A note on gamma factors for pairs
In this short note we observe that the gamma factor defined by Gelfand and Kazhdan coincides with the Rankin-Selberg root number defined by Jacquet, Piatetskii-Shapiro and Shalika.
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arXiv subjects
Omer Offen
Publications and source records attributed to Omer Offen.
Intertwining periods, L-functions and local-global principles for distinction of automorphic representations
We provide a criterion for non-vanishing of period integrals on automorphic representations of a general linear group over a division algebra. We consider three different periods: linear periods, twisted-linear periods and Galois periods. Our criterion is a local-global principle, which is stated in terms of local distinction, a further local obstruction, and poles of certain global L-functions associated to the underlying involution via the Jacquet-Langlands correspondence. Our local-global principle follows from a new method, relying on the Maass-Selberg relations and a careful analysis of singularities of local and global intertwining periods. Our results generalize to inner forms, known results for split general linear groups. Moreover, our result for twisted linear periods is new even in the split situation. As a consequence of our local-global principle, we complete the proof of one direction of the Guo-Jacquet conjecture.
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On the cubic Shimura lift to $PGL(3)$: Hecke correspondences
In this paper we establish a new Fundamental Lemma for Hecke correspondences. Let $F$ be a local field containing the cube roots of unity. We exhibit an algebra isomorphism of the spherical Hecke algebra of $PGL_3(F)$ and the spherical Hecke algebra of anti-genuine functions on the cubic cover $G'$ of $SL_3(F)$. Then we show that there is a matching (up to a specific transfer factor) of distributions on the two groups for all functions that correspond under this isomorphism. On $PGL_3(F)$ the distributions are relative distributions attached to a period involving the minimal representation on $SO_8$, while on $G'$ they are metaplectic Kuznetsov distributions. This Fundamental Lemma is a key step towards establishing a relative trace formula that would give a new global Shimura lift from genuine automorphic representations on the triple cover of $SL_3$ to automorphic representations on $PGL_3$, and also characterize the image of the lift by means of a period. It extends the matching for the unit elements of the Hecke algebras established by the authors in prior work.
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On residual automorphic representations and period integrals for symplectic groups
We construct new irreducible components in the discrete automorphic spectrum of symplectic groups. The construction lifts a cuspidal automorphic representation of $\mathrm{GL}_{2n}$ with a linear period to an irreducible component of the residual spectrum of the rank $k$ symplectic group $\mathrm{Sp}_k$ for any $k\ge 2n$. We show that this residual representation admits a non-zero $\mathrm{Sp}_n\times \mathrm{Sp}_{k-n}$-invariant linear form. This generalizes a construction of Ginzburg, Rallis and Soudry, the case $k=2n$, that arises in the descent method.
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On local intertwining periods
We prove the absolute convergence, functional equations and meromorphic continuation of local intertwining periods on parabolically induced representations of finite length for certain symmetric spaces over local fields of characteristic zero, including Galois pairs as well as pairs of Prasad and Takloo-Bighash type. Furthermore, for a general symmetric space we prove a sufficient condition for distinction of an induced representation in terms of distinction of its inducing data. Both results generalize previous results of the first two named authors. In particular, for both we remove a boundedness assumption on the inducing data and for the second we further remove any assumption on the symmetric space. Moreover, when the inducing representation is uniformly bounded, we extend the field of cofficients from p-adic to any local field of characteristic zero. In fact this extension holds for all finite length representations under a natural generic irreducibility assumption for parabolic induction. In the case of p-adic symmetric spaces, combined with the necessary conditions for distinction that follow from the geometric lemma, this provides a necessary and sufficient condition for distinction of representations induced from cuspidal.
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On the Cubic Shimura lift to $PGL(3)$: The Fundamental Lemma
The classical Shimura correspondence lifts automorphic representations on the double cover of $SL_2$ to automorphic representations on $PGL_2$. Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of $SL_3$ to $PGL_3$, and also characterize the image of the lift. The characterization would be through the nonvanishing of a certain global period involving a function in the space of the automorphic minimal representation $\Theta_{SO_8}$ for split $SO_8({\mathbb{A}})$, consistent with a 2001 conjecture of Bump, Friedberg and Ginzburg. In this paper, we first analyze a global distribution on $PGL_3({\mathbb{A}})$ involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of $SL_3({\mathbb{A}})$. We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the Fundamental Lemma.
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Intertwining periods and distinction for p-adic Galois symmetric pairs
We consider distinction of representations in the context of $p$-adic Galois symmetric spaces. We provide new sufficient conditions for distinction of parabolically induced representations in terms of similar conditions on the inducing data and deduce a characterization for distinction of representations parabolically induced from cuspidal. We explicate the results further for classical groups and give several applications, in particular, concerning the preservation of distinction via Langlands functoriality. We relate our results with a conjecture of Dipendra Prasad.
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Explicit Plancherel formula for the space of symplectic forms
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 $GL_n(F)$.
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On ${\rm Sp}$-distinguished representations of the quasi-split unitary groups
We study ${\rm Sp}_{2n}(F)$-distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Moeglin-Tadic classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$-packets with no distinguished members that transfer under stable base change to ${\rm Sp}_{2n}(E)$-distinguished representations of ${\rm GL}_{2n}(E)$.
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Gamma factors root numbers and distinction
We study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of $p$-adic fields. We show that the local Rankin-Selberg root number of any pair of distinguished representation is trivial and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at $1/2$ is trivial for distinguished representations as well as the converse problem.
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Klyachko models for ladder representations
We give a new proof of the existence of Klyachko models for unitary representations of ${\rm GL}_{n}(F)$ over a non-archimedean local field $F$. Our methods are purely local and are based on studying distinction within the class of ladder representations introduced by Lapid and Minguez. We classify those ladder representations that are distinguished with respect to Klyachko models. We prove the hereditary property of these models for induced representations from arbitrary finite length representations. Finally, in the other direction and in the context of admissible representations induced from ladder, we study the relation between distinction of the parabolic induction with respect to the symplectic groups and distinction of the inducing data.
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A criterion for integrability of matrix coefficients with respect to a symmetric space
Let $G$ be a reductive group and $θ$ an involution on $G$, both defined over a $p$-adic field. We provide a criterion for $G^θ$-integrability of matrix coefficients of representations of $G$ in terms of their exponents along $θ$-stable parabolic subgroups. The group case reduces to Casselman's square-integrability criterion. As a consequence we assert that certain families of symmetric spaces are strongly tempered in the sense of Sakellaridis and Venkatesh. For some other families our result implies that matrix coefficients of all irreducible, discrete series representations are $G^θ$-integrable.
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On the distinguished spectrum of $Sp_{2n}$ with respect to $Sp_n\times Sp_n$
Given a reductive group $G$ and a reductive subgroup $H$, both defined over a number field $F$, we introduce the notion of the $H$-distinguished automorphic spectrum of $G$ and analyze it for the pairs $(GL_{2n},Sp_n)$ and $(Sp_{2n},Sp_n\times Sp_n)$. In the first case we give a complete description using results of Jacquet--Rallis, Offen and Yamana. In the second case we give an upper bound, generalizing vanishing results of Ash--Ginzburg--Rallis and a lower bound, extending results of Ginzburg--Rallis--Soudry.
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The SL(2)-type and Base Change
The SL(2)-type of any smooth, irreducible and unitarizable representation of GL(n) over a p-adic field was defined by Venkatesh. We provide a natural way to extend the definition to all smooth and irreducible representations. For unitarizable representations we show that the SL(2)-type of a representation is preserved under base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation πof GL(n) depends only on the SL(2)-type of π. As a consequence we observe that the Klyachko model of πand of its base-change are of the same type.
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On Unitary Representations of GL2n Distinguished by the Symplectic Group
We provide a family of representations of GL(2n) over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp(2n)- distinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, re- lies on the theory of automorphic forms. The results of this paper together with later works by the authors imply that the family of representations studied in this paper contains all irreducible, unitary representations of the general linear group that are distin- guished by the symplectic group.
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Uniqueness and disjointness of Klyachko models
We show the uniqueness and disjointness of Klyachko models for GL(n,F) over a non-archimedean local field F. This completes, in particular, the study of Klyachko models on the unitary dual. Our local results imply a global rigidity property for the discrete automorphic spectrum.
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Global Mixed Periods and local Klyachko models for the general linear group
We show that every irreducible representation in the discrete automorphic spectrum of GL(n) admits a non vanishing mixed (Whittaker-symplectic) period integral. The analog local problem is a study of models first considered by Klyachko over a finite field. Locally, we show that for a p-adic field F every irreducible, unitary representation of GL(n,F) has a Klyachko model.
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