Harish-Chandra Class Research Articles

Hilbert-substructure of Real Measurable Spaces on Reductive Groups, I; Basic Theory

This paper reconsiders the age-long problem of normed linear spaces which do not admit inner product and shows that, for some subspaces, Fn(G), of real Lp(G)−spaces (when G is a reductive group in the Harish-Chandra class and p=2n), the situation may be rectified, via an outlook which generalizes the fine structure of the Hilbert space, L2(G). This success opens the door for harmonic analysis of unitary representations, G→End(Fn(G)), of G on the Hilbert-substructure Fn(G), which has hitherto been considered impossible.

The Borel–Weil theorem for reductive Lie groups

In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomolgy of an equivariant holomorphic line bundle defined on an open invariant submanifold of a complex flag space. Our main result is the following: suppose $G_{0}$ is a real reductive group of Harish-Chandra class and let $X$ be the associated full complex flag space. Suppose $\mathcal{O}_{\lambda}$ is the sheaf of sections of a $G_{0}$-equivariant holomorphic line bundle on $X$ whose parameter $\lambda$ (in the usual twisted $\mathcal{D}% $-module context) is antidominant and regular. Let $S\subseteq X$ be a $G_{0}% $-orbit and suppose $U\supseteq S$ is the smallest $G_{0}$-invariant open submanifold of $X$ that contains $S$. From the analytic localization theory of Hecht and Taylor one knows that there is a nonegative integer $q$ such that the compactly supported sheaf cohomology groups $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ vanish except in degree $q$, in which case $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ is the minimal globalization of an associated standard Beilinson-Bernstein module. In this study we show that the $q$-th compactly supported cohomolgy group $H_{\text{c}}^{q}(U,\mathcal{O}_{\lambda}\mid_{U})$ defines, in a natural way, a nonzero submodule of $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$, which is irreducible (i.e. realizes the unique irreducible submodule of $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$) when an associated algebraic variety is nonsingular. By a tensoring argument, we can show that the result holds, more generally (for nonsingular Schubert variety), when the representation $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ is what we call a classifying module.

THE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS

If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π|H contains each irreducible representation of H with finite multiplicity. In addition, if G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of π|H is constant along "continuous parameters". In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction π|H via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.

Radon transformation on reductive symmetric spaces: Support theorems

We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms.A reductive symmetric space is a homogeneous space G/H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G. Let P be a parabolic subgroup such that σ(P) is opposite to P and let NP be the unipotent radical of P. For a compactly supported smooth function ϕ on G/H, we define RP(ϕ)(g) to be the integral of NP∋n↦ϕ(gn⋅H) over NP. The Radon transform RP thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions.For these transforms we derive support theorems in which the support of ϕ is (partially) characterized in terms of the support of RPϕ. The proof is based on the relation between the Radon transform and the Fourier transform on G/H, and a Paley–Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.

Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups

Let $Q\rightarrow X$ be a principal bundle having as structuralgroup $G$ areductive Lie group in the Harish-Chandra class that includes the case when $G$ is semi-simple with finite center. A semiflow $\phi _{k}$ ofendomorphisms of $Q$ induces a semiflow $\psi _{k}$ on theassociated bundle $\mathbb{E}=Q\times _{G}\mathbb{F}$, where$\mathbb{F}$ is the maximal flag bundle of $G$. The $A$-componentof the Iwasawa decomposition $G=KAN$ yields an additive vectorvalued cocycle $\mathsf{a}\left( k,\xi \right) $, $\xi\in \mathbb{E}$, over $\psi _{k}$ with values in the Lie algebra $\mathfrak{a}$ of $A$. We prove the Multiplicative Ergodic Theorem ofOseledets for this cocycle: If $\nu $ is a probability measureinvariant by the semiflow on $X$ then the $\mathfrak{a}$-Lyapunovexponent $\lambda \left( \xi \right) =\lim\frac{1}{k}\mathsf{a}\left( k,\xi \right) $ exists for every $\xi$ on the fibers above a set of full $\nu $-measure. The level setsof $\lambda \left( \cdot \right) $ on the fibers are described inalgebraic terms. When $\phi _{k}$ is a flow the description of thelevel sets is sharpened. We relate the cocycle $\mathsf{a}\left(k,\xi \right) $ with the Lyapunov exponents of a linear flow on avector bundle and other growth rates.

Kazhdan-Lusztig algorithms for nonlinear groups and applications to Kazhdan-Patterson lifting

We establish an algorithm to compute characters of irreducible Harish-Chandra modules for a large class of nonalgebraic Lie groups. (Roughly speaking the class of groups consists of those obtained as nonlinear double covers of linear groups in Harish-Chandra's class.) We then apply this theory to study a particular group (the universal cover of the real general linear group), and discover a symmetry of the character computations encoded in a character multiplicity duality. Using this duality theory, we reinterpret a kind of representation-theoretic Shimura correspondence for the general linear group geometrically, and find that it is dual to an analogous lifting for indefinite unitary groups. It seems likely that this example is illustrative of a general framework for studying similar correspondences.

Approximating Fourier transformation of orbital integrals

In this paper, we are concerned with orbital integrals on a class C of real reductive Lie groups with non-compact Iwasawa K-component. The class C contains all connected semisimple Lie groups with infinite center. We establish that any given orbital integral over general orbits with compactly supported continuous functions for a group G in C is convergent. Moreover, it is essentially the limit of corresponding orbital integrals for its quotient groups in Harish-Chandra's class. Thus the study of orbital integrals for groups in class C reduces to those of Harish-Chandra's class. The abstract theory for this limiting technique is developed in the general context of locally compact groups and linear functionals arising from orbital integrals. We point out that the abstract theory can be modified easily to include weighted orbital integrals as well. As an application of this limiting technique, we deduce the explicit Plancherel formula for any group in class C .

A complete analogue of Hardy's theorem on semisimple Lie groups

A result by G. H. Hardy ([11]) says that if f and its Fourier transform f are O(|x|e 2 ) and O(|x|e 2/(4α)) respectively for some m,n ≥ 0 and α > 0, then f and f are P (x)e 2 and P (x)e /(4α) respectively for some polynomials P and P . If in particular f is as above, but f is o(e 2/(4α)), then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

The action of intertwining operators on spherical vectors in the minimal principal series of a reductive symmetric space

We study the action of standard intertwining operators on H-fixed generalized vectors in the minimal principal series of a reductive symmetric space G H of Harish-Chandra's class. The main result is that — after an appropriate normalization — this action is unitary for the unitary principal series. This is an extension of previous work under more restrictive hypotheses on G and H. The present result implies the Maass-Selberg relations for Eisenstein integrals of the minimal principal series. These play a fundamental role in the most-continuous part of the Plancherei decomposition for G H .

On the Decomposition of Langlands Subrepresentations for a Group in the Harish-Chandra Class

When a group G is in the Harish-Chandra class, the goal of classifying its tempered representations and the goal of decomposing the Langlands subrepresentation for any of its standard representations are equivalent. The main result of this work is given in Theorem (5.3.5) that consists of a formula for decomposing any Langlands subrepresentation for the group G . The classification of tempered representations is a consequence of this theorem (Corollary (5.3.6)).

On the decomposition of Langlands subrepresentations for a group in the Harish-Chandra class

When a group G G is in the Harish-Chandra class, the goal of classifying its tempered representations and the goal of decomposing the Langlands subrepresentation for any of its standard representations are equivalent. The main result of this work is given in Theorem (5.3.5) that consists of a formula for decomposing any Langlands subrepresentation for the group G G . The classification of tempered representations is a consequence of this theorem (Corollary (5.3.6)).

A convexity theorem for semisimple symmetric spaces

In this paper we prove a generalization of the convexity theorem in [3] for a symmetric space G/H. Here G is a connected semisimple Lie group with finite center, σ an involution of G and H an open subgroup of the group G of fixed points for σ. The generalization involves Iwasawa decompositions related to minimal parabolic subgroups of G of arbitrary type instead of the particular type of parabolic subgroup considered in [3]. From now on we assume more generally that G is a real reductive group of the HarishChandra class; this will allow an inductive argument relative to the real rank of G. Let θ : G → G be a Cartan involution of G that commutes with σ; for its existence, see [20, Thm. 6.16]. The associated group K := G of fixed points is a maximal compact subgroup of G. For the infinitesimal involutions determined by σ and θ we use the same symbols: θ, σ : g → g; here g denotes the Lie algebra of G. With respect to the infinitesimal involutions, g decomposes as