Minimal Parabolic Subgroup Research Articles

Erratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle $${\mathcal {E}}$$ on X. Let $$\mathfrak {h}$$ be the Lie algebra of H. Let $$\mathcal {S}(X,{\mathcal {E}})$$ be the space of Schwartz sections of $${\mathcal {E}}$$ . We prove that $$\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})$$ is a closed subspace of $$\mathcal {S}(X,{\mathcal {E}})$$ of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let $$\pi $$ be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants $$V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}$$ is an isomorphism if and only if any linear $$\mathfrak {h}$$ -invariant functional on V is continuous in the topology induced from $$\pi $$ . The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.

Finite orbit decomposition of real flag manifolds

Let G be a connected real semi-simple Lie group and H a closed connected subgroup. Let P be a minimal parabolic subgroup of G . It is shown that H has an open orbit on the flag manifold G/P if and only if it has finitely many orbits on G/P . This confirms a conjecture by T. Matsuki.

Congruence Subgroups of Lattices in Rank 2 Kac–Moody Groups over Finite Fields

Let G be a rank 2 complete affine or hyperbolic simply-connected Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let be the negative standard parabolic subgroup of G corresponding to w1. It is known that the subgroups U−, B− and , are nonuniform lattice subgroups of G. Here we construct an infinite sequence of congruence subgroups of as natural generalizations of the corresponding notions for lattices in Lie groups. We also show that the group U− contains analogous congruence subgroups. Our technique involves determining graphs of groups presentations for U−, B−, and with the fundamental apartment of the Bruhat–Tits tree X a quotient graph for U− and for B− on X. When k = 𝔽q and q = 2s, the graph of groups for has the the positive half of the fundamental apartment as quotient graph. We explicitly construct the graphs of groups for the principal (level 1) congruence subgroup of and the analogous subgroups of U− giving generalized amalgam presentations for them.

Convexity theorems for semisimple symmetric spaces

Abstract We prove a convexity theorem for semisimple symmetric spaces G / H ${G/H}$ which generalizes an earlier theorem of the second named author to a setting without restrictions on the minimal parabolic subgroup involved. The new more general result specializes to Kostant’s non-linear convexity theorem for a real semisimple Lie group G in two ways, firstly by taking H maximal compact and secondly by viewing G as a symmetric space for G × G ${G\times G}$ .

The local structure theorem for real spherical varieties

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q\times _{L}S$. Here $Q$ is a parabolic subgroup with Levi decomposition $L\ltimes U$, and $S$ is a homogeneous space for a quotient $D=L/L_{n}$ of $L$, where $L_{n}\subseteq L$ is normal, such that $D$ is compact modulo center.

STEPWISE SQUARE INTEGRABLE REPRESENTATIONS FOR LOCALLY NILPOTENT LIE GROUPS

In a recent paper we found conditions for a nilpotent Lie group N to have a filtration by normal subgroups whose successive quotients have square integrable representations, and such that these square integrable representations fit together nicely to give an explicit construction of Plancherel for almost all representations of N. The prototype for this sort of group is the group of upper triangular real matrices with 1’s down the diagonal. More generally, this class of groups contains the nilradicals of minimal parabolic subgroups of all (finite-dimensional) reductive real or complex Lie groups, in other words, all groups N in Iwasawa decompositions of reductive real or complex Lie groups.

Simple compactifications and polar decomposition of homogeneous real spherical spaces

Let Z be an algebraic homogeneous space Z=G/H attached to real reductive Lie group G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper.

Multiplicity bounds and the subrepresentation theorem for real spherical spaces

Let G be a real semi-simple Lie group and H a closed subgroup which admits an open orbit on the flag manifold of a minimal parabolic subgroup. Let V be a Harish-Chandra module. A sharp finite bound is given for the dimension of the space of H-fixed distribution vectors for V and a related subrepresentation theorem is derived. Extended final version. To appear in Trans. Amer. Math. Soc.

Decay of matrix coefficients on reductive homogeneous spaces of spherical type

Let \(Z\) be a homogeneous space \(Z=G/H\) of a real reductive Lie group \(G\) with a reductive subgroup \(H\). The investigation concerns the quantitative decay of matrix coefficients on \(Z\) under the assumption that \(Z\) is of spherical type, that is, minimal parabolic subgroups have open orbits on \(Z\).

CLASSIFICATION OF FINITE-MULTIPLICITY SYMMETRIC PAIRS

We give a complete classification of the reductive symmetric pairs (G, H) for which the homogeneous space (G × H)/ diag H is real spherical in the sense that a minimal parabolic subgroup has an open orbit. Combining with a criterion established in T. Kobayashi, T. Oshima, Adv. Math. 2013, we give a necessary and sufficient condition for a reductive symmetric pair (G, H) such that the multiplicities for the branching law of the restriction of any admissible smooth representation of G to H have finiteness/boundedness property.

A Satake isomorphism for representations modulo p of reductive groups over local fields

Abstract Let F be a local field with finite residue field of characteristic p. Let G be a connected reductive group over F and B a minimal parabolic subgroup of G with Levi decomposition B = Z U $B=ZU$ . Let K be a special parahoric subgroup of G, in good position relative to (Z,U). Fix an absolutely irreducible smooth representation of K on a vector space V over some field C of characteristic p. Writing ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ for the intertwining Hecke algebra of V in G, we define a natural algebra homomorphism from ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ to ℋ ( Z , Z ∩ K , V U ∩ K ) $\mathcal {H}(Z,Z\cap K,V^{U\cap K})$ , we show it is injective and identify its image. We thus generalize work of F. Herzig, who assumed F of characteristic 0, G unramified and K hyperspecial, and took for C an algebraic closure of the prime field 𝔽 p . We show that in the general case ℋ ( G , K , V ) $\mathcal {H}(G,K,V)$ need not be commutative; that is in contrast with the cases Herzig considers and with the more classical situation where V is trivial and the field of coefficients is the field of complex numbers.

Stepwise square integrable representations of nilpotent Lie groups

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of “stepwise square integrable” representations on which Plancherel measure is concentrated. Further, we work out the character formulae for those stepwise square integrable representations, and we give an explicit Plancherel formula. Next, we use some structure theory to check that all these constructions and results apply to nilradicals of minimal parabolic subgroups of real reductive Lie groups. Finally, we develop multiplicity formulae for compact quotients $$N/\varGamma $$ where $$\varGamma $$ respects the filtration.

On a lemma of Samokhin

Let G be a reductive algebraic group over an algebraically closed field \(\mathbb{k}\) of positive characteristic p, B a Borel subgroup of G, P a minimal parabolic subgroup of G containing B, and π: G/B→G/P the natural morphism. Using Orlov’s semiorthogonal decomposition of the bounded derived category of coherent sheaves on G/B to those on G/P, Samokhin derived a short exact sequence relating the Frobenius direct image of the structure sheaf of G/B to that of G/P, which was a key to his proof of the D-affinity of G/B for the symplectic group G of degree 4 over \(\mathbb{k}\). In this note we obtain his exact sequence from a short exact sequence of G n B-modules, G n the n-th Frobenius kernel of G, using representation theory.

Théorème de Paley—Wiener pour les fonctions de Whittaker sur un groupe réductif p-adique

RésuméSoit G un groupe réductif p-adique et U0 le radical unipotent d'un sous-groupe parabolique minimal de G. Nous introduisons une transformation de Fourier pour l'espace des fonctions de Whittaker lisses sur G et à support compact modulo U0. Nous en déterminons l'image. La preuve suit celle d'Heiermann pour les fonctions sur le groupe.Au cours de la preuve, une formule d'inversion est prouvée. Celle-ci permet de montrer qu'une représentation lisse irréductible de G, qui possède modèle de Whittaker dans les fonctions de Whittaker à support compact modulo U0, est cuspidale.Ce travail nous a donné l'opportunité de préparer un cadre pour l'analyse harmonique sur les espaces symétriques réductifs p-adiques: B-matrices et terme constant, propriétés des paquets d'ondes.

Spherical gradient manifolds

We study the action of a real-reductive group G=Kexp(𝔭) on a real-analytic submanifold X of a Kähler manifold. We suppose that the action of G extends holomorphically to an action of the complexified group G ℂ on this Kähler manifold such that the action of a maximal compact subgroup is Hamiltonian. The moment map induces a gradient map μ 𝔭 :X→𝔭. We show that μ 𝔭 almost separates the K–orbits if and only if a minimal parabolic subgroup of G has an open orbit. This generalizes Brion’s characterization of spherical Kähler manifolds with moment maps.

GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY

Abstract

On the residual spectrum of Hermitian quaternionic inner form of SO_8

In this paper we decompose the residual spectrum sup- ported in the minimal parabolic subgroup of an inner form of the split group SO8. The approach is the Langlands spectral theory. However, since the group is non-quasi-split, it is out of the scope of the Langlands-Shahidi method and the new technique for the normalization of standard intertwin- ing operators is developed. The decomposition shows interesting parts of the residual spectrum not appearing in the case of quasi-split groups.

The residual spectrum of an inner form of 𝑆𝑝₈ supported in the minimal parabolic subgroup

The part of the residual spectrum of an inner form of the split group S p 8 Sp_8 supported in the minimal parabolic subgroup is decomposed. Since the considered inner form is not quasi–split, the normalization of the standard intertwining operators, required for the calculation of the poles of the Eisenstein series, is out of the reach of the Langlands–Shahidi method. Hence, a normalization technique, based on the transfer of the Plancherel measure between the split group and its inner form, is applied. The obtained decomposition reveals certain features of the residual spectrum of the inner form which do not appear for the split group.

A remark on the identification of Lie-type groups as amalgams of minimal parabolic subgroups

A remark on the identification of Lie-type groups as amalgams of minimal parabolic subgroups

Proximality and Equidistribution on the Furstenberg Boundary

Let G be a connected semisimple Lie group with finite center and without compact factors, P a minimal parabolic subgroup of G, and Γ a lattice in G. We prove that every Γ-orbits in the Furstenberg boundary G/P is equidistributed for the averages over Riemannian balls. The proof is based on the proximality of the action of Γ on G/P.