Real Reductive Group Research Articles

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.

Finite Multiplicities Beyond Spherical Spaces

Abstract Let $G$ be a real reductive algebraic group, and let $H\subset G$ be an algebraic subgroup. It is known that the action of $G$ on the space of functions on $G/H$ is “tame” if this space is spherical. In particular, the multiplicities of the space ${\mathcal {S}}(G/H)$ of Schwartz functions on $G/H$ are finite in this case. In this paper, we formulate and analyze a generalization of sphericity that implies finite multiplicities in ${\mathcal {S}}(G/H)$ for small enough irreducible representations of $G$.

Properties of gradient maps associated with action of real reductive group

Let [Formula: see text] be a Kähler manifold and let [Formula: see text] be a compact connected Lie group with Lie algebra [Formula: see text] acting on [Formula: see text] and preserving [Formula: see text]. We assume that the [Formula: see text]-action extends holomorphically to an action of the complexified group [Formula: see text] and the [Formula: see text]-action on [Formula: see text] is Hamiltonian. Then there exists a [Formula: see text]-equivariant momentum map [Formula: see text]. If [Formula: see text] is a closed subgroup such that the Cartan decomposition [Formula: see text] induces a Cartan decomposition [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the Lie algebra of [Formula: see text], there is a corresponding gradient map [Formula: see text]. If [Formula: see text] is a [Formula: see text]-invariant compact and connected real submanifold of [Formula: see text] we may consider [Formula: see text] as a mapping [Formula: see text] Given an [Formula: see text]-invariant scalar product on [Formula: see text], we obtain a Morse like function [Formula: see text] on [Formula: see text]. We point out that, without the assumption that [Formula: see text] is a real analytic manifold, the Lojasiewicz gradient inequality holds for [Formula: see text]. Therefore, the limit of the negative gradient flow of [Formula: see text] exists and it is unique. Moreover, we prove that any [Formula: see text]-orbit collapses to a single [Formula: see text]-orbit and two critical points of [Formula: see text] which are in the same [Formula: see text]-orbit belong to the same [Formula: see text]-orbit. We also investigate convexity properties of the gradient map [Formula: see text] in the Abelian case. In particular, we study two-orbit variety [Formula: see text] and we investigate topological and cohomological properties of [Formula: see text].

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.

Galois cohomology and component group of a real reductive group

Galois cohomology and component group of a real reductive group

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.

Partial Dirac cohomology and tempered representations

The tempered representations of a real reductive Lie group G are naturally partitioned into series associated with conjugacy classes of Cartan subgroups H of G. We define partial Dirac cohomology, apply it for geometric construction of various models of these H–series representations, and show how this construction fits into the framework of geometric quantization and symplectic reduction.

A short proof of Timashev’s theorem on the real component group of a real reductive group

Using results of Cartan, Matsumoto, and Casselman, we give a short proof of Timashev’s theorem computing the real component group \(\pi _0G({\mathbb {R}})\) of a connected reductive \({\mathbb {R}}\)-group G in terms of a maximal torus of G containing a maximal split torus.

Unipotent ideals for spin and exceptional groups

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.

Generalized Whittaker Quotients of Schwartz Functions onG-Spaces

AbstractLet $G$ be a reductive group over a local field $F$ of characteristic zero. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$, considered as a representation of $G$. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying our theorems to symmetric pairs $(G,H)$ we show that there exists an infinite-dimensional $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is non-compact. For quasi-split $G$ we also extend to the Archimedean case the theorem of Prasad stating that there exists a generic $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is quasi-split. In the non-Archimedean case our result also gives rather sharp bounds on the wave-front sets of distinguished representations. Finally, we deduce a corollary on vanishing of period integrals of automorphic forms with certain Whittaker supports. This corollary, when combined with the restrictions on the Whittaker support of cuspidal automorphic representations proven by Gomez–Gourevitch–Sahi, implies many of the vanishing results on periods of automorphic forms proved by Ash–Ginzburg–Rallis.

The dependence on parameters of the inverse functor to the 𝐾-finite functor

An interpretation of the Casselman-Wallach Theorem is that the K K -finite functor is an isomorphism of categories from the category of finitely generated, admissible smooth Fréchet modules of moderate growth to the category of Harish-Chandra modules for a real reductive group, G G (here K K is a maximal compact subgroup of G G ). In this paper we study the dependence of the inverse functor to the K K -finite functor on parameters. Our main result implies that holomorphic dependence implies holomorphic dependence. The work uses results from the excellent thesis of van der Noort. Also a remarkable family of universal Harish-Chandra modules, developed in this paper, plays a key role.

Vanishing of certain equivariant distributions on spherical spaces for quasi-split groups

We prove the vanishing of z -eigen distributions on a quasi-split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup.

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.

Mackey analogy as deformation of $${\mathcal {D}}$$-modules

Given a real reductive linear Lie group \(G_{\mathbb {R}}\), the Mackey analogy is a bijection between the set of irreducible tempered representations of \(G_{\mathbb {R}}\) and the set of irreducible unitary representations of its Cartan motion group, established by Higson and Afgoustidis. We show that this bijection arises naturally from families of twisted \({\mathcal {D}}\)-modules over the ag variety of \(G_{\mathbb {R}}\).

A Paley–Wiener theorem for Harish–Chandra modules

We formulate and prove a Paley-Wiener theorem for Harish-Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture.

COMPACT ORBITS OF PARABOLIC SUBGROUPS

AbstractWe study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected 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 exists a gradient map $\mu _{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$ . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map $\mu _{\mathfrak p}$ .

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]).

Ellipticity and discrete series

Abstract We explain by elementary means why the existence of a discrete series representation of a real reductive group G implies the existence of a compact Cartan subgroup of G. The presented approach has the potential to generalize to real spherical spaces.

A generalization of the Kobayashi–Oshima uniformly bounded multiplicity theorem

Let [Formula: see text] be a minimal parabolic subgroup of a real reductive Lie group [Formula: see text] and [Formula: see text] a closed subgroup of [Formula: see text]. Then it is proved by Kobayashi and Oshima that the regular representation [Formula: see text] contains each irreducible representation of [Formula: see text] at most finitely many times if the number of [Formula: see text]-orbits on [Formula: see text] is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite, where [Formula: see text] are complexifications of [Formula: see text], respectively, and [Formula: see text] is a Borel subgroup of [Formula: see text]. In this paper, we prove that the multiplicities of the representations of [Formula: see text] induced from a parabolic subgroup [Formula: see text] in the regular representation on [Formula: see text] are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite. For the proof of this claim, we also show the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic [Formula: see text]-modules.

Unipotent representations attached to the principal nilpotent orbit

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 theK\mathbf {K}-types, associated varieties, and Langlands parameters of all such representations.