Unipotent Representations and Microlocalization

Mason-Brown, Lucas

doi:10.48550/arxiv.1805.12038

Cited by 4 publications

(7 citation statements)

References 12 publications

(34 reference statements)

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“…In particular, Proposition 6.7.1 provides an affirmative answer to a conjecture of Vogan ([Vog91, Conj 12.1]) in the case of complex groups. Special cases of this conjecture were previously established by [Mas18], [Bar17], and [Won18a]. Proposition 6.7.1 is the 'complex' case of a more general conjecture about irreducible representations of real reductive Lie groups, see [Vog91,Sec 12].…”

Section: By the Construction Of Qmentioning

confidence: 92%

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Unipotent Ideals and Harish-Chandra Bimodules

Losev¹,

Mason-Brown²,

Matvieievskyi³

2021

Preprint

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Let G be a real reductive Lie group. One of the most basic unsolved problems in representation theory and abstract Harmonic analysis is the classification of the set Irr u pGq of irreducible unitary G-representations. Although this problem remains unsolved, some general patterns have emerged. For each group G there should be a finite set of 'building blocks' UnippGq Ă Irr u pGq called unipotent representations with an array of distinguishing properties. Every representation B P Irr u pGq should be obtained through one of several procedures (like parabolic induction) from a unipotent representation B L P UnippLq of a suitable Levi subgroup L Ă G. Furthermore, the representations in UnippGq should be indexed by nilpotent co-adjoint G-orbits and their equivariant covers. This roadmap has emerged over many decades (see [Vog87, for an overview) and is supported by numerous successes in important special cases (e.g. [Vog86b] and [Bar89]). A crucial problem with this approach is that the set UnippGq has not yet been defined in the appropriate generality. The main contribution of this paper is a definition of 'unipotent' in the case when G is complex. Our definition generalizes the notion of special unipotent, due to Barbasch-Vogan and Arthur ([BV85],[Art83]).The representations we define arise from finite equivariant covers of nilpotent co-adjoint G-orbits. To each such cover r O, we attach a distinguished filtered algebra A 0 equipped with a graded Poisson isomorphism grpA 0 q » Cr rOs. The existence of this algebra follows from the theory of filtered quantizations of conical symplectic singularities, see [Los16]. The algebra A 0 receives a distinguished homomorphism from the universal enveloping algebra U pgq, and the kernel of this homomorphism is a completely prime primitive ideal in U pgq with associated variety O. A unipotent ideal is any ideal in U pgq which arises in this fashion. A unipotent representation is an irreducible Harish-Chandra bimodule which is annihilated (on both sides) by a unipotent ideal.Our definitions are vindicated by the many favorable properties which these ideals and bimodules enjoy. First of all, we show that both unipotent ideals and bimodules have nice geometric classifications. Unipotent ideals are classified by certain geometrically-defined equivalence classes of covers of nilpotent orbits. Unipotent bimodules are classified by irreducible representations of certain finite groups. For classical groups, we show that all unipotent ideals are maximal and all unipotent bimodules are unitary. In addition, we show that all unipotent bimodules are, as G-representations, of a very special form, proving a conjecture of Vogan ([Vog91]). Finally, we show that all special unipotent bimodules are unipotent. The final assertion is proved using a certain refinement of Barbasch-Vogan-Lusztig-Spaltenstein duality, inspired by the symplectic duality of [BLPW16b]. Along the way, we establish combinatorial algorithms (in classical and exceptional types) for computing the infinitesimal characters of unip...

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Section: By the Construction Of Qmentioning

confidence: 92%

“…The CrO K s-module ΓpO K , L B q is Cohen-Macaulay. Conjecture 6.7.4 was proved for complex groups in [Mas18]. Some additional (real) cases were established in [Mas20].…”

Section: By the Construction Of Qmentioning

confidence: 99%

Unipotent Ideals and Harish-Chandra Bimodules

Losev¹,

Mason-Brown²,

Matvieievskyi³

2021

Preprint

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“…Preserve the notation of 5.1, so G X is a chain. Let {U j } j∈SU , {Z i } i∈SZ , { Zk } k∈S Z be the irreducible components 13 of U , Z, and Z, respectively. Here S U , S Z , S Z are sets that index the irreducible components.…”

Section: 21mentioning

confidence: 99%

“…The first claim is a conjecture of Vogan in [15]. Evidence for the second and third claims comes from [15], [13], and many low-rank examples (including the two given below).…”

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confidence: 94%

Equivariant Vector Bundles on Varieties with Codimension-one Orbits

Mason-Brown¹,

Tao²

2022

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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. LUCAS MASON-BROWN AND JAMES TAO 4. Specialization maps 20 4.1. Line bundles 20 4.2. Factoring through a quotient of Gs 22 4.3. Local systems 23 4.4. The universal tame quotient 25 4.5. The subcategory associated to a tame quotient 29 4.6. Application to equivariant twisted D-modules 30 5. Equivariant sheaves on weakly normal chains 33 5.1. Weak normality 33 5.2. Patching vector bundles on chains 34 5.3. The case of line bundles 36 6. An application to the representation theory of real reductive groups 38 6.1. Admissibility 40 6.2. The chain associated to an irreducible (g, K)-module 40 6.3. The Slodowy slice 42 6.4. Unipotent sheaves 45 References 47

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“…In a previous paper ( [14]), we have made some encouraging progress towards (1). In this paper, we will turn our attention towards (2).…”

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confidence: 94%

Upper Triangularity for Unipotent Representations

Mason-Brown

2019

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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 wellunderstood 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.

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Unipotent Representations and Microlocalization

Cited by 4 publications

References 12 publications

Unipotent Ideals and Harish-Chandra Bimodules

Unipotent Ideals and Harish-Chandra Bimodules

Equivariant Vector Bundles on Varieties with Codimension-one Orbits

Upper Triangularity for Unipotent Representations

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