Theta Correspondence Research Articles

Bessel integral operators

We define an algebra of linear operators whose kernels are essentially Bessel functions. These operators act on spaces of entire functions of exponential type and are endowed with a complete and explicit symbolic calculus that we will describe. We explain how these operators were designed to be models for Toeplitz operators on the unit sphere in the n-complex domain. As an application we give a symbolic calculus for the Boutet de Monvel – Howe correspondence.

The geometric theta correspondence for Hilbert modular surfaces

In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real differential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case. In this paper we study in detail the geometric theta lift for Hilbert modular surfaces. In particular, we will give a new proof and an extension (to all finite index subgroups of the Hilbert modular group) of the celebrated theorem of Hirzebruch and Zagier that the generating function for the intersection numbers of the Hirzebruch-Zagier cycles is a classical modular form of weight 2. In our approach we replace Hirzebuch's smooth complex analytic compactification $\tilde{X}$ of the Hilbert modular surface $X$ with the (real) Borel-Serre compactification $\bar{X}$. The various algebro-geometric quantities are then replaced by topological quantities associated to 4-manifolds with boundary. In particular, the "boundary contribution" in Hirzebruch-Zagier is replaced by sums of linking numbers of circles (the boundaries of the cycles) in the 3-manifolds of type Sol (torus bundle over a circle) which comprise the Borel-Serre boundary.

Geometric local theta correspondence for dual reductive pairs of type II at the Iwahori level

In this paper we are interested in the geometric local theta correspondence at the Iwahori level for dual reductive pairs ( G , H ) (G,H) of type II over a non-Archimedean field of characteristic p ≠ 2 p\neq 2 in the framework of the geometric Langlands program. We consider the geometric version of the I H × I G I_{H}\times I_{G} -invariants of the Weil representation S I H × I G \mathcal {S}^{I_{H}\times I_{G}} as a bimodule under the action of Iwahori-Hecke algebras H I G \mathcal {H}_{I_{G}} and H I H \mathcal {H}_{I_{H}} and we give some partial geometric description of the corresponding category under the action of Hecke functors. We also define geometric Jacquet functors for any connected reductive group G G at the Iwahori level and we show that they commute with the Hecke action of the H I L \mathcal {H}_{I_{L}} -subelgebra of H I G \mathcal {H}_{I_{G}} for a Levi subgroup L L .

Local root numbers, Bessel models, and a conjecture of Guo and Jacquet

Let E/F be a quadratic extension of number fields and D a quaternion algebra over F containing E. Let πD be a cuspidal automorphic representation of GL(n,D) and π its Jacquet–Langlands transfer to GL(2n). Guo and Jacquet conjectured that if πD is distinguished by GL(n,E), then π is symplectic and L(1/2,πE)≠0, where πE is the base change of π to E. When n is odd, Guo and Jacquet also conjectured a converse. The converse does not always hold when n is even, but we conjecture it holds if and only if certain local root number conditions are satisfied, which is if and only if the corresponding generic representation of the split special orthogonal group SO(2n+1) has a special E-Bessel model. We use the theta correspondence to relate E-Bessel periods on SO(5) with GL(2,E)-periods on GL(2,D), and deduce part of our conjecture when n=2.

L-functions and theta correspondence for classical groups

Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the L-functions and local theta correspondence.

A note on the local theta correspondence for unitary similitude dual pairs

Following Robertsʼ work in the case of orthogonal-symplectic similitude dual pairs, we study the local theta correspondence for unitary similitude dual pairs over a p -adic field.

Split metaplectic groups and their L-groups

Abstract We adapt the conjectural local Langlands parameterization to split metaplectic groups over local fields. When G˜ is a central extension of a split connected reductive group over a local field (arising from the framework of Brylinski and Deligne), we construct a dual group 𝐆 ˜ ∨ $\tilde{\mathbf {G}}^{\vee }$ and an L-group L 𝐆 ˜ ∨ ${\,}^L \tilde{\mathbf {G}}^{\vee }$ as group schemes over ℤ. Such a construction leads to a definition of Weil–Deligne parameters (Langlands parameters) with values in this L-group, and to a conjectural parameterization of the irreducible genuine representations of G˜. This conjectural parameterization is compatible with what is known about metaplectic tori, Iwahori–Hecke algebra isomorphisms between metaplectic and linear groups, and classical theta correspondences between 𝑀𝑝 2 n $\textit {Mp}_{2n}$ and special orthogonal groups.

Abelian varieties and Weil representations

The main goal of this article is to construct and study a family of Weil representations over an arbitrary locally noetherian scheme without restriction on characteristic. The key point is to recast the classical theory in the scheme-theoretic setting. As in work of Mumford, Moret-Bailly and others, a Heisenberg group (scheme) and its representation can be naturally constructed from a pair of an abelian scheme and a nondegenerate line bundle, replacing the role of a symplectic vector space. Once enough is understood about the Heisenberg group and its representations (e.g., the analogue of the Stone–von Neumann theorem), it is not difficult to produce the Weil representation of a metaplectic group (functor) from them. As an interesting consequence (when the base scheme is SpecF¯p), we obtain the new notion of mod p Weil representations of p-adic metaplectic groups on F¯p-vector spaces. The mod p Weil representations admit an alternative construction starting from a p-divisible group with a symplectic pairing. We have been motivated by a few possible applications, including a conjectural mod p theta correspondence for p-adic reductive groups and a geometric approach to the (classical) theta correspondence.

Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence

AbstractUsing theta correspondence, we classify the irreducible representations ofMp2nin terms of the irreducible representations ofSO2n+1and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger forn=1.

The Conservation Relation for Discrete Series Representations of Metaplectic Groups

Let F denote a nonarchimedean local field of characteristic zero with odd residual characteristic and let denote the rank n metaplectic group over F. If r±(σ) denotes the first occurrence index of the irreducible genuine representation σ of in the theta correspondence for the dual pair ⁠, the conservation relation, conjectured by Kudla and Rallis, states that r+(σ)+r−(σ)=2n. The purpose of this paper is to prove this conjecture for discrete series which appear as subquotients of generalized principle series where the representation on the metaplectic part is strongly positive. Assuming the basic assumption, we also prove the conservation relation for general discrete series of metaplectic groups by explicitly determining the first occurrence indices.

La fonctorialité dʼArthur–Langlands locale géométrique et la correspondance de Howe au niveau Iwahori

We state a conjecture on the geometric local Arthur–Langlands functoriality at the Iwahori level. Given a morphism Gˇ×SL2→Hˇ of Langlands dual groups over Q¯ℓ, we construct a bimodule over the affine extended Hecke algebras of G and H which should realize the geometric local Langlands functoriality at the Iwahori level. We state a second conjecture relating this bimodule and the bimodule arising in the geometric Howe correspondence at the Iwahori level. This conjecture is verified for all dual reductive pairs (G=GL1,H=GLm).

A Bessel identity for the theta correspondence

We establish a spectral identity between global Bessel distributions with respect to generic cuspidal representations of an odd orthogonal group and the metaplectic cover of a symplectic group which are related by the theta correspondence. We also provide analogous local identities for square-integrable representations.

Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups. As an application of our formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.

Normalized local theta correspondence and the duality of inner product formulas

We conjecture that local theta correspondence can be normalized by the leading coefficient of a weighted local period integral, and that there exists a duality of local and global inner product formulas. The conjecture is verified for the pair (\(\widetilde{SL}_2 \), PGL 2) and (SL 2, SO(2, 2)). As an application, global inner product formulas are obtained for liftings in the directions PGL 2 → \(\widetilde{SL}_2 \), GSO(2, 2) → GL 2.

Rank one reducibility for unitary groups

Let (G,G ' ) denote a dual reductive pair consisting of two unitary groups over a nonarchimedean local field of characteristic zero. We relate the reducibility of the parabolically induced representations of these two groups if the inducing data is cuspidal and related to each other by theta correspondence. We calculate theta lifts of the irreducible subquotients of these parabolically induced representations. To obtain these results, we explicitly calculate filtration of Jacquet modules of the appropriate Weil representation (as Kudla did for the orthogonal- symplectic dual pairs), but keeping in mind the explicit splittings of covers of these two unitary groups, also obtained by Kudla.

Rank One Reducibility for Metaplectic Groups via Theta Correspondence

Abstract We calculate reducibility for the representations of metaplectic groups induced from cuspidal representations of maximal parabolic subgroups via theta correspondence, in terms of the analogous representations of the odd orthogonal groups. We also describe the lifts of all relevant subquotients.

On zeta function identities involving sums of squares and the zeta–theta correspondence

We prove two identities involving Dirichlet series, in the denominators of whose terms sums of two, three and four squares appear. These follow from two classical identities of Jacobi involving the four Jacobian Theta Functions θ1(z;q), θ2(z;q), θ3(z;q) and θ4(z;q), by the application of the Mellin transform. These results motivate the well-known correspondence between the set of the four Jacobian Theta Functions and the set of four classical zeta functions of which the Riemann Zeta Function is the third, and the Dirichlet Beta Function is the first.

Dual pairs and contragredients of irreducible representations

Let G be one of the classical groups GL(n), U(n), O(n) or Sp(2n), over a nonarchimedean local field of characteristic zero. It is well known that the contragredient of an irreducible admissible smooth representation of G is isomorphic to a twist of it by an automorphism of G. We prove that similar results hold for double covers of G that occur in the study of local theta correspondences.

Compatibility of the theta correspondence with the Whittaker functors

Nous demontrons que le foncteur geometrique de theta-lifting pour la paire duale (H, G) est compatible avec la normalisation de Whittaker, ou (H, G) est l'une des paires (SO 2n ,Sp 2n ), (Sp 2n ,SO 2n+2 ) ou (GL n ,GL n+1 ). Plus precisement, le compose du foncteur de theta-lifting de H vers G et du foncteur de Whittaker pour G est isomorphe au foncteur de Whittaker pour H.

An inequality for local unitary Theta correspondence

Given a representation π of a local unitary group G and another local unitary group H, either the Theta correspondence provides a representation θ H (π) of H or we set θ H (π)=0. If G is fixed and H varies in a Witt tower, a natural question is: for which H is θ H (π)≠0 ? For given dimension m there are exactly two isometry classes of unitary spaces that we denote H m ± . For ϵ∈{0,1} let us denote m ϵ ± (π) the minimal m of the same parity of ϵ such that θ H m ± (π)≠0, then we prove that m ϵ + (π)+m ϵ - (π)≥2n+2 where n is the dimension of π.