Graded characters of induced representations for real reductive Lie groups I

This is a semi–expository update and rewrite of my 1974 AMS AMS Memoir describing Plancherel formulae and partial Dolbeault cohomology realizations for standard tempered representations for general real reductive Lie groups. Even after so many years, much of that Memoir is up to date, but of course there have been a number of refinements, advances and new developments, most of which have applied to smaller classes of real reductive Lie groups. Here we rewrite that AMS Memoir in in view of these advances and indicate the ties with some of the more recent (or at least less classical) approaches to geometric realization of unitary representations.

Discrete Series Representations for the Orbit Spaces Arising from Two Involutions of Real Reductive Lie Groups

Let H⊂G be real reductive Lie groups and π an irreducible unitary representation of G. We introduce an algebraic formulation (discretely decomposable restriction) to single out the nice class of the branching problem (breaking symmetry in physics) in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction π| H . This paper offers basic algebraic properties of discretely decomposable restrictions, especially for a reductive symmetric pair (G,H) and for the Zuckerman-Vogan derived functor module , and proves that the sufficient condition [Invent. Math. '94] is in fact necessary. A finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum. An application to the restriction π| H of discrete series π for a symmetric space G/H is also given.

In recent years there has developed a satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters. These polynomials include as special cases: symmetric functions; zonal spherical functions on real and p-adic reductive Lie groups; the Jacobi polynomials of Heckman and Opdam; and the Askey–Wilson polynomials, which themselves include as special or limiting cases all the classical families of orthogonal polynomials in one variable. This book, first published in 2003, is a comprehensive and organised account of the subject aims to provide a unified foundation for this theory, to which the author has been a principal contributor. It is an essentially self-contained treatment, accessible to graduate students familiar with root systems and Weyl groups. The first four chapters are preparatory to Chapter V, which is the heart of the book and contains all the main results in full generality.

This book is an expanded version of the Hermann Weyl Lectures given at the Institute for Advanced Study in January 1986. It outlines some of what is now known about irreducible unitary representations of real reductive groups, providing fairly complete definitions and references, and sketches (at least) of most proofs. The first half of the book is devoted to the three more or less understood constructions of such representations: parabolic induction, complementary series, and cohomological parabolic induction. This culminates in the description of all irreducible unitary representation of the general linear groups. For other groups, one expects to need a new construction, giving "unipotent representations." The latter half of the book explains the evidence for that expectation and suggests a partial definition of unipotent representations.

Let G be a real reductive algebraic group with maximal compact subgroup K, and let Fr be a rank r free group. We show that the space of closed orbits in Hom(Fr ,G)/G admits a strong deformation retraction to the orbit space Hom(Fr ,K)/K. In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups G, such as Sp(4,ℝ) and U(2,2). We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

Approximating Fourier transformation of orbital integrals

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.

Representation theory in complex rank, II

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

Weights for symmetric and general linear groups

All the Lie algebras and Lie groups considered in this chapter are finite-dimensional; sometimes without mentioning this specifically we confine ourselves to a reductive Lie group, i.e., to a direct product of a simple group by a 1-dimensional center.

The asymptotic root distribution is computed for systems of matrix functions associated with finite-dimensional holomorphic representations of a Lie group. This distribution can be expressed via the increments of the representations involved. If the group is reductive, then the number of equations in the system can be arbitrary, from 1 to the dimension of the group. In this case, the computation results are stated in the language of convex geometry. These computations imply the previously known formulas for the density of the solution variety of a system of exponential equations as well as for the number of solutions of a polynomial system and, more generally, of a system formed by matrix functions of representations of a complex reductive Lie group.

One approach to constructing unitary representations for semisimple Lie groups utilizes analytic cohomology on open orbits of generalized flag manifolds. This work gives explicit formulas for harmonic cocycles associated to certain holomorphic homogeneous vector bundles, extending previous results of the author (Intertwining operators into cohomology representations for semisimple Lie groups, J. Funct. Anal. 151 (1997), 138–165). The key step shows that holomorphic discrete series representations and their limits are well-behaved with respect to restriction to certain submanifolds.