arXiv subjects

David Vogan

Publications and source records attributed to David Vogan.

6 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 $SL_2(\mathbb{C})$.

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arXiv2023

Disconnected reductive groups

In this paper, we describe the possible disconnected complex reductive algebraic groups $E$ with component group $\Gamma = E/E_0$. We show that there is a natural bijection between such groups $E$ and algebraic extensions of $\Gamma$ by $Z(E_0)$.

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arXiv2019

Compactification of semi-simple Lie groups

We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary face fibering over two copies of the corresponding flag variety with fiber modeled on the (compactification of the) reductive part. On the hd-compactification Harish-Chandra's Schwartz space is identified with a space of conormal functions of rapid-logarithmic decay relative to square-integrable functions.

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arXiv2017

Dirac Index and associated cycles of Harish-Chandra modules

Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_\mathbb{R})={\rm rank}(K_\mathbb{R})$. For any representation $X$ of Gelfand-Kirillov dimension $\frac{1}{2} {\rm dim}(G_{\mathbb{R}}/K_{\mathbb{R}})$, we consider the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing $X$. Under a technical condition involving the Springer correspondence, we establish an explicit relationship between this polynomial and the multiplicities of the irreducible components occurring in the associated cycle of $X$. This relationship was conjectured in \cite{MehdiPandzicVogan15}.

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arXiv2017

Computing the associatied cycles of certain Harish-Chandra modules

Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_\mathbb{R})={\rm rank}(K_\mathbb{R})$. In \cite{MPVZ} we proved that for any representation $X$ of Gelfand-Kirillov dimension $\frac{1}{2}\dim(G_{\mathbb{R}}/K_{\mathbb{R}})$, the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing $X$ is a linear combination, with integer coefficients, of the multiplicities of the irreducible components occurring in the associated cycle. In this paper we compute these coefficients explicitly.

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arXiv2012

Involutions in Coxeter groups

This paper constructs a representation of a Hecke algebra on a vector space spanned by the involutions in a Coxeter group.

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