Big Theta Lift in Theta Correspondence

• Big Theta Lift is the full theta lift derived from the Weil representation for a reductive dual pair, capturing the entire module before irreducible quotienting.
• It differentiates from the small theta lift by retaining the complete output, enabling deeper analysis of nilpotent orbits, Whittaker models, and period relations.
• Applications include global automorphic constructions, explicit arithmetic realizations in similitude settings, and profound links with modern analytic theta lifts.

Big theta lift is the standard name, in much of the local theta-correspondence literature, for the full theta lift attached to the Weil representation of a reductive dual pair. If $(G,G')\subset \mathrm{Sp}(W)$ is a local dual pair and $\omega$ is the Weil representation, then for an irreducible representation $\pi$ of $G$ the full lift is characterized by

$$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$

and several sources explicitly identify this full lift with what is colloquially called the big theta lift (Gomez et al., 2013, Xue, 2023). In parallel, the same kernel-lifting paradigm also appears in global automorphic constructions, in type II dual pairs, and in generalized metaplectic settings where the classical oscillator kernel is replaced by small residual representations rather than the Weil representation (Leslie, 2017).

1. Terminology and formal definition

In the classical local theory over a local field of characteristic $0$, one fixes a reductive dual pair $G(W)\times H(V)$, chooses splitting data and an additive character, and obtains a Weil representation $\Omega_{V,W,\psi_F}$ or $\omega$ of $G(W)\times H(V)$. For an irreducible smooth representation $\omega$0 of $\omega$1, the maximal $\omega$2-isotypic quotient of $\omega$3 is

$\omega$4

and $\omega$5 is the big theta lift to $\omega$6 (Chen et al., 2023). In the non-Archimedean formulation emphasized for type I dual pairs, the same object is the full theta lift $\omega$7, while the small theta lift is obtained from it by passage to a distinguished irreducible quotient (Gomez et al., 2013).

The precise convention for the small lift varies slightly. One formulation defines

$\omega$8

and then uses Howe duality to conclude that this quotient is irreducible when nonzero (Chen et al., 2023). Another formulation states directly that $\omega$9 has a unique irreducible quotient $\pi$0 (Gomez et al., 2013, Loke et al., 2013). In either convention, the structural problem is the same: the big lift is the full module produced by the Weil representation, and the small lift is the canonical irreducible object extracted from it.

The same terminology extends to type II dual pairs. For $\pi$1, the Weil representation on $\pi$2 yields

$\pi$3

so the big theta lift is again the full coinvariant module rather than merely its irreducible cosocle (Chen et al., 10 Jul 2025). This makes the phrase “big theta lift” uniform at the formal level: it denotes the full output of the theta functor before irreducibility is imposed.

2. Irreducibility and the problem “big equals small”

A central question is when the full lift is already irreducible. In stable range for type I real dual pairs, Loke and Ma proved that for an irreducible unitarizable genuine $\pi$4-module $\pi$5, one has

$\pi$6

except in the special case

$\pi$7

with $\pi$8 the one-dimensional genuine representation of $\pi$9 (Loke et al., 2013). Over non-Archimedean local fields, a complementary generic irreducibility theorem shows that if the critical factors

$G$0

are excluded from the factor set of $G$1, then the big lift is irreducible when nonzero; in stable range with $G$2 the smaller member, every irreducible unitary $G$3 satisfies this condition, so $G$4 is irreducible (Chen et al., 2023).

More recent work proves irreducibility in several finer regimes. For symplectic–even orthogonal $G$5-adic dual pairs, the Muić conjecture was established in the discrete-series case: if $G$6 is a discrete series representation and $G$7, then the big lift is irreducible, hence equal to the small lift (Hanzer, 12 Oct 2025). For real unitary groups, tempered lifts are completely determined: if

$G$8

is irreducible tempered and $G$9, then

$$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$0

and for $$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$1 discrete series the lift is expressed as a cohomologically induced module in the weakly fair range (Ichino, 2020).

Type II dual pairs exhibit a particularly explicit version of the same problem. For $$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$2, the big lift is completely determined when the Godement–Jacquet $$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$3-function is holomorphic at

$$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$4

Under that hypothesis,

$$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$5

and

$$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$6

according to whether $$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$7 or $$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$8 is holomorphic there; in particular, tempered $$\omega/N(\pi)\cong \pi\boxtimes \Theta(\pi), \qquad \Theta(\pi)=(\omega\otimes \pi^\vee)_G,$$9 have irreducible big lift for all $0$0 (Chen et al., 10 Jul 2025). In low rank, the full structure can be exceptional: for $0$1, every nontrivial irreducible representation has irreducible full lift, but the trivial representation satisfies

$0$2

an induced representation of length $0$3 with subrepresentation $0$4 and quotient $0$5 (Xue, 2023).

3. Geometric invariants, nilpotent orbits, and Whittaker data

The big theta lift is also studied through nilpotent-orbit geometry and generalized Whittaker models. For reductive dual pairs over local fields of characteristic $0$6, a nilpotent orbit $0$7 in the image of the moment map from full-rank elements of $0$8 lifts to a nilpotent orbit $0$9, and the full theta lift satisfies

$G(W)\times H(V)$0

In stable range with the smaller member on the source side, every nilpotent orbit occurs in the image of the moment map, so the theorem applies to all generalized Whittaker types (Gomez et al., 2013).

Loke and Ma further showed that in stable range the associated variety and associated cycle of the full lift are controlled exactly by theta-lifting the corresponding invariants of the contragredient representation: $G(W)\times H(V)$1 They also prove the upper bound

$G(W)\times H(V)$2

without stable-range assumptions (Loke et al., 2013). This places the big theta lift squarely inside the orbit-method perspective: it is not only a transfer of representations, but also a transfer of nilpotent support and isotropy data.

A global variant of this structural viewpoint appears in the period theory of theta lifts on orthogonal groups. For anisotropic $G(W)\times H(V)$3 of signature $G(W)\times H(V)$4 and $G(W)\times H(V)$5, a new period relation identifies a weighted orthogonal period on $G(W)\times H(V)$6 with a Bessel period on the theta lift to $G(W)\times H(V)$7. Nonvanishing of the orthogonal period forces the form on $G(W)\times H(V)$8 to lie in the theta-lift image from $G(W)\times H(V)$9 (Brumley et al., 2023). This gives a period-theoretic criterion distinguishing automorphic theta lifts inside the orthogonal spectrum.

4. Global lifts, towers, and first occurrence

At the global automorphic level, the standard theta lift is defined by integrating the theta kernel against a cusp form. In the stable-range setting of cohomological applications, if $\Omega_{V,W,\psi_F}$0 is a cuspidal automorphic representation on the smaller member $\Omega_{V,W,\psi_F}$1, then the global theta lift $\Omega_{V,W,\psi_F}$2 is nonzero for suitable data, square-integrable, irreducible, and factors as

$\Omega_{V,W,\psi_F}$3

with $\Omega_{V,W,\psi_F}$4 the local small lifts (Cossutta et al., 2011). This global factorization is one of the standard bridges between the big automorphic kernel construction and local irreducible correspondents.

A non-classical generalization arises on higher metaplectic covers. Leslie constructs a generalized theta lift on the $\Omega_{V,W,\psi_F}$5-fold Brylinski–Deligne cover of a symplectic group by replacing the oscillator representation with the global theta representation $\Omega_{V,W,\psi_F}$6 on the bigger cover. The lift is defined by the kernel integral

$\Omega_{V,W,\psi_F}$7

and in the tower notation by

$\Omega_{V,W,\psi_F}$8

This lift satisfies a tower property analogous to Rallis theory: vanishing persists downward, some higher lift is eventually nonzero, and first occurrence is cuspidal. Moreover, the first nontrivial lift is CAP, hence non-tempered, and its conjectural Arthur parameter is

$\Omega_{V,W,\psi_F}$9

(Leslie, 2017). The same paper identifies the maximal nilpotent orbit of the kernel representation as

$\omega$0

which is the “smallness” input driving the tower analysis.

In the generalized metaplectic theta tower of Friedberg–Ginzburg type, genericity behaves differently from the classical case. For lifts from the $\omega$1-fold cover of a symplectic group, the Whittaker range contains

$\omega$2

orthogonal groups rather than the two groups occurring in the classical tower. Outside the range

$\omega$3

the lift is not generic, while inside that range genericity is characterized by explicit period integrals (Friedberg et al., 2021). This shows that tower/first-occurrence phenomena survive far beyond the classical double-cover setting, but with substantially richer behavior.

5. Similitude variants and explicit arithmetic realizations

The big-theta formalism is also realized in explicit similitude correspondences. For a real quadratic extension $\omega$4, an irreducible cuspidal automorphic representation of $\omega$5 is first transferred to a representation on a four-dimensional quadratic space $\omega$6, and then lifted through the similitude pair

$\omega$7

The global theta lift is

$\omega$8

and the local model is realized by explicit Bessel-type integrals $\omega$9. For each finite place, one can choose local Schwartz data producing a vector invariant under the paramodular subgroup $G(W)\times H(V)$0, with

$G(W)\times H(V)$1

except in the wildly ramified case (Johnson-Leung et al., 15 Jan 2025). This is a particularly concrete realization of the big lift: the full theta construction is paired with explicit local newvectors.

Pollack’s quaternionic Saito–Kurokawa-type lift gives another arithmetic instantiation. Starting from a cuspidal Siegel modular form on $G(W)\times H(V)$2, the global theta lift to $G(W)\times H(V)$3 is defined by

$G(W)\times H(V)$4

With a special archimedean Schwartz function

$G(W)\times H(V)$5

the lifted form is quaternionic at infinity; in the $G(W)\times H(V)$6 case it is cuspidal, and its restriction to $G(W)\times H(V)$7 gives cuspidal modular forms on $G(W)\times H(V)$8 with algebraic, and in the level-one case integral, Fourier coefficients (Pollack, 2019). Although the paper does not formalize a big/small distinction, it is a genuine global theta lift built from the Weil representation.

A third application uses theta distinction rather than construction alone. On anisotropic $G(W)\times H(V)$9, a new orthogonal period detects forms in the image of the theta lift from $\omega$00, and this is combined with counting on the source side to produce large sup norms on the orthogonal side (Brumley et al., 2023). In this sense the big theta lift functions as a spectral subspace with rigid period-theoretic characterization.

6. Broader analytic uses and terminological variation

The phrase “theta lift” also appears in several analytic contexts adjacent to, but not identical with, the standard big theta lift. In genus two, the Kawazumi–Zhang invariant is shown to be the theta lift of the unique weak Jacobi form of weight $\omega$01, via a $\omega$02-signature Siegel–Narain kernel: $\omega$03 From this, a parallel theta-lift representation of the genus-two Faltings invariant is derived (Pioline, 2015). Here the target is a real-analytic invariant on Siegel space rather than a representation in the local Howe-duality sense.

For harmonic Maass forms, a regularized Shintani theta lift maps

$\omega$04

in signature $\omega$05, with Fourier coefficients described by traces of CM values and regularized cycle integrals, and with a precise relation to the Millson theta lift via the $\omega$06-operator (Alfes-Neumann et al., 2017). An earlier explicit Maass lifting of $\omega$07 constructs a harmonic weak Maass form $\omega$08 satisfying

$\omega$09

so that the coefficients of the holomorphic part of $\omega$10 are governed by real quadratic class numbers and logarithms of units, in contrast to the imaginary quadratic data encoded by $\omega$11 itself (Rhoades et al., 2011).

By contrast, the “massive theta lift” is explicitly a different object: it deforms Rankin–Selberg/Siegel–Narain theta integrals by a mass parameter $\omega$12, yielding exponentially decaying lifts, but it is not the big theta lift in the standard Howe-duality sense (Berg et al., 2022). This suggests that “big theta lift” is a context-sensitive term. In the strict representation-theoretic literature it denotes the full theta module attached to the Weil representation, whereas in broader automorphic and analytic practice it can designate any large-scale theta-kernel transform patterned on the same mechanism.

The modern picture is therefore plural rather than singular. In the narrow classical sense, the big theta lift is the full local or global theta module before irreducible quotienting. In generalized metaplectic and higher-cover theories, the same structural role can be played by a residual theta representation rather than the oscillator representation (Leslie, 2017). In arithmetic and analytic applications, theta lifts often preserve the kernel-integral paradigm while targeting invariants, periods, or special-function expansions rather than irreducible correspondents.