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Let |${\textsf k}$| be a |$p$| -adic field, and let |$\textbf {G}({\textsf k})$| be the |${\textsf k}$| -points of a connected reductive group, inner to split. The set of Aubert–Zelevinsky duals of the constituents of a tempered L-packet forms an Arthur packet for |$\textbf {G}({\textsf k})$|⁠. 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 weak Arthur packets and describe their constituents in terms of their Langlands parameters. [ABSTRACT FROM AUTHOR]

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_{n}\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_{n}\mathcal {R}$| takes unipotent representations of |$ G $| to unipotent representations of |$ G_{{\mathbb {R}}} $|⁠. Taking the alternating sum of |$ L_{n}\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. [ABSTRACT FROM AUTHOR]

We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev [Duke Math. J. 159 (2011), pp. 99–143]. We explore the applications of this theory to unipotent representations of real reductive groups. For a unipotent representation of a complex group, we deduce a formula for the restriction to a maximal compact subgroup, proving an old conjecture of Vogan [ Associated varieities and unipotent representations , Birkhäuser Boston, Boston, MA, 1991] in a large family of cases. [ABSTRACT FROM AUTHOR]

In the monograph [27] , 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 [4] and form the (conjectural) building blocks of the unitary dual. In [27] 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 [2]) we show that if G is a real form of a simple group of exceptional type, all such representations are unitary. [ABSTRACT FROM AUTHOR]

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 [ Hodge theory and unitary representations of reductive Lie groups, Frontiers of Mathematical Sciences , Int. Press, Somerville, MA, 2011, pp. 397–420], 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 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. [ABSTRACT FROM AUTHOR]

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 K-types, associated varieties, and Langlands parameters of all such representations. [ABSTRACT FROM AUTHOR]

For more than a century, the John Carter Brown Library has been in possession of a rare seventeenth-century book. Its margins are filled with cryptic shorthand writing, long believed to be the work of Roger Williams, the seventeenth-century theologian and founder of Rhode Island. Despite several attempts to decode it, this writing remained stubbornly indecipherable until 2012, when a group of undergraduate researchers at Brown University, advised and supported by an interdisciplinary team of scholars, finally cracked the code. Contained within these margins is Williams’s last major piece of writing, an unpublished shorthand treatise titled “A Brief Reply to a Small Book Written by John Eliot.” In one short section of this treatise, Williams took on the question of Indian conversion, breaking a thirty-year silence on the issue. This passage is important because it came so late in Williams’s life; it was written in the precarious years after King Philip’s War; it provides a new window into Williams’s rare engagement with Eliot on Indian conversion; and it reveals a basic continuity in Williams’s thinking between the 1640s and the 1680s regarding the enormous difficulty of producing “true” conversions among the Native peoples of New England. [ABSTRACT FROM AUTHOR]