We prove several multiplicity one theorems in this paper. For k a local field not of characteristic two, and V a symplectic space over k, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), where V is an orthogonal space over k. When k is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4).
Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we are able to prove the multiplicity one theorem for the Ginzburg-Rallis model over the Vogan packets in the tempered case. In some cases, we can also relate the multiplicity to the epsilon factor. This is a sequel of our work \cite{Wan15} in which we consider the supercuspidal case.
In this work we prove the local multiplicity at most one theorem underlying the definition and theory of local $γ$-, $ε$- and $L$-factors, defined by virtue of the generalized doubling method, over any local field of characteristic 0. We also present two applications: one to the existence of local factors for genuine representations of covering groups, the other to the global unfolding argument of the doubling integral.
Let G be a special orthogonal group or an inner form of a symplectic group over a number field F such that there exists a non-empty set S of real places of F at which G has discrete series and outside of which G is quasi-split. We prove Arthur's multiplicity formula for automorphic representations of G having algebraic regular infinitesimal character at all places in S.
Each orthogonal group $\OO(n)$ has a nontrivial $\GL(1)$-extension, which we call $\GPin(n)$. The identity component of $\GPin(n)$ is the more familiar $\GSpin(n)$, the general Spin group. We prove that the restriction to $\GPin(n-1)$ of an irreducible admissible representation of $\GPin(n)$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for $\GSpin(n)$. Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for $\OO(n)$, and Waldspurger, who proved that for $\SO(n)$. We also …
Let $π_1$ be a standard representation of $\mathrm{GL}_{n+1}(F)$ and let $π_2$ be the smooth dual of a standard representation of $\mathrm{GL}_n(F)$. When $F$ is non-Archimedean, we prove that $\mathrm{Ext}^i_{\mathrm{GL}_n(F)}(π_1, π_2)$ is $\cong \mathbb C$ when $i=0$ and vanishes when $i \geq 1$. The main tool of the proof is a notion of left and right Bernstein-Zelevinsky filtrations. An immediate consequence of the result is to give a new proof on the multiplicity at most one theorem. Along the way, we also study an application of an Euler-Poincaré pairing formula of D. Prasad on the coef…
The main result in this paper is the generalization of the Kostant-Rallis multiplicity formula to general θ--groups (in the sense of Vinberg). The special cases of the two most interesting examples one for $E_6$ (three qubits) and one for $E_8$ are given explicit formulas.
It is shown for all local fields $\mathbb{F}$ which are of characteristic different from $2$ that any distribution on $GL_{n+1}(\mathbb{F})$ which is invariant under conjugation by $GL_n(\mathbb{F})$ is also invariant under transposition. In this paper we give an adaptation of the proof of this theorem to fields of characteristic 2.
We show that certain paramodular cuspidal automorphic irreducible representations of $\mathrm{GSp}(4,\mathbb{A}_\mathbb{Q})$, which are not CAP, are globally generic. This implies a multiplicity one theorem for paramodular cuspidal automorphic representations. Our proof relies on a reasonable hypothesis concerning the non-vanishing of central values of automorphic $L$-series.
We set up a general framework to compute the exact multiplicity with which certain automorphic representations appear in both the cuspidal and Eisenstein cohomology of locally symmetric spaces. We apply this machinery to Eisenstein series on $GSp_4$ and split $G_2$. In the case of $G_2$, we will also obtain new information about the archimedean components of certain CAP representations using Arthur's conjectures.
In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an admissible irreducible representation of GL(n+1), when restricted to GL(n) decomposes with multiplicity one. Similar Theorems are obtained for orthogonal or unitary groups.
For every genuine irreducible admissible smooth representation $π$ of the metaplectic group $\widetilde{\Sp}(2n)$ over a p-adic field, and every smooth oscillator representation $ω_ψ$ of $\widetilde{\Sp}(2n)$, we prove that the tensor product $π\otimes ω_ψ$ is multiplicity free as a smooth representation of the symplectic group $\Sp(2n)$. Similar results are proved for general linear groups and unitary groups.
We prove a geometric criterion for the bounded multiplicity property of "small" infinite-dimensional representations of real reductive Lie groupsin both induction and restrictions. Applying the criterion to symmetric pairs, we give a full description of the triples $H \subset G \supset G'$ such that any irreducible admissible representations of $G$ with $H$-distinguished vectors have the bounded multiplicity property when restricted to the subgroup $G'$. This article also completes the proof of the general results announced in the previous paper [Adv. Math. 2021, Section 7].
We prove spectral analogues of the classical strong multiplicity one theorem for newforms. Let $Γ_1$ and $Γ_2$ be uniform lattices in a semisimple group $G$. Suppose all but finitely many irreducible unitary representations (resp. spherical) of $G$ occur with equal multiplicities in $L^{2}(Γ_1 \backslash G)$ and $L^{2}(Γ_2 \backslash G)$. Then $L^{2}(Γ_1 \backslash G) \cong L^{2}(Γ_2 \backslash G)$ as $G$ - modules (resp. the spherical spectra of $L^{2}(Γ_1 \backslash G)$ and $L^{2}(Γ_2 \backslash G)$ are equal).
This paper computes the Dirac cohomology $H_D(π)$ of irreducible unitary Harish-Chandra modules $π$ of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are shown to be unitarily induced from unipotent representations. When nonzero, there is a unique, multiplicity free $K-$type in $π$ contributing to $H_D(π)$. This confirms conjectures formulated by the first named author and Pandzic in 2011.
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.
Let $\mathbb{F}$ be a non-archimedean local field of positive characteristic different from 2. We consider distributions on $\mathrm{GL}(n+1,\mathbb{F})$ which are invariant under the adjoint action of $\mathrm{GL}(n,\mathbb{F})$. We prove that any such distribution is invariant with respect to transposition. This implies that the restriction to $\mathrm{GL}(n,\mathbb{F})$ of any irreducible smooth representation of $\mathrm{GL}(n+1,\mathbb{F})$ is multiplicity free.
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 show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).
We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.
We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various multiplicity-free theorems in representation theory for both finite and infinite dimensional cases in a systematic and synthetic manner. The key geometric condition in our theorem is an orbit-preserving anti-holomorphic diffeomorphism on the base space, which brings us to the concept of visible actions on complex manifolds.
We find upper and lower bounds of the multiplicities of irreducible admissible representations $π$ of a semisimple Lie group $G$ occurring in the induced representations $Ind_H^Gτ$ from irreducible representations $τ$ of a closed subgroup $H$. As corollaries, we establish geometric criteria for finiteness of the dimension of $Hom_G(π,Ind_H^G τ)$ (induction) and of $Hom_H(π|_H,τ)$ (restriction) by means of the real flag variety $G/P$, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.
Let F be either R or C. Consider the standard embedding GL(n,F)<GL(n+1,F) and the action of GL(n,F) on GL(n+1,F) by conjugation. In this paper we show that any GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. We show that this implies that for any irreducible admissible smooth Frechet representations $π$ of GL(n+1,F) and $τ$ of GL(n,F), $$dim Hom_{GL(n,F)}(π,τ) \leq 1.$$ For p-adic fields those results were proven in [AGRS].
Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with the maximal ideal $\wp$ and the finite residue field of characteristic $p.$ Let $\mathbf{G}$ be the General Linear or Special Linear group with entries from the finite quotients $\mathfrak{o}/\wp^\ell$ of $\mathfrak{o}$ and $\mathbf{U}$ be the subgroup of $\mathbf{G}$ consisting of upper triangular unipotent matrices. We prove that the induced representation $\mathrm{Ind}^{\mathbf{G}}_{\mathbf{U}}(θ)$ of $\mathbf{G}$ obtained from a ${\it non-degenerate}$ character $θ$ of $\mathbf{U}$ is multiplicity free for all …
This is my PhD thesis submitted to the Weizmann Institute of Science. It is based on the papers [AG08c], [AG08d], [AGRS07], [AGS08], [AGS09], [Aiz08] and [SZ08]. This thesis includes an introduction to Gelfand pairs and invariant distributions, a list of tools to work with invariant distributions oriented towards proving Gelfand property and a proof that the pair (GL(n+1,F),GL(n,F)) is a strong Gelfand pair. Namely, we prove that if $π$ is an irreducible admissible smooth representation of GL(n+1,F) and $ρ$ is an irreducible admissible smooth representation of GL(n,F) then $$dim Hom_{GL(n,F)}(…
Let $G$ be one of the classical Lie groups $\GL_{n+1}(\R)$, $\GL_{n+1}(\C)$, $\oU(p,q+1)$, $\oO(p,q+1)$, $\oO_{n+1}(\C)$, $\SO(p,q+1)$, $\SO_{n+1}(\C)$, and let $G'$ be respectively the subgroup $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\oO(p,q)$, $\oO_n(\C)$, $\SO(p,q)$, $\SO_n(\C)$, embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\GL_{n}(\R)\ltimes \oH_{2n+1}(\R)$, $\GL_{n}(\C)\ltimes…
We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.
We call the dimension data $\mathscr{D}_{H_{1}}$ and $\mathscr{D}_{H_{2}}$ of two closed subgroups $H_{1}$ and $H_{2}$ of a given compact Lie group $G$ {\it almost equal} if $\mathscr{D}_{H_{1}}(ρ)=\mathscr{D}_{H_{2}}(ρ)$ for all but finitely many irreducible complex linear representations $ρ$ of $G$ up to equivalence. When $G$ is connected, we show that: if $\mathscr{D}_{H_{1}}$ and $\mathscr{D}_{H_{2}}$ are almost equal, then they are equal. When $G$ is non-connected, $G^{0}\subset H_{1}\cap H_{2}$ is a trivial sufficient condition for $\mathscr{D}_{H_{1}}$ and $\mathscr{D}_{H_{2}}$ to be al…
We present a conjecture on multiplicity of irreducible representations of a subgroup $H$ contained in the irreducible representations of a group $G$, with $G$ and $H$ having the same derived groups. We point out some consequences of the conjecture, and verification of some of the consequences. We give an explicit example of multiplicity $2$ upon restriction, as well as certain theorems in the context of classical groups where the multiplicity is $1$.
Let F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G:=GL(n+k,F) and let M:=GL(n,F) x GL(k,F)<G be a maximal Levi subgroup. Let U< G be the corresponding unipotent subgroup and let P=MU be the corresponding parabolic subgroup. Let J denote the Jacquet functor from representations of G to representations of M (i.e. the functor of coinvariants w.r.t. U). In this paper we prove that J is a multiplicity free functor, i.e. dim Hom(J(π),ρ)<= 1, for any irreducible representations πof G and ρof M. To do that we adapt the classical method of Ge…
Let $P$ be a minimal parabolic subgroup of a real reductive Lie group $G$ and $H$ a closed subgroup of $G$. Then it is proved by T. Kobayashi and T. Oshima that the regular representation $C^{\infty}(G/H)$ contains each irreducible representation of $G$ at most finitely many times if the number of $H$-orbits on $G/P$ is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of $H_{\mathbb C}$-orbits on $G_{\mathbb C}/B$ is finite, where $G_{\mathbb C}, H_{\mathbb C}$ are complexifications of $G, H$, respectively, and $B$ is a Borel subgroup of $G_{\mathb…
Publications on automorphic representations of the group U(3) assumed the validity of multiplicity one theorem since I claimed it in 1982. But the argument, published 1988, was based on a misinterpretation of a claim of Gelbart and Piatetski-Shapiro, SLN 1041 (1984), Prop. 2.4(i): ``L^2_{0,1} has multiplicity 1'', as meaning that each irreducible in the space L^2_{0,1} of generic cusp forms has multiplicity one in the space L^2_0 of cusp forms. The statement meant in SLN 1041, ``each irreducible in L^2_{0,1} has multiplicity 1 in L^2_{0,1}'' is too weak to be useful for nongeneric representati…
In the first part, in the local non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We conjecture that such distributions are invariant by transposition. This would imply multiplicity at most one for restrictions from GL(n+1) to GL(n). We reduce ourselves to distributions with "singular" support and then finish the proof for n< 9. In the second part we show that similar Theorems for orthogonal or unitary groups follow from the case of GL(n)
Let $G$ be a compact connected semisimple Lie group, let $K$ be a closed subgroup of $G$, let $Γ$ be a finite subgroup of $G$, and let $τ$ be a finite-dimensional representation of $K$. For $π$ in the unitary dual $\widehat G$ of $G$, denote by $n_Γ(π)$ its multiplicity in $L^2(Γ\backslash G)$. We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the $n_Γ(π)$ for $π$ in the set $\widehat G_τ$ of irreducible $τ$-spherical representations of $G$. More precisely, for $Γ$ and $Γ'$ finite subgroups of $G$, we prove that if $n_Γ(π)= n_{Γ'}(π)$ for all but finitely many …
Divergence-free Lie algebras (also known as the special Lie algebras of Cartan type) are Lie algebras of volume-preserving transformation groups. They are simple in generic case. Dokovic and Zhao found a certain graded generalization of them. In this paper, we classify all the irreducible and indecomposable multiplicity-free modules of the simple generalized divergence-free Lie algebras.
Branching of symplectic groups is not multiplicity-free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra $B$. The algebra $B$ is a graded algebra whose components encode the multiplicities of irreducible representations of $Sp_{2n-2}$ in irreducible representations of $Sp_{2n}$. Our first theorem states that the map taking an element of $Sp_{2n}$ to its principal $n \times (n+1)$ submatrix induces an isomorphism of $\B$ to a different branching algebra $\B'$. The algebra $\B'$ encodes multiplicities of irreducible represen…
In this note we prove a certain multiplicity formula regarding the restriction of an irreducible admissible genuine representation of a 2-fold cover $\widetilde{\rm GL}_{2}(E)$ of ${\rm GL}_{2}(E)$ to the 2-fold cover $\widetilde{\rm SL}_{2}(E)$ of ${\rm SL}_{2}(E)$, and find in particular that this multiplicity may not be one, a result that seems to have been noticed before. The proofs follow the standard path via Waldspurger's analysis of theta correspondence between $\widetilde{\rm SL}_{2}(E)$ and ${\rm PGL}_{2}(E)$.
Let F be a non-archimedean local field of characteristic zero. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that any such distribution is invariant with respect to transposition. This implies that the restriction to GL(n) of any irreducible smooth representation of GL(n+1) is multiplicity free. Our paper is based on the recent work [RS] of Steve Rallis and Gerard Schiffmann where they made a remarkable progress on this problem. In [RS], they also show that our result implies multiplicity one theorem for restrictions from the orthogona…
We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated to Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.
