Real Reductive Groups Research Articles

Recent advances in branching problems of representations

How does an irreducible representation of a group behave when restricted to a subgroup? This is part of branching problems, which are one of the fundamental problems in representation theory, and also interact naturally with other fields of mathematics. This expository paper is an up-to-date account on some new directions in representation theory highlighting the branching problems for real reductive groups and their related topics ranging from global analysis of manifolds via group actions to the theory of discontinuous groups beyond the classical Riemannian setting. This article is an outgrowth of the invited lecture that the author delivered at the commemorative event for the 70th anniversary of the re-establishment of the Mathematical Society of Japan, and originally appeared in Japanese in Sugaku 71 (2019).

Correction: Stability, Analytic Stability for Real Reductive Lie Groups

Correction: Stability, Analytic Stability for Real Reductive Lie Groups

Bessel functions on GL(n), I

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on GL(n) as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on GL(n) and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on GL(n) with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of GL(4) Bessel functions.

A Hilbert–Mumford criterion for polystability for actions of real reductive Lie groups

AbstractWe study a Hilbert–Mumford criterion for polystablility associated with an action of a real reductive Lie group G on a real submanifold X of a Kähler manifold Z. Suppose the action of a compact Lie group with Lie algebra $$\mathfrak {u}$$ u extends holomorphically to an action of the complexified group $$U^{\mathbb {C}}$$ U C and that the U-action on Z is Hamiltonian. If $$G\subset U^{\mathbb {C}}$$ G ⊂ U C is compatible, there is a corresponding gradient map $$\mu _\mathfrak {p}: X\rightarrow \mathfrak {p}$$ μ p : X → p , where $$\mathfrak {g}= \mathfrak {k}\oplus \mathfrak {p}$$ g = k ⊕ p is a Cartan decomposition of the Lie algebra of G. Under some mild restrictions on the G-action on X, we characterize which G-orbits in X intersect $$\mu _\mathfrak {p}^{-1}(0)$$ μ p - 1 ( 0 ) in terms of the maximal weight functions, which we viewed as a collection of maps defined on the boundary at infinity ($$\partial _\infty G/K$$ ∂ ∞ G / K ) of the symmetric space G/K. We also establish the Hilbert–Mumford criterion for polystability of the action of G on measures.

L-packets over strong real forms

Langlands [On the classification of irreducible representations of real algebraic groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170] defined L L -packets for real reductive groups. In order to refine the local Langlands correspondence, Adams-Barbasch-Vogan [The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992] combined L-packets over all real forms belonging to an inner class. In the tempered setting, using different methods, Kaletha [Ann. of Math. (2) 184 (2016), pp. 559–632] also defines such combined L-packets with a refinement to the local Langlands correspondence. We prove that the tempered L-packets of Adams-Barbasch-Vogan and Kaletha are the same and are parameterized identically.

Dominant weight associated with actions of real reductive groups

We study the action of a real reductive Lie group G on a real submanifold X of a Kähler manifold Z. Suppose the action of a compact Lie group U with Lie algebra u extends holomorphically to an action of the complexified group UC and that the U-action on Z is Hamiltonian. If G⊂UC is compatible, there is a corresponding G-gradient map μp:X→p, where g=k⊕p is a Cartan decomposition of the Lie algebra of G, k is the Lie algebra of K=U∩G, and p=iu∩g. One can associate a G-invariant maximal weight function to this action. To each x∈X, there is a dominant weight that minimizes the maximal weight function. We study the dominant weight's existence and uniqueness under various stability conditions.

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.

Epsilon Dichotomy for Linear Models: The Archimedean Case

Abstract Let $G=\textrm {GL}_{2n}({\mathbb {R}})$ or $G=\textrm {GL}_n({\mathbb {H}})$ and $H=\textrm {GL}_n({\mathbb {C}})$ regarded as a subgroup of $G$. Here, ${\mathbb {H}}$ is the quaternion division algebra over ${\mathbb {R}}$. For a character $\chi $ on ${\mathbb {C}}^\times $, we say that an irreducible smooth admissible moderate growth representation $\pi $ of $G$ is $\chi _H$-distinguished if $\operatorname {Hom}_H(\pi , \chi \circ \det _H)\neq 0$. We compute the root number of a $\chi _H$-distinguished representation $\pi $ twisted by the representation induced from $\chi $. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology $H_\ast (H, \pi \otimes \chi )$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $\chi $ of $H$.

On the Notion of Metaplectic Barbasch–Vogan Duality

Abstract In analogy with the Barbasch–Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this duality notion with the theory of primitive ideals and extend the notion of special unipotent representations to the real metaplectic groups. We also interpret the duality map in terms of double cells of Weyl group representations.

Local and multilinear noncommutative de Leeuw theorems

Let Γ<G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma < G$$\\end{document} be a discrete subgroup of a locally compact unimodular group G. Let m∈Cb(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\\in C_b(G)$$\\end{document} be a p-multiplier on G with 1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le p < \\infty $$\\end{document} and let Tm:Lp(G^)→Lp(G^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{m}: L_p({\\widehat{G}}) \\rightarrow L_p({\\widehat{G}})$$\\end{document} be the corresponding Fourier multiplier. Similarly, let Tm|Γ:Lp(Γ^)→Lp(Γ^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{m \\vert _\\Gamma }: L_p({\\widehat{\\Gamma }}) \\rightarrow L_p({\\widehat{\\Gamma }})$$\\end{document} be the Fourier multiplier associated to the restriction m|Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m|_{\\Gamma }$$\\end{document} of m to Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. We show that c(supp(m|Γ))‖Tm|Γ:Lp(Γ^)→Lp(Γ^)‖≤‖Tm:Lp(G^)→Lp(G^)‖,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} c( {{\\,\ extrm{supp}\\,}}( m|_{\\Gamma } ) ) \\Vert T_{m \\vert _\\Gamma }: L_p({\\widehat{\\Gamma }}) \\rightarrow L_p({\\widehat{\\Gamma }}) \\Vert \\le \\Vert T_{m }: L_p({\\widehat{G}}) \\rightarrow L_p({\\widehat{G}}) \\Vert , \\end{aligned}$$\\end{document}for a specific constant 0≤c(U)≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le c(U) \\le 1$$\\end{document} that is defined for every U⊆Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U \\subseteq \\Gamma $$\\end{document}. The function c quantifies the failure of G to admit small almost Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, c(Γ)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(\\Gamma ) =1$$\\end{document} when G has small almost Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-invariant neighbourhoods. Our result thus extends the de Leeuw restriction theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well as de Leeuw’s classical theorem (Ann Math 81(2):364–379, 1965). For real reductive Lie groups G we provide an explicit lower bound for c in terms of the maximal dimension d of a nilpotent orbit in the adjoint representation. We show that c(BρG)≥ρ-d/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(B_\\rho ^G) \\ge \\rho ^{-d/4}$$\\end{document} where BρG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_\\rho ^G$$\\end{document} is the ball of g∈G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\in G$$\\end{document} with ‖Adg‖<ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert {{\\,\ extrm{Ad}\\,}}_g \\Vert < \\rho $$\\end{document}. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs Γ<G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma <G$$\\end{document} with c(Γ)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(\\Gamma ) = 1$$\\end{document}. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

Stability, Analytic Stability for Real Reductive Lie Groups

We presented a systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group G on a real submanifold X of a Kähler manifold Z. More precisely, we suppose that the action of a compact Lie group U with Lie algebra $${\mathfrak {u}}$$ extends holomorphically to an action of the complexified group $$U^\mathbb {C}$$ and that the U-action on Z is Hamiltonian. If $$G\subset U^\mathbb {C}$$ is compatible, there is a corresponding gradient map $$\mu _{\mathfrak {p}} : X\rightarrow {\mathfrak {p}}$$ , where $${\mathfrak {g}}= {\mathfrak {k}}\oplus {\mathfrak {p}}$$ is a Cartan decomposition of the Lie algebra of G. The concept of energy complete action of G on X is introduced. For such actions, one can characterize stability, semistability and polystability of a point by a numerical criteria using a G-equivariant function called maximal weight. We also prove the classical Hilbert–Mumford criteria for semistability and polystability conditions.

A generalization of combinatorial identities for stable discrete series constants

This article is concerned with the constants that appear in Harish-Chandra’s character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra’s work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky–Kottwitz–MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel (2011). We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Goresky– Kottwitz–MacPherson’s 1997 paper. In an appendix, we extend Herb’s notion of 2-structures to pseudo-root systems.

HORN PROBLEM FOR QUASI-HERMITIAN LIE GROUPS

AbstractIn this paper, we prove some convexity results associated to orbit projection of noncompact real reductive Lie groups.

Dirac series for complex classical Lie groups: A multiplicity-one theorem

This paper computes the Dirac cohomology HD(π) of irreducible unitary Harish-Chandra modules π of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are shown to be unitarily induced from unipotent representations. When nonzero, there is a unique, multiplicity free K-type in π contributing to HD(π). This confirms conjectures formulated by the first named author and Pandžić in 2011.

Multiplicity in restricting small representations

We give a geometric criterion for the bounded multiplicity property of “small” infinite-dimensional representations of real reductive Lie groups in both induction and restrictions. In particular, for a reductive symmetric pair $(G,H)$, we determine the reductive subgroups $G'$ having the property that any irreducible $H$-distinguished admissible representations of $G$ are of bounded multiplicity when restricted to $G'$.

Non-spherical Harish-Chandra Fourier Transforms on Real Reductive Groups

The Harish-Chandra Fourier transform, $$f\mapsto \mathcal {H}f,$$ is a linear topological algebra isomorphism of the spherical (Schwartz) convolution algebra $$\mathcal {C}^{p}(G//K)$$ (where K is a maximal compact subgroup of any arbitrarily chosen group G in the Harish-Chandra class and $$0<p\le 2$$ ) onto the (Schwartz) multiplication algebra $$\bar{\mathcal {Z}}({\mathfrak {F}}^{\epsilon })$$ (of $$\mathfrak {w}$$ -invariant members of $$\mathcal {Z}({\mathfrak {F}}^{\epsilon }),$$ with $$\epsilon =(2/p)-1$$ ). This is the well-known Trombi–Varadarajan theorem for spherical functions on the real reductive group, G. Even though $$\mathcal {C}^{p}(G//K)$$ is a closed subalgebra of $$\mathcal {C}^{p}(G),$$ a similar theorem has not however been successfully proved for the full Schwartz convolution algebra $$\mathcal {C}^{p}(G)$$ except; for $$\mathcal {C}^{p}(G/K)$$ (whose method is essentially that of Trombi–Varadarajan, as shown by M. Eguchi); for few specific examples of groups (notably $$G=SL(2,\mathbb {R})$$ ) and; for some notable values of p (with restrictions on G and/or on members of $$\;\mathcal {C}^{p}(G)$$ ). In this paper, we construct an appropriate image of the Harish-Chandra Fourier transform for the full Schwartz convolution algebra $$\mathcal {C}^{p}(G),$$ without any restriction on any of G, p and members of $$\;\mathcal {C}^{p}(G).$$ Our proof, that the Harish-Chandra Fourier transform, $$f\mapsto \mathcal {H}f,$$ is a linear topological algebra isomorphism on $$\mathcal {C}^{p}(G),$$ equally shows that its image $$\mathcal {C}^{p}(\widehat{G})$$ can be nicely decomposed, that the full invariant harmonic analysis is available and implies that the definition of the Harish-Chandra Fourier transform may now be extended to include all p-tempered distributions on G and to the zero-Schwartz spaces.

Galois cohomology of real quasi-connected reductive groups

By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set \(H^1({\mathbb {R}},G)\) of a quasi-connected reductive group G over the field \({\mathbb {R}}\) of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.

A proof of Casselman’s comparison theorem

Let G G be a real linear reductive group and K K be a maximal compact subgroup. Let P P be a minimal parabolic subgroup of G G with complexified Lie algebra p \mathfrak {p} , and n \mathfrak {n} be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation V V of G G , the inclusion V K ⊂ V V_{K}\subset V induces isomorphisms H i ( n , V K ) ≅ H i ( n , V ) H_{i}(\mathfrak {n},V_{K})\cong H_{i}(\mathfrak {n},V) ( i ≥ 0 i\geq 0 ), where V K V_{K} denotes the ( g , K ) (\mathfrak {g},K) module of K K finite vectors in V V . This is called Casselman’s comparison theorem (see Henryk Hecht and Joseph L. Taylor [A remark on Casselman’s comparison theorem, Birkhäuser Boston, Boston, Ma, 1998, pp. 139–146]). As a consequence, we show that: for any k ≥ 1 k\geq 1 , n k V \mathfrak {n}^{k}V is a closed subspace of V V and the inclusion V K ⊂ V V_{K}\subset V induces an isomorphism V K / n k V K = V / n k V V_{K}/\mathfrak {n}^{k}V_{K}= V/\mathfrak {n}^{k}V . This strengthens Casselman’s automatic continuity theorem (see W. Casselman [Canad. J. Math. 41 (1989), pp. 385–438] and Nolan R. Wallach [Real reductive groups, Academic Press, Boston, MA, 1992]).

Equivariant deformation quantization and coadjoint orbit method

Our goal here is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive Lie groups. We first prove some general results on the existence of equivariant deformation quantizations of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to the orbit method and construct nontrivial irreducible Harish-Chandra modules for certain nilpotent coadjoint orbits. Our examples include new geometric construction of representations associated to a large class of nilpotent orbits of real exceptional Lie groups.

Cohomological representations for real reductive groups

For a connected reductive group $G$ over ${\mathbb R}$, we study cohomological $A$-parameters, which are Arthur parameters with the infinitesimal character of a finite-dimensional representation of $G({\mathbb C})$. We prove a structure theorem for such $A$-parameters, and deduce from it that a morphism of $L$-groups which takes a regular unipotent element to a regular unipotent element respects cohomological $A$-parameters. This is used to give complete understanding of cohomological $A$-parameters for all classical groups. We review the parametrization of Adams-Johnson packets of cohomological representations of $G({\mathbb R})$ by cohomological $A$-parameters and discuss various examples. We prove that the sum of the ranks of cohomology groups in a packet on any real group (and with any infinitesimal character) is independent of the packet under consideration, and can be explicitly calculated. This result has a particularly nice form when summed over all pure inner forms.