Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover,we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by Atobe-Gan for quasi-split even orthogonal groups.
In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.
Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which was established in an early paper of Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this provides an alternative approach to the works of Kaletha-Minguez-Shin-White and Moeglin-Renard.
In this paper, we extend our result on a depth preserving property of the local Langlands correspondence for quasi-split unitary groups (arXiv:1804.10901) to non-quasi-split unitary groups by using the local theta correspondence. The key ingredients are a depth preserving property of the local theta correspondence proved by Pan and a description of the local theta correspondence via the local Langlands correspondence established by Gan--Ichino. To combine them, we compare splittings for metaplectic covers of unitary groups constructed by Kudla with those constructed by Pan.
We explicitly determine the theta correspondences for $\GSp_4$ and orthogonal similitude groups associated to various quadratic spaces of rank $4$ and $6$. The results are needed in our proof of the local Langlands correspondence for $\GSp_4$.
We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
This is an expanded set of notes based on two lectures given by the author at the 2022 IHES summer school on the Langlands program, on explicit techniques for the construction of automorphic representations.
We study the theta lifting for real unitary groups and completely determine the theta lifts of tempered representations. In particular, we show that the theta lifts of (limits of) discrete series representations can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
This paper deals with the geometric local theta correspondence at the Iwahori level for dual reductive pairs of type II over a non Archimedean field $F$ of characteristic $p\neq 2$ in the framework of the geometric Langlands program. First we construct and study the geometric version of the invariants of the Weil representation of the Iwahori-Hecke algebras. In the particular case of $(GL_1, GL_m)$ we give a complete geometric description of the corresponding category. The second part of the paper deals with geometric local Langlands functoriality at the Iwahori level in a general setting. Giv…
We, firstly, improve a theorem of B. Roberts which characterizes non-vanishing of a global theta lift from O(X) to Sp(n) in terms of non-vanishing of local theta lifts. In particular, we will remove all the archimedean conditions imposed upon his theorem. Secondly, following Roberts, we will apply our theorem to theta lifting of low rank similitude groups. Namely we characterize the non-vanishing condition of a global theta lift from GO(4) to GSp(2) in our improved setting. Also we consider non-vanishing conditions of a global theta lift from GO(4) to GSp(1) and explicitly compute the lift whe…
We study the generalized theta lifting between the double covers of split special orthogonal groups, which uses the non-minimal theta representations constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of non-generic representations and make a conjecture that gives an upper bound of the first non-zero occurrence of the liftings, depending only on the unipotent orbit. We prove both global and local results that support the conjecture.
The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
In this paper, we give an explicit determination of the non-vanishing of the theta liftings for unitary dual pairs (U(p,q), U(r,s)). Assuming the local Gan-Gross-Prasad conjecture, we determine when theta lifts of tempered representations are nonzero in terms of the local Langlands correspondence.
In this note, we show that the metaplectic theta correspondence is compatible with the tempered condition by directly estimating the matrix coefficients, without using the classification theorem.
We establish the Fourier-Jacobi case of the local Gross-Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier-Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of D. Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence.
The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to automorphic representations on special orthogonal groups. It is of interest to vary the orthogonal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. …
Following the approach of B. Roberts, we characterize the non-vanishing of global theta lifts for symplectic-orthogonal dual pairs in terms of its local counterpart. In particular, we replace the temperedness assumption present in Robert's work by a certain weaker assumption, and apply our results to small rank similitude groups. Among our applications is a certain instance of Langlands functorial transfer of a (non-generic) cuspidal automorphic representation of $GSp(4)$ to GL(4).
We study a new lifting of automorphic representations using the theta representation $Θ$ on the $4$-fold cover of the symplectic group, $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to $Θ$. We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. Assuming this, we compute the effect of our lift on Arthur parameters and show th…
This paper proves the local Langlands conjecture for the non quasi-split inner form Sp(1,1) of Sp(4) over a p-adic field of characteristic 0, by studying the restriction of representations from the non quasi-split inner form GSp(1,1) of GSp(4) to Sp(1,1). The L-packets for Sp(1,1) are constructed based on the earlier work on the local Langlands correspondence for GSp(1,1) by Gan and Tantono. To parameterize them in terms of so-called S-groups, we establish and utilize the local Langlands correspondence for reductive dual groups which participate in the theta correspondence with Sp(1,1) and GSp…
We derive a (weak) second term identity for the regularized Siegel-Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the "second term range". As an application, we show the following non-vanishing result of global theta lifts from orthogonal groups. Let $π$ be a cuspidal automorphic representation of an orthogonal group $O(V)$ with $\dim V=m$ even and $r+1\leq m\leq 2r$. Assume further that there is a place $v$ such that $π_v\congπ_v\otimes\det$. Then the global theta lift of $π$ to $Sp_{2r}$ does not vanish up to twisting by automorphic dete…
The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issu…
For every irreducible Harish-Chandra module of $O^{*}(4)$, we determine its theta lift to $Sp(p,q)$ in terms of the Langlands parameter, for all non-negative integers $p$ and $q$. Our strategy is to determine the desired theta lifts by their infinitesimal characters and lowest $K$-types.
We relate poles of local Godement-Jacquet L-functions to distributions on matrix spaces with singular supports. As an application, we show the irreducibility of the full theta lifts to $GL_n(F)$ of generic irreducible representations of $GL_n(F)$, where $F$ is an arbitrary local field.
In this paper we give the description of generic representations of metaplectic groups over p-adic fields in terms of their Langlands parameters and calculate their theta lifts on all levels for any tower of odd orthogonal groups. We also describe precisely all the occurrences of the failure of the standard module conjecture for metaplectic groups.
In this paper we study the theta lifting for loop groups and extend the classical tower property established by S. Rallis to the loop setting. As an application we obtain cusp forms on loop groups, and we give the first example where the cusp forms constructed using this method are nonvanishing.
In this paper, we completely describe the Howe correspondence for the dual pairs from the title over a nonarchimedean local field of characteristic zero. More specifically, for every irreducible admissible representation of these groups, we find its first occurrence index in the theta correspondence and we describe, in terms of their Langlands parameters, the small theta lifts on all levels.
This article has a twofold purpose. First, by recent works of Kaletha and Loke-Ma, we give an explicit description of the local theta correspondence between regular supercuspidal representations in the equal rank symplectic-orthogonal case. Second, based on this description, we show that the local theta correspondence preserves distinction with respect to unramified Galois involutions.
This paper describes the lifting of automorphic characters of $\GO(3)(\A)$ to $\SLT(\A)$. It does so by matching the image of this lift with the lift of automorphic characters from $\GO(1)(\A)$ to $\SLT(\A)$. Our matching actually gives a matching of individual automorphic forms, and not just of representation spaces. Let $\V$ be a $3-$ dimensional quadratic vector space and $\U$ a certain $1-$ dimensional quadratic space. To an automorphic form $I_{\V}(χ,ϕ)$ determined by the Schwartz function $ϕ\in \Sc(\V(\A))$ in the lift of the character $χ$ we match an automorphic form $I_{\U}(μ,ϕ_{0})$ d…
We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations for the dual pair of groups $(\mathrm{Sp}_{2n}, \mathrm{O}(V))$ defined over a local nonarchimedean field $\mathbb{F}$ of characteristic $0$. As a direct application of our results, we are able to produce a series of non-generic unitarizable representations of these groups.
In this paper we study representations of the indefinite orthogonal group O(n,m) which are local theta lifts of one dimensional characters or unitary lowest weight modules of the double covers of the symplectic groups. We apply the transfer of K-types on these representations of O(n,m), and we study their effects on the dual pair correspondences. These results provide examples that the theta lifting is compatible with the transfer of K-types. Finally we will use these results to study subquotients of some cohomologically induced modules.
We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $ρ: W_K \to G_2(\CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional th…
Let $G$ be a split simply laced group defined over a $p$-adic field $F$. In this paper we study the restriction of the minimal representation of $G$ to various dual pairs in $G$. For example, the restriction of the minimal representation of $E_7$ to the dual pair $G_2 \times{}$Sp(6) gives the non-endoscopic Langlands lift of irreducible representations of $G_2$ to Sp(6).
The Langlands correspondence of GL(2,F) over a non-Archimedean local field F of characteristic 0 has been well studied. The construction uses the theta correspondence. In this paper, we are going to describe explicitly how this construction is done. We will then provide new proofs to several properties of the correspondence. Including that the correspondence does not depend on the additive character introduced by the Weil representation.
In this paper we first construct natural filtrations on the full theta lifts for any real reductive dual pairs. We will use these filtrations to calculate the associated cycles and therefore the associated varieties of Harish-Chandra modules of the indefinite orthogonal groups which are theta lifts of unitary lowest weight modules of the metaplectic double covers of the real symplectic groups. We will show that some of these representations are special unipotent and satisfy a K-type formula in a conjecture of Vogan.
We study three exceptional theta correspondences for p-adic groups, where one member of the dual pair is the exceptional group G2. We prove the Howe duality conjecture for these dual pairs and a dichotomy theorem, and determine explicitly the theta lifts of all non-cuspidal representations.
Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual group of F one can associate certain meromorphic function on the set of isomorphism classed of irreducible representations of G (which we call gamma-functions). In this paper we describe a general framework for an explicit construction of these gamma-functions. We make this idea precise in certain special cases. As a byproduct we give explicit conjectural for…
During the last two decades, great efforts have been devoted to the calculation of the local theta correspondence for reductive dual pairs. However, uniform formulas remain elusive for real dual pairs of type I. The purpose of this paper is twofold: to formulate an explicit version of induction principle for dual pairs $(O(p,q),Sp(2n,\mathbb{R}))$ with $p + q$ even, and to apply it to obtain a complete and explicit description of the local theta correspondence when $p + q = 4$. Our approach is very elementary by analysis on the infinitesimal characters and K-types under the correspondence.
We give a proof of the Howe duality conjecture for the (almost) equal rank dual pairs in full generality. For arbitrary dual pairs, we prove the irreducibility of the (small) theta lifts for all tempered representations. Our proof works for any nonarchimedean local field of characteristic not 2 and in arbitrary residual characteristic.
Let $π$ be a simple supercuspidal representation of the symplectic group $Sp_{2l}(F)$, over a $p$-adic field $F$. In this work, we explicitly compute the Rankin-Selberg $γ$-factor of rank-$1$ twists of $π$. We then completely determine the Langlands parameter of $π$, if $p \neq 2$. In the case that $F = \mathbb{Q}_2$, we give a conjectural description of the functorial lift of $π$, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.
Let $(G,G')$ be a type I irreducible reductive dual pair in $\mathrm{Sp}(W_{\mathbb{R}})$. We assume that $(G,G')$ is in the stable range where $G$ is the smaller member. Let $K$ and $K'$ be maximal compact subgroups of $G$ and $G'$ respectively. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and $\mathfrak{g}' = \mathfrak{k}' \oplus \mathfrak{p}'$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G'$ respectively. Let ${\widetilde{K}}$ and ${\widetilde{K}}'$ be the inverse images of $K$ and $K'$ in the metaplectic double cover $\widetilde{\mathrm{Sp}}(W_\mathbb{…
In this paper, we establish the second term identity of the Siegel-Weil formula in full generality, and derive the Rallis inner product formula for global theta lifts for any dual pair. As a corollary, we resolve the non-vanishing problem of global theta lifts initiated by Steve Rallis.
We show that if an irreducible admissible representation of $\mathrm{SO}_{4n}(F)$ has a generalized Shalika model, then its small theta lift to $\mathrm{Sp}_{4n}(F)$ has the symplectic linear model, thus answering a question posed by D. Jiang. Here $F$ is a non-archimedean field of characteristic zero.
Let $\ell$ and $p$ be distinct primes, $n$ a positive integer, $F_\ell$ an $\ell$-adic local field of characteristic $0,$ and let $W(k)$ denote the ring of Witt vectors over an algebraically closed field of characteristic $p$. Work of Emerton-Helm, Helm and Helm-Moss defines and constructs a smooth $A[{GL}_n(F_\ell)]$-module $\tildeπ(ρ_A)$ for a continuous Galois representation $ρ_A : G_{F_\ell} \to {GL}_n(A)$ over a $p$-torsionfree reduced complete local $W(k)$-algebra $A$ interpolating the local Langlands correspondence. However, since $\tildeπ$ is not a functor, there is no clear way to spe…
