Representations Of Real Reductive Groups Research Articles

Unipotent representations and microlocalization

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.

Annihilator varieties of distinguished modules of reductive Lie algebras

AbstractWe provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs.Let${\mathbf {G}}$be a complex algebraic reductive group and${\mathbf {H}}\subset {\mathbf {G}}$be a spherical algebraic subgroup. Let${\mathfrak {g}},{\mathfrak {h}}$denote the Lie algebras of${\mathbf {G}}$and${\mathbf {H}}$, and let${\mathfrak {h}}^{\bot }$denote the orthogonal complement to${\mathfrak {h}}$in${\mathfrak {g}}^*$. A${\mathfrak {g}}$-module is called${\mathfrak {h}}$-distinguished if it admits a nonzero${\mathfrak {h}}$-invariant functional. We show that the maximal${\mathbf {G}}$-orbit in the annihilator variety of any irreducible${\mathfrak {h}}$-distinguished${\mathfrak {g}}$-module intersects${\mathfrak {h}}^{\bot }$. This generalises a result of Vogan [Vog91].We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – whenHis a symmetric subgroup of a real reductive groupG, the existence of a temperedH-distinguished representation ofGimplies the existence of a genericH-distinguished representation ofG.Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup$\bf H$, and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules overW-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20].Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.

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.

Unitary representations of real reductive groups

We present an algorithm for computing the 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 a unitary representation from Harish-Chandra’s Plancherel formula. The behavior of these deformations was approximately determined in the KazhdanLusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. Our algorithm follows the signature of the form through this deformation, counting changes at reducibility points. An important tool is Weyl’s “unitary trick:” replacing the classical invariant Hermitian form (where Lie(G) acts by skew-adjoint operators) by a new one where a compact form of Lie(G) acts by skew-adjoint operators. ∗All of the authors were supported in part by NSF grant DMS-0968275. The first author was supported in part by NSF grant DMS-0967566. †The third author was supported in part by NSF grant DMS-0968060. ‡The fourth author was supported in part by NSF grant DMS-0967272.

Regular supercuspidal representations

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p p -adic group G G arise from pairs ( S , θ ) (S,\theta ) , where S S is a tame elliptic maximal torus of G G , and θ \theta is a character of S S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p p -adic groups.

Derived functor modules, dual pairs and [formula omitted]-actions

Derived functors (or Zuckerman functors) play a very important role in the study of unitary representations of real reductive groups. These functors are usually applied on highest weight modules in the so-called good range and the theory is well-understood. On the other hand, there were several studies on the irreducibility and unitarizability, in which derived functors are applied to singular modules. See Enright et al. (Acta Math. 1985), for example. In this article, we apply derived functors to certain modules arising from the formalism of local theta lifting, and investigate the irreducible constituent of resulting modules. The key technique is to understand U(g)K-actions in the setting of a see-saw pair. Our results suggest that derived functor constructions are compatible with local theta liftings.

Dirac cohomology for symplectic reflection algebras

We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld (Anal i Prilozhen 20(1):69–70, 1986). We generalize in this way, the Dirac cohomology theory for Lusztig’s graded affine Hecke algebras defined in Barbasch et al. (Acta Math 209(2):197–227, 2012) and further developed in Barbasch et al. (Acta Math 209(2):197–227, 2012), Ciubotaru et al. (J Inst Math Jussieu 13(3):447–486, 2014), Ciubotaru (J Reine Angew Math 671:199–222, 2012), Ciubotaru and He (Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index, Adv. Math. 283:1–50, 2015), Chan (On a twisted Euler-Poincare pairing for graded affine Hecke algebras, preprint 2014, arXiv:1407.0956). We apply these constructions to the case of the symplectic reflection algebras defined by Etingof and Ginzburg (Invent Math 147:243–348, 2002), particularly to rational Cherednik algebras for real or complex reflection groups. As applications, we give criteria for unitarity of modules in category \({\mathcal {O}}\) and we show that the 0-fiber of the Calogero–Moser space admits a description in terms of a certain “Dirac morphism” originally defined by Vogan for representations of real reductive groups.

On representations of real Jacobi groups

We consider a category of continuous Hilbert space representations and a category of smooth Fréchet representations, of a real Jacobi group G. By Mackey’s theory, they are respectively equivalent to certain categories of representations of a real reductive group $\tilde L$ . Within these categories, we show that the two functors that take smooth vectors for G and for $\tilde L$ are consistent with each other. By using Casselman-Wallach’s theory of smooth representations of real reductive groups, we define matrix coefficients for distributional vectors of certain representations of G. We also formulate Gelfand-Kazhdan criteria for real Jacobi groups which could be used to prove multiplicity one theorems for Fourier-Jacobi models.

Quasi-Conformal Actions, Quaternionic Discrete Series and Twistors: SU(2, 1) and G 2(2)

Quasi-conformal actions were introduced in the physics literature as a generalization of the familiar fractional linear action on the upper half plane, to Hermitian symmetric tube domains based on arbitrary Jordan algebras, and further to arbitrary Freudenthal triple systems. In the mathematics literature, quaternionic discrete series unitary representations of real reductive groups in their quaternionic real form were constructed as degree 1 cohomology on the twistor spaces of symmetric quaternionic-Kahler spaces. These two constructions are essentially identical, as we show explicitly for the two rank 2 cases SU(2, 1) and G 2(2). We obtain explicit results for certain principal series, quaternionic discrete series and minimal representations of these groups, including formulas for the lowest K-types in various polarizations. We expect our results to have applications to topological strings, black hole micro-state counting and to the theory of automorphic forms.

Automorphism-invariant representations of real reductive groups

We characterize representations of a connected real reductive group which are invariant under an automorphism of finite order. The intertwining operators constructed in this characterization are then used to prove a Paley-Wiener theorem for a class of real reductive groups twisted by involutions.

On Admissibility and Temperedness of Representations of Real Reductive Groups

Let G be a real Lie group with reductive Lie algebra g. We call a (g, K)-module weakly admissible if its elements are K- and 3-finite, where 3 is the center of the enveloping algebra of [g, g]C. We prove that the finitely generated weakly admissible (g, K)-modules are exactly the submodules of the "almost principal" (g, K)-modules (i.e., the K-finite subspaces of representations induced continuously from finite dimensional continuous representations of a minimal parabolic subgroup). We call attention that K is not necessarily compact; moreover, the center of the (semi-simple) connected Lie subgroup with Lie algebra [g, g] may be infinite. We redefine admissibility by calling a weakly admissible (g, K)-module admissible if its K-structure is unitarizable (then a (g, K)-module is admissible in our sense if and only if its finitely generated parts are admissible in the old sense).

Projections of measures on nilpotent orbits and asymptotic multiplicities ofK-types in rings of regular functions. I

Let G be the adjoint group of a real semi-simple Lie algebra g and let Kbea maximal compact subgroup of G. Kc, the complexification of K, acts on p£, the complexified cotangent space of G/K at eK. If O is a nilpotent KQ orbit inp£, we study the asymptotic behavior of the K-types in the module ϋ[O], the regular functions on the Zariski closure of O. We show that in many cases this asymptotic behavior is determined precisely by the canonical Liouville measure on a nilpotent G orbit in g* which is naturally associated to O. We provide evidence for a conjecture of Vogan stating that this relationship is true in general. Vogan's conjecture is consistent with the philosophy of the orbit method for representations of real reductive groups.

Continuous Cohomology and Unitary Representations of Real Reductive Groups

Continuous Cohomology and Unitary Representations of Real Reductive Groups