Let $G$ be a reductive group in the Harish-Chandra class e.g. a connected semisimple Lie group with finite center, or the group of real points of a connected reductive algebraic group defined over $\R$. Let $σ$ be an involution of the Lie group $G$, $H$ an open subgroup of the subgroup of fixed points of $σ$. One decomposes the elements of $L^2(G/H)$ with the help of joint eigenfunctions under the algebra of left invariant differential operators under $G$ on $G/H$.
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…
The c-functions, related to a reductive symmetric space G/H and a fixed representation of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions.
We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and study some of their properties. In particular we obtain a support theorem that generalizes Helgason's support theorem for the horospherical transform on a Riemannian symmetric space.
Let G/H be a reductive symmetric space and K a maximal compact subgroup of G. We study Fourier transforms of compactly supported K-finite distributions on G/H and characterize the image of the space of such distributions.
Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-finite compactly supported smooth functions on X is characterized.
We give a classification of the triples (g,g',q) such that Zuckerman's derived functor (g,K)-module A_q(λ) for a θ-stable parabolic subalgebra q is discretely decomposable with respect to a reductive symmetric pair (g,g'). The proof is based on the criterion for discretely decomposable restrictions by the first author and on Berger's classification of reductive symmetric pairs.
For reductive symmetric spaces G/H of split rank one we identify a class of minimal parabolic subgroups for which certain cuspidal integrals of Harish-Chandra - Schwartz functions are absolutely convergent. Using these integrals we introduce a notion of cusp forms and investigate its relation with representations of the discrete series for G/H.
We construct minimal Eisenstein integrals for a reductive symmetric space G/H as matrix coefficients of the minimal principal series of G. The Eisenstein integrals thus obtained include those from the σ-minimal principal series. In addition, we obtain related Eisenstein integrals, but with different normalizations. Specialized to the case of the group, this wider class includes Harish-Chandra's minimal Eisenstein integrals.
The asymptotic behavior of holomorphic families of generalized eigenfunctions on a reductive symmetric space is studied. The family parameter is a complex character on the split component of a parabolic subgroup. The main result asserts that the family vanishes if a particular asymptotic coefficient does. This allows an induction of relations between families that will be applied in forthcoming work on the Plancherel and the Paley-Wiener theorem.
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.
We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space. Our starting point is an inversion formula for spherical smooth compactly supported functions. The latter formula was earlier obtained from the most continuous part of the Plancherel formula by means of a residue calculus. In the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized Eisenstein integrals and of the Maass-Selberg relations satisfied by the associated C-functions.
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 $G/H$.
Let $G$ be a complex reductive group and $H=G^θ$ be its fixed point subgroup under a Galois involution $θ$. We show that any $H$-distinguished representation $π$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\neq0$) satisfies: 1) $π^θ\simeq\tildeπ$, where $\tildeπ$ is the contragredient representation and $π^θ$ is the twist of $π$ under $θ$. 2) $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.
We form wave packets in the Schwartz space of a reductive p-adic symmetric space for certain famillies of tempered functions. We show how to construct such families from Eisenstein integrals.
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…
In this paper, we established the relationship among left-invariant flat connections on Lie groups, left-symmetric algebras, Frobenius Lie algebras and étale prehomogeneous vector spaces, gave a one-to-one correspondence between the left-symmetric Lie algebras with a right identity and the étale prehomogeneous vector spaces for a Lie group, and proved that, in essence, any left-symmetric structure on a reductive Lie algebra has a right identity, which implies that the classification of flat connections on a reductive Lie group $G$ amounts to that of étale prehomogeneous vector spaces for $G$. …
We show that Arthur's Paley-Wiener theorem for K-finite compactly supported smooth functions on a real reductive Lie group G of the Harish-Chandra class can be deduced from the Paley-Wiener theorem we established in the more general setting of a reductive symmetric space. In addition, we formulate an extension of Arthur's theorem to K-finite compactly supported generalized functions (distributions) on G and show that this result follows from the analogous result for reductive symmetric spaces as well.
We study a notion of cusp forms for the symmetric spaces G/H with G = SL(n,R) and H = S(GL(n-1,R) x GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series of representations of G/H coincides with the space of cusp forms.
This article is an expository paper. We first survey developments over the past three decades in the theory of harmonic analysis on reductive symmetric spaces. Next we deal with the particular homogeneous space of non-reductive type, the so called Siegel-Jacobi space that is important arithmetically and geometrically. We present some new results on the Siegel-Jacobi space.
Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of $L^2(X)$. Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on $X$. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.
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…
Let $G$ be a finite reductive group defined over a finite field $F_q$. In the case where $G$ is a special linear group, we compute the multiplicities of irreducible characters of $G(F_{q^2})$ with the character of $G(F_{q^2})$ induced from the trivial character of $G(F_q)$. We discuss the relationship between these multiplicities with the theory of Shintani descent for finite reductive groups in general. We also give some formula concerning the decomposition of the character of $G(F_{q^r})$ induced from the trivial character of $G(F_q)$ for reductive groups $G$ with any positive integer $r$.
The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain som…
We consider symmetric pairs of Lie superalgebras which are strongly reductive and of even type, and introduce a graded Harish-Chandra homomorphism. We prove that its image is a certain explicit filtered subalgebra of the Weyl invariants on a Cartan subspace whose associated graded is the image of Chevalley's restriction map on symmetric invariants. This generalises results of Harish-Chandra and V. Kac, M. Gorelik.
We define and study the variety of reductions for a reductive symmetric pair (G,theta), which is the natural compactification of the set of the Cartan subspaces of the symmetric pair. These varieties generalize the varieties of reductions for the Severi varieties studied by Iliev and Manivel, which are Fano varieties. We develop a theoretical basis to the study these varieties of reductions, and relate the geometry of these variety to some problems in representation theory. A very useful result is the rigidity of semi-simple elements in deformations of algebraic subalgebras of Lie algebras. We…
For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the alternating or symmetric square representation is irreducible or decomposes into an irreducible representation and a trivial representation.
Yiannis Sakellaridis and Akshay Venkathesh have determined, when the group $G$ is split and the field $\F$ is of characteristic zero, the Plancherel formula for any spherical space $X$ for $G$ modulo the knowledge of the discrete spectrum. The starting point is the determination of good neighborhoods at infinity of $X/J$, where $J$ is a small compact open subgroup of $G$. These neighborhoods are related to "boundary degenerations" of $X$. The proof of their existence is made by using wonderful compactifications. In this article we will show the existence of such neighborhoods assuming that $\F…
Let F be a non Archimedean locally compact field of residue characteristic different from 2, let G be a connected reductive group defined over F, let s be an involutive F-automorphism of G and H an open F-subgroup of the fixed points group of s. We denote by G(F) (resp. H(F)) the group of F-points of G (resp. H). In this paper, we obtain an analogue of the Cartan decomposition for the reductive symmetric space G(F)/H(F). More precisely, we obtain a decomposition of G(F) as a union of H(F)-cosets which is related to the H(F)-conjugacy classes of maximal s-anti-invariant F-split tori in G. When …
The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh--Gordan--Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua--Kostant--Schmid $K$-type formula, and the …
In this paper we examine various properties/constructions which are known for reductive groups and we do some experiments to see to what extent they generalize to symmetric spaces.
This is a survey article on the Springer correspondence for symmetric spaces. We discuss various generalization of the theory of the Springer correspondence for reductive groups to symmetric spaces and exotic symmetric spaces associated to classical groups.
We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces $Z=G/H$ attached to a real reductive Lie group $G$. A special emphasis is made to the case where $Z$ is real spherical.
Let $g_e$ be the centraliser of a nilpotent element $e$ in a finite dimensional simple Lie algebra $g$ of rank $l$ over an algebraically closed field of characteristic 0. We investigate the algebra $S(g_e)^{g_e}$ of symmetric invariants of $g_e$ and prove that if $g$ is of type $A$ or $C$, then $S(g_e)^{g_e}$ is always a graded polynomial algebra in $l$ variables. We show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type $A$ we prove that $S(g_e)^{g_e}$ is freely generated by a regular sequence in $S(g_e)$ and describe the tangent cone at $e$ t…
We study the coadjoint representation of contractions of reductive Lie algebras associated with symmetric decompositions. Let $\frak g=\frak g_0\oplus \frak g_1$ be a symmetric decomposition of a reductive Lie algebra $\frak g$. Then the semi-direct product of $\frak g_0$ and the $\frak g_0$-module $\frak g_1$ is a contraction of $\frak g$. We conjecture that these contractions have many properties in common with reductive Lie algebras. In particular, it is proved that in many cases the algebra of invariants is polynomial. We also discuss the so-called "codim--2 property" for coadjoint represe…
Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if $K$ has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in $g(K)$ by locally constant functions, which, extended by zero to all of $g(K)$, are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability [R. Cluckers, J. Gordon, I. Halupcz…
If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $τ$ is a unitary representation of $H$, then the authors give a sufficient condition on $ξ\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^Gτ$. In the special case where $τ$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $π$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $\operatorname{WF}(π)$ in terms of the direct integral decomposition of $π$ int…
We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive Lie algebras.
Let (g,k) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a in p. The restriction map S(p^*)^k->S(a^*)^W where W=W(g_0:a) is the Weyl group, is injective. We determine its image explicitly. In particular, our theorem applies to the case of a symmetric superpair of group type, i.e. (k+k,k) with the flip involution where k is a classical Lie superalgebra with a non-degenerate invariant even form (equivalently, a finite-dimensional contragredient Lie superalgebra). Thus, we obtain a new proof of the generalisation of Chevalley's restriction theor…
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…
The author classifies Klein four symmetric pairs of holomorphic type for non-compact Lie group $\mathrm{E}_{6(-14)}$, which gives a class of pairs $(G,G')$ of real reductive Lie group $G$ and its reductive subgroup $G'$ such that there exist irreducible unitary representations $π$ of $G$, which are admissible upon restriction to $G'$.
Let $G$ be a connected semisimple noncompact real Lie group and let $ρ: G \longrightarrow \mathrm{SL}(V)$ be a representation on a finite dimensional vector space $V$ over $\mathbb R$, with $ρ(G)$ closed in $\mathrm{SL}(V)$. Identifying $G$ with $ρ(G)$, we assume there exists a $K$-invariant scalar product $\mathtt g$ such that $G=K\exp(\mathfrak p)$, where $K=\mathrm{SO}(V,\mathtt g)\cap G$, $\mathfrak p=\mathrm{Sym}_o (V,\mathtt g)\cap \mathfrak g$ and $\mathfrak g$ denotes the Lie algebra of $G$. Here $\mathrm{Sym}_o (V,\mathtt g)$ denotes the set of symmetric endomorphisms with trace zero.…
In this paper we study the Plancherel formula for a new class of homogeneous spaces for real reductive Lie groups; these spaces are fibered over non-Riemannian symmetric spaces, and they exhibit a phenomenon of uniform infinite multiplicities. They also provide examples of non-tempered representations of the group appearing in the Plancherel formula. Several classes of examples are given.
We consider a see-saw pair consisting of a Hermitian symmetric pair (G_R, K_R) and a compact symmetric pair (M_R, H_R), where (G_R, H_R) and (K_R, M_R) form real reductive dual pairs in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain K_C-invariant elements in U(Lie(G)_C) in terms of H_C-invariant elements in U(Lie(M)_C). The corresponding H_C-invariant elements are called Capelli elements. We also give a decomposition of the intersection of O_{2n}-harmonics and Sp_{2n}-harmonics as a module of GL_n = O_{2n} \cap Sp_{2n}, and construct a b…
The main result of this paper is a new parameterization of the orbits of a symmetric subgroup on a partial flag variety. The parameterization is in terms of certain Spaltenstein varieties, on one hand, and certain nilpotent orbits, on the other. One of the motivations is related to enumerating special unipotent representations of real reductive groups. Another motivation is understanding (a portion of) the closure order on the set of nilpotent coadjoint orbits.
The coadjoint representation of a connected algebraic group $Q$ with Lie algebra $\mathfrak q$ is a thrilling and fascinating object. Symmetric invariants of $\mathfrak q$ (= $\mathfrak q$-invariants in the symmetric algebra $S(\mathfrak q)$) can be considered as a first approximation to the understanding of the coadjoint action $(Q:\mathfrak q^*)$ and coadjoint orbits. In this article, we study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. If $G$ is a se…
We give an overview on the c-function of a non-compactly causal symmetric space G/H and explain its interplay with harmonic analysis and representation theory.
We give a complete classification of the reductive symmetric pairs (G,H) for which the homogeneous space $(G \times H)/diag(H)$ 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 any admissible smooth representation of $G$ to $H$ have finiteness/boundedness property.
In this article we study, in the context of complex representations of symmetric groups, some aspects of the Heisenberg product, introduced by Marcelo Aguiar, Walter Ferrer Santos, and Walter Moreira in 2017. When applied to irreducible representations, this product gives rise to the Aguiar coefficients. We prove that these coefficients are in fact also branching coefficients for representations of connected complex reductive groups. This allows to use geometric methods already developped in a previous article, notably based on notions from Geometric Invariant Theory, and to obtain some stabil…
Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in [math.RT/0112251]. That paper also introduced the micro-support of an L-module, a combinatorial invariant that to a great extent characterizes the cohomology of the associated sheaf. The theory has been successfully applied to solve a number of problems concerning the intersection cohomology and weighted cohomology of the reductive Borel-Serre compactification [m…
