Search

In this paper, we prove the FPP conjecture, giving a strong upper bound on the unitary dual of a real reductive group. Our proof is an application of the global generation properties of $\mathcal{D}$-modules on the flag variety and their Hodge filtrations.

Let $G$ be a complex reductive algebraic group. In arXiv:2108.03453, we have defined a finite set of irreducible admissible representations of $G$ called `unipotent representations', generalizing the special unipotent representations of Arthur and Barbasch-Vogan. These representations are defined in terms of filtered quantizations of symplectic singularities and are expected to form the building blocks of the unitary dual of $G$. In this paper, we provide a description of these representations in terms of the Langlands dual group $G^{\vee}$. To this end, we construct a duality map $D$ from the set of pairs $(\mathbb{O}^{\vee},\bar{C})$ consisting of a nilpotent orbit $\mathbb{O}^{\vee} \subset \mathfrak{g}^{\vee}$ and a conjugacy class $\bar{C}$ in Lusztig's canonical quotient $\bar{A}(\mathbb{O}^{\vee})$ to the set of finite covers of nilpotent orbits in $\mathfrak{g}^*$., Comment: comments welcome!

The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra-Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig's category of unipotent representations in terms of the Deligne-Langlands-Lusztig correspondence. Our formula generalizes the main result of arXiv:2112.14354, where this formula was obtained in the Iwahori-spherical case. We deduce that for any irreducible unipotent representation with real infinitesimal character, the algebraic wavefront set is a singleton, verifying a conjecture of M\oeglin and Waldspurger. In the process, we establish new properties of the generalized Springer correspondence in relation to Lusztig's families of unipotent representations of finite reductive groups., Comment: added some remarks about independence of isogeny

Let $k$ be a $p$-adic field and let $\mathbf{G}(k)$ be the $k$-points of a connected reductive group, inner to split. The set of Aubert-Zelevinsky duals of the constituents of a tempered L-packet form an Arthur packet for $\mathbf{G}(k)$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang-Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets which we call \emph{weak} Arthur packets and describe their constituents in terms of their Langlands parameters., Comment: arXiv admin note: text overlap with arXiv:2112.14354

Let $\mathbf{G}(\mathsf{k})$ be a semisimple $p$-adic group, inner to split. In this article, we compute the algebraic and canonical unramified wavefront sets of the irreducible supercuspidal representations of $\mathbf{G}(\mathsf{k})$ in Lusztig's category of unipotent representations.

Let $G$ be a complex connected reductive algebraic group and let $G_{\mathbb{R}}$ be a real form of $G$. We construct a sequence of functors $L_i\mathcal{R}$ from admissible (resp. finite-length) representations of $G$ to admissible (resp. finite-length) representations of $G_{\mathbb{R}}$. We establish many basic properties of these functors, including their behavior with respect to infinitesimal character, associated variety, and restriction to a maximal compact subgroup. We deduce that each $L_i\mathcal{R}$ takes unipotent representations of $G$ to unipotent representations of $G_{\mathbb{R}}$. Taking the alternating sum of $L_i\mathcal{R}$, we get a well-defined homomorphism on the level of characters. We compute this homomorphism in the case when $G_{\mathbb{R}}$ is split., Comment: Comments are welcome!

Let $G$ be a complex reductive algebraic group with Lie algebra $\mathfrak{g}$ and let $G_{\mathbb{R}}$ be a real form of $G$ with maximal compact subgroup $K_{\mathbb{R}}$. Associated to $G_{\mathbb{R}}$ is a $K \times \mathbb{C}^{\times}$-invariant subvariety $\mathcal{N}_{\theta}$ of the (usual) nilpotent cone $\mathcal{N} \subset \mathfrak{g}^*$. In this article, we will derive a formula for the ring of regular functions $\mathbb{C}[\mathcal{N}_{\theta}]$ as a representation of $K \times \mathbb{C}^{\times}$. Some motivation comes from Hodge theory. In arXiv:1206.5547, Schmid and Vilonen use ideas from Saito's theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If $G_{\mathbb{R}}$ is split, and $X$ is the spherical principal series representation of infinitesimal character $0$, then conjecturally $\mathrm{gr}(X) \simeq \mathbb{C}[\mathcal{N}_{\theta}]$ as representations of $K \times \mathbb{C}^{\times}$. So a formula for $\mathbb{C}[\mathcal{N}_{\theta}]$ is an essential ingredient for computing Hodge filtrations., Comment: added a reference to Kostant-Rallis

The first half of this article is expository -- I will review, with examples, the main statements of the Langlands classification and Arthur's conjectures for real reductive groups as formulated by Adams, Barbasch, and Vogan. In the second half, I will turn my attention to the Orbit Method, a conjectural scheme for classifying irreducible unitary representations of a real reductive group. I will give a definition of the Orbit Method in the case when the group is complex. The main input is the theory of unipotent ideals and Harish-Chandra bimodules, developed in arXiv:2108.03453. I will show that the Orbit Method I define is related to Arthur's conjectures via a natural duality map. Finally, I will sketch a possible generalization of this Orbit Method for arbitrary real groups., Comment: Article prepared for the Proceedings of the IHES 2022 summer school on the Langlands program. Comments welcome!

Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild technical hypothesis. We deduce simpler classifications in the special cases of line bundles and vector bundles which are generically local systems. We apply our results to the study of admissible representations of semisimple Lie groups. Our main result gives a new set of constraints on the associated cycles of unipotent representations.

The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an explicit description of the wavefront set of an irreducible spherical representation with real Satake parameter., Comment: Changed title

In the monograph arXiv:2108.03453, we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of Barbasch-Vogan and form the (conjectural) building blocks of the unitary dual. In arXiv:2108.03453 we provide combinatorial formulas for the infinitesimal characters of all unipotent representations of linear classical groups. In this paper, we establish analogous formulas for spin and exceptional groups, thus completing the determination of the infinitesimal characters of all unipotent ideals. Using these formulas, we prove an old conjecture of Vogan: all unipotent ideals are maximal. For $G$ a real reductive Lie group (not necessarily complex), we introduce the notion of a unipotent representation attached to a rigid nilpotent orbit (in the complexified Lie algebra of $G$). Like their complex group counterparts, these representations form the (conjectural) building blocks of the unitary dual. Using the atlas software (and the work of Adams-Miller-van Leeuwen-Vogan) we show that if $G$ is a real form of a simple group of exceptional type, all such representations are unitary.

Let $G$ be a complex reductive algebraic group. In this paper, we give a geometric definition of a unipotent representation of $G$. Our definition generalizes the notion of a special unipotent representation, due to Barbasch-Vogan and Arthur. The representations we define arise from finite equivariant covers of nilpotent co-adjoint $G$-orbits. To each such cover $\tilde{\mathbb{O}}$, we attach a distinguished filtered algebra $\mathcal{A}_0$ equipped with a graded Poisson isomorphism $\mathrm{gr}(\mathcal{A}_0)\simeq \mathbb{C}[\tilde{\mathbb{O}}]$. The algebra $\mathcal{A}_0$ receives a distinguished homomorphism from the universal enveloping algebra $U(\mathfrak{g})$, and the kernel of this homomorphism is a completely prime primitive ideal in $U(\mathfrak{g})$ with associated variety $\overline{\mathbb{O}}$. A unipotent ideal is any ideal in $U(\mathfrak{g})$ which arises in this fashion. A unipotent representation is an irreducible Harish-Chandra bimodule which is annihilated (on both sides) by such an ideal. Our unipotent ideals and representations have all of the expected properties: the unipotent representations attached to $\tilde{\mathbb{O}}$ are parameterized by irreducible representations of a certain finite group (generalizing Lusztig's canonical quotient) and, when restricted to $K$, are of the form conjectured by Vogan. In classical types, all unipotent ideals are maximal, and all unipotent representations are unitary (we expect these properties to hold for arbitrary groups). Finally, all special unipotent representations are unipotent. To prove the last assertion, we introduce a refinement of Barbasch-Vogan-Lusztig-Spaltenstein duality, inspired by the symplectic duality of Braden, Licata, Proudfoot, and Webster., Comment: 191 pages

Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary representation of $G$ can be constructed through a sequence of well-understood operations from a finite set of building blocks, called the unipotent representations. These representations are `attached' (in a certain mysterious sense) to the nilpotent orbits of $G$ on the dual space of its Lie algebra. Inside this finite set is a still smaller set, consisting of the unipotent representations attached to non-induced nilpotent orbits. In this paper, we prove that in many cases this smaller set generates (through a suitable kind of induction) all unipotent representations.

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations., Comment: revised exposition

We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev (\cite{Losev2011}). We explore the applications of this theory to unipotent representations of real reductive groups. For complex groups, we deduce a formula for the $K$-multiplicities of unipotent representations attached to a nilpotent orbit $\OO$, proving an old conjecture of Vogan (\cite{Vogan1991}) in a large family of cases., Comment: Changed title, revised abstract, revised exposition throughout, added "Organization" section, updated references