal order of a lift of $w$ has order $d$ or $2d$, but it can be a subtle question which holds. We first consider the question of when $W$ itself lifts to a subgroup of $N$ (in which case every element of $W$ lifts to an element of $N$ of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of $w$ are conjugate, and therefore have the same order. We also consider the twisted case.

math.RT↗

arXiv2016

Euler Poincare Characteristic for the Oscillator Representation

Suppose $(G,G')$ is a dual pair of subgroups of a metaplectic group. The dual pair correspondence is a bijection between (subsets of the) irreducible representations of $G$ and $G'$, defined by the non-vanishing of Hom$(ω,π\timesπ')$, where $ω$ is the oscillator representation. Alternatively one considers Hom$_G(ω,π)$ as a $G'$-module. It is fruitful to replace Hom with Ext$^i$, and general considerations suggest that the Euler-Poincare characteristic EP$(ω,π)$, the alternating sum of Ext$^i(ω,π)$, will be a more elementary object. We restrict to the case of $p$-adic groups, and prove that EP$(ω,π)$ is a well defined element of the Grothendieck group of finite length representations of $G'$, and show that it is indeed more elementary than Hom$(ω,π)$. We expect that computation of EP, together with vanishing results for higher Ext groups, will be a useful tool in computing the dual pair correspondence, and will help to elucidate the structure of Hom$(ω,π)$.

math.RT↗

arXiv2015

Contragredient representations and characterizing the local Langlands correspondence

We consider the question: what is the contragredient in terms of L-homomorphisms? We conjecture that it corresponds to the Chevalley automorphism of the L-group, and prove this in the case of real groups. The proof uses a characterization of the local Langlands correspondence over R. We also consider the related notion of Hermitian dual, in the case of GL(n,R).

math.RT↗

arXiv2015

Parameters for Twisted Representations

The study of Hermitian forms on a real reductive group $G$ gives rise, in the unequal rank case, to a new class of Kazhdan-Lusztig-Vogan polynomials. These are associated with an outer automorphism $δ$ of $G$, and are related to representations of the extended group $ $. These polynomials were defined geometrically by Lusztig and Vogan in "Quasisplit Hecke Algebras and Symmetric Spaces", Duke Math. J. 163 (2014), 983--1034. In order to use their results to compute the polynomials, one needs to describe explicitly the extension of representations to the extended group. This paper analyzes these extensions, and thereby gives a complete algorithm for computing the polynomials. This algorithm is being implemented in the Atlas of Lie Groups and Representations software.

math.RT↗

arXiv2014

Galois Cohomology of Real Groups

Real forms of a complex reductive group are classified in terms of Galois cohomology $H^1(Γ,G_{ad})$ where $G_{ad}$ is the adjoint group. Alternatively, the theory of the Cartan involution gives a description in terms of cohomology with respect to a holomorphic involution: $H^1(\mathbb Z/2\mathbb Z,G_{ad})$ where the non trivial element acts by a holomorphic involution $θ$. The main theorem is that in general, if $θ$ is the Cartan involution of a real form $σ$, there is a canonical isomorphism $H^1(Γ,G)\simeq H^1(\mathbb Z/2\mathbb Z,G)$. This has applications to the structure and representation theory of real groups. We give two such applications. The first is a simple proof of Matsuki's result on conjugacy classes of tori in real groups. The second is a computation of $H^1(Γ,G)$ in general. The answer is expressed in terms of the notion of strong real forms. We include tables for all simply connected simple groups.

math.GR↗