We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I (math.RT/0107063). The formula for Schwartz functions involves Eisenstein integrals obtained by a residual calculus. In the present paper we identify these integrals as matrix coefficients of the generalized principal series.
The aim of this article is to present a simple generalized Plancherel formula for a locally compact unimodular topological group G of type I. This formula specializes to the Whittaker-Plancherel formula for a split reductive p-adic group of Sakellaridis-Venkatesh and differs from that of a quasi-split p-adic group due to Delorme. Furthermore, it also applies to certain metaplectic groups and other interesting situations where the local theory of distinguished representations has been studied.
Given a number field $F$ and a reductive group $G$ over $F$, the unitary dual $\hat{G(\mathbb{A}_F)}$ of the adelic group $G(\mathbb{A}_F)$ and the Placherel measure $ν_{G(\mathbb{A}_F)}$ on it can be determined by the Plancherel measure of its local groups $G(F_v)$. Given a subset $X\subset \hat{G(\mathbb{A}_F)}$ of finite Plancherel measure, let $H_X$ be the direct integral of the irreducible representations in $X$. Besides a $G(\mathbb{A}_F)$-module and a $G(F)$-module, $H_X$ is also a module over the group von Neumann algebra $\mathcal{L}(G(F))$, hence there is a canonical dimension $\dim_…
In this article we give a precise description of the Plancherel decomposition of the most continuous part of $L^{2}(Z)$ for a real spherical homogeneous space $Z$. Our starting point is the recent construction of Bernstein morphisms by Delorme, Knop, Krötz and Schlichtkrull. The most continuous part decomposes into a direct integral of unitary principal series representations. We give an explicit construction of the $H$-invariant functionals on these principal series. We show that for generic induction data the multiplicity space equals the full space of $H$-invariant functionals. Finally, we …
Let $G/H$ be a reductive symmetric space of split rank $1$ and let $K$ be a maximal compact subgroup of $G$. In a previous article the first two authors introduced a notion of cusp forms for $G/H$. We show that the space of cusp forms coincides with the closure of the $K$-finite generalized matrix coefficients of discrete series representations if and only if there exist no $K$-spherical discrete series representations. Moreover, we prove that every $K$-spherical discrete series representation occurs with multiplicity $1$ in the Plancherel decomposition of $G/H$.
There are several global functionals on irreducible automorphic representations which are Eulerian, that is: pure tensors of local functionals, when the representation is written as an Euler product $π= \otimes'_v π_v$ of local representations. The precise factorization of such functionals is of interest to number theorists and is -- naturally -- very often related to special values of $L$-functions. The purpose of this paper is to develop in full generality the Plancherel formula for the Weil or oscillator representation, considered as a unitary representation of a reductive dual pair, and to…
In this paper, we establish a Whittaker-Plancherel inversion formula for $\mathrm{SL}_2(\mathbb{C})$ from the analytic perspective of the Bessel transform of Bruggeman and Motohashi. The formula gives a decomposition of the Whittaker-Fourier coefficient of a compactly supported function on $\mathrm{SL}_2(\mathbb{C})$ in terms of its Bessel coefficients attached to irreducible unitary tempered representations of $\mathrm{SL}_2(\mathbb{C})$.
We study the principal series representations of central extensions of a split reductive algebraic group by a cyclic group of order $n$. We compute the Plancherel measure of the representation using Eisenstein series and a comparison method. In addition, we construct genuine central characters of the metaplectic torus in the simply-laced case.
We present a finite algorithm for computing the set of irreducible unitary representations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant Hermitian form as a deformation of one of the unitary representations in Harish-Chandra's Plancherel formula. The behavior of these deformations was determined to a first approximation in the Kazhdan-Lusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. The basic idea of our algor…
We provide a Plancherel decomposition for the space of symplectic bilinear forms of rank $2n$ over a local non-archimedean field $F$ in terms of that of $GL_n(F)$.
A Cartan decomposition for symmetric pairs plays an important role to study not only orbit geometry of the symmetric spaces but also harmonic analysis on them. For non-symmetric reductive pairs, there are examples of generalizations of Cartan decompositions for some spherical complex homogeneous spaces such as complex line bundles over the complexified Hermitian symmetric spaces and triple spaces. This paper provides new examples of a Cartan decomposition for non-symmetric reductive pairs, namely, reductive non-symmetric spherical pairs of rank-one type. We also show that the action of some co…
This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the associated $L^2$ space thus confirming a conjecture of Sakellaridis-Venkatesh in this particular case. The second is a formula for the multiplicities of generic representations in the $p$-adic case that extends previous work of Feigon-Lapid-Offen. Both results are stated in terms of Arthur-Clozel's quadratic local base-change and the proofs are based on local …
Cocenters of Hecke algebras $\mathcal H$ play an important role in studying mod $\ell$ or $\mathbb C$ harmonic analysis on connected $p$-adic reductive groups. On the other hand, the depth $r$ Hecke algebra $\mathcal H_{r^+}$ is well suited to study depth $r$ smooth representations. In this paper, we study depth $r$ rigid cocenters $\overline{\mathcal H}^{\mathrm{rig}}_{r^+}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\ell\neq p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an exp…
The restriction of an irreducible unitary representation $π$ of a real reductive group $G$ to a reductive subgroup $H$ decomposes into a direct integral of irreducible unitary representations $τ$ of $H$ with multiplicities $m(π,τ)\in\mathbb{N}\cup\{\infty\}$. We show that on the smooth vectors of $π$, the direct integral is pointwise defined. This implies that $m(π,τ)$ is bounded above by the dimension of the space $\operatorname{Hom}_H(π^\infty|_H,τ^\infty)$ of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between t…
We establish an explicit Plancherel decomposition for $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$ where $E/F$ is a quadratic extension of local fields of characteristic zero by making use of a local functional equation for Asai $γ$-factors. We also give two applications of this Plancherel formula: first to the global Ichino-Ikeda conjecture for unitary groups by completing a comparison between local relative characters that was left open by W. Zhang and secondly to the Hiraga-Ichino-Ikeda conjecture on formal degrees in the case of unitary groups.
Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda conjectured that the formal degree of a square-integrable G-representation $π$ can be expressed in terms of the adjoint $γ$-factor of the enhanced L-parameter of $π$. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations. We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $γ$-factors.
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…
We prove the existence of Bialynicki-Birula decomposition for Kempf monoids, which is a large class that contains for example monoids with reductive unit group in all characteristics. This extends the existence statements from previous works of Alper-Hall-Rydh and Jelisiejew-Sienkiewicz.
Starting with a highest weight representation of a Kac-Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac-Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highes…
Starting with a highest weight representation of a Kac-Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac-Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highes…
We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic $0$ and those in characteristic $p$ (for groups defined over discrete valuation rings). For each of these topics, we obtain natural generalizations of the well-known results for connected reductive groups.
We generalize the Białynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Białynicki-Birula decomposition functorially: for a fixed ${\bf G}$-scheme $X$ and a monoid $\overline{\bf G}$ which partially compactifies ${\bf G}$, the BB decomposition parameterizes ${\bf G}$-schemes over $X$ for which the ${\bf G}$-action extends to the $\overline{\bf G}$-action. The freedom of choice of $\ov…
Let G be a simple complex algebraic group and let K be a reductive subgroup of G such that the coordinate ring of G/K is a multiplicity free G-module. We consider the G-algebra structure of C[G/K], and study the decomposition into irreducible summands of the product of irreducible G-submodules in C[G/K]. When the spherical roots of G/K generate a root system of type A we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system gen…
Let $\mathbf{G}$ be a connected reductive group with connected center defined over $\mathbb{F}_q$, with Frobenius morphism F. Given an irreducible complex character $χ$ of $\mathbf{G}^F$ with its Jordan decomposition, and a Galois automorphism $σ\in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we give the Jordan decomposition of the image ${^σχ}$ of $χ$ under the action of $σ$ on its character values.
Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $σ$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $σ.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.
The theory of almost characters which is closely related to character sheaves is proposed by Lusztig to study the representation theory of finite reductive groups. In this article we show that the decomposition of the Weil character for finite reductive dual pairs $(\Sp_{2n},\rmO^\pm_{2n'})$ or $(\Sp_{2n},\SO_{2n'+1})$ with respect to the almost characters is exactly the same as the decomposition with respect to the irreducible characters. As a consequence, the finite theta correspondence on almost characters is established.
We show that, for an arbitrary quasi-reductive Lie superalgebra with a triangular decomposition and a character $ζ$ of the nilpotent radical, the associated Backelin functor $Γ_ζ$ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category $\mathcal O$. In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor $Γ_ζ$ and its target category, respect…
