Local Theta Correspondence

Updated 13 November 2025

Local theta correspondence is a framework connecting smooth representations of reductive dual pairs via the oscillator (Weil) representation, ensuring multiplicity-one correspondence.
It employs explicit constructions, conservation relations, and Langlands parameter lifts to classify representations and analyze invariant cycles within both algebraic and analytic settings.
The theory extends to modular and C*-algebraic frameworks, integrating harmonic analysis with automorphic forms and noncommutative geometry.

Local theta correspondence is a fundamental theory in the representation theory of p-adic and real reductive groups, relating the smooth representations of two groups forming a reductive dual pair via the oscillator (Weil) representation of the metaplectic group. It is characterized by several rigorous properties—multiplicity-one, bijection on relevant spectra, conservation relations, and compatibility with Langlands parameters—and admits variant formulations in both algebraic and analytic settings. The modern development encompasses explicit constructions for representations of classical groups, generalizations to modular and geometric settings, as well as categorical and C*-algebraic perspectives, forming a unified structure within noncommutative harmonic analysis.

1. Reductive Dual Pairs and the Weil Representation

Let $V$ and $V'$ be finite-dimensional right modules over a division algebra $\mathfrak{D}$ ( $\mathbb{R}$ , $\mathbb{C}$ , or $\mathbb{H}$ ), equipped with nondegenerate $\epsilon$ -Hermitian forms $\langle\cdot,\cdot\rangle_V$ and $\langle\cdot,\cdot\rangle_{V'}$ with $\epsilon\epsilon'=-1$ . Their isometry groups $G(V)$ and $G(V')$ act as mutual centralizers embedded in the symplectic group $\mathrm{Sp}(W)$ , where $W = \operatorname{Hom}_{\mathfrak{D}}(V,V')$ . Such pairs fall into the 'type I' or 'type II' categories, comprising classic families:

Orthogonal/symplectic: $(O(p,q), Sp(2n))$
Unitary/unitary: $(U(p,q), U(r,s))$
Quaternionic/unitary pairs, and
Type II: $(GL_m(\mathbb{C}), GL_n(\mathbb{C}))$ , etc.

The group $\mathrm{Sp}(W)$ acts on the Heisenberg group $H(W) = W \times F$ via automorphisms. The Weil (oscillator) representation $\omega$ is then a unitary module of the metaplectic double cover $\mathrm{Mp}(W)$ , realized as the unique irreducible representation with prescribed central character (Stone–von Neumann). In the Schrödinger model, the action of $\omega$ on the Schwartz space $\mathcal{S}(L)$ is explicitly described by:

$\begin{aligned} \left(\omega(w,0)\varphi\right)(x) &= e^{2\pi i\frac12\langle w_L,w_{L^*}\rangle}\varphi(x+w_L),\ \left(\omega(0,t)\varphi\right)(x) &= e^{2\piit}\varphi(x),\ \left(\omega(s)\varphi\right)(x) &= \int_{L}e^{2\pi i\langle x,sy\rangle}\varphi(y)dy, \end{aligned}$

where the last term involves lifting $s\in \mathrm{Sp}(W)$ to $\mathrm{Mp}(W)$ . The oscillator representation is the vehicle for defining the local theta correspondence.

2. The Howe Duality Theorem and Multiplicity-One

Given the metaplectic preimages $\widetilde{G} \rightarrow G(V)$ , $\widetilde{G'} \rightarrow G(V')$ , and oscillator representation $\omega$ of $\widetilde{G}\times \widetilde{G'}$ , one defines for a Casselman–Wallach representation $\pi$ of $\widetilde{G}$ its theta lift by

$\Theta(\pi) = \operatorname{Hom}_{\widetilde{G}}(\omega,\, \pi^\vee)$

as a smooth $\widetilde{G'}$ -module.

Howe Duality Theorem (Zhu, 11 Nov 2025, Sun et al., 2020): For (almost all) irreducible admissible representations $\pi$ of $\widetilde{G}$ , there is a unique irreducible quotient $\theta(\pi)$ of $\Theta(\pi)$ . Moreover, the set of $(\pi,\pi')$ appearing in $\omega$ forms the graph of a bijection between irreducible representations with nonvanishing theta lift—establishing multiplicity-one

$\dim \operatorname{Hom}_{\widetilde{G}\times\widetilde{G'}}(\omega,\, \pi\otimes\pi') \le 1.$

In both archimedean (Zhu, 11 Nov 2025) and non-archimedean (Chen et al., 2023) settings, proofs use the Fock model, seesaw argument, and invariant theory, yielding functorial and continuous assignments between irreducible spectra.

3. Conservation Relations and First Occurrence

The essential "conservation relation" of Kudla–Rallis records the sum of the first occurrence indices of an irreducible representation $\pi$ and its twist in complementary Witt towers (e.g., the determinant or sign representation):

$n(\pi) + n(\pi\otimes\det) = \dim V,$

for orthogonal-symplectic pairs (Zhu, 11 Nov 2025, Sun et al., 2012). Analogous formulas hold for other types:

For symplectic towers,

$m_+(\pi') + m_-(\pi') = 4n + 4$

For general type I pairs with division algebra $\mathfrak{D}$ ,

$m_{t_1}(\pi) + m_{t_2}(\pi) = 2\dim_{\mathfrak{D}} U + d_{\mathfrak{D,\epsilon}}$

where $d_{\mathfrak{D,\epsilon}$ is the anisotropic dimension (Sun et al., 2012).

These relations underpin nonvanishing results, the fine structure of theta correspondences, and their compatibility with global automorphic lifts.

4. Explicit Descriptions via Langlands, Arthur, and Nilpotent Invariants

Langlands and Arthur Parameters

The local theta correspondence respects and transforms Langlands–Vogan and Arthur parameters (Bakic et al., 2022, Atobe et al., 2016, Gan et al., 2014, Kakuhama, 1 Sep 2024):

For tempered $\pi$ with parameter $\varphi_\pi$ , the lifted representation $\sigma=\theta_{V_m,W_n}(\pi)$ has parameter:

$\varphi_\sigma = (\varphi_\pi\otimes\chi_V\chi_W) \ominus \chi_WS_\ell$

for "going down" case, or

$\varphi_\sigma = (\varphi_\pi\otimes\chi_V\chi_W) \oplus \chi_WS_{\ell+2}$

for "going up" case (Atobe et al., 2016).

Matching of characters in component groups is prescribed by explicit epsilon-factors or root numbers (see Prasad's conjectures proved in (Gan et al., 2014)). For Arthur packets, the local theta lift of an $\pi\in\Pi_\psi$ lands in $\Pi_{\psi_\alpha}$ , with precise descriptions at every occurrence index and combinatorial invariants controlling which parameter lifts nontrivially (Bakic et al., 2022).

Nilpotent-Orbit Invariants

Generalized Whittaker models and associated cycles transfer under theta correspondence according to a double-fibration structure of moment maps:

If $\gamma=\{X,H,Y\}$ is a $\mathfrak{sl}_2$ -triple in $\mathfrak{g}$ , then (Zhu, 11 Nov 2025, Zhu, 2018):

$\Wh_\gamma(\pi) \cong \Wh_{\gamma'}(\theta(\pi^{\vee}))$

where $\gamma'$ is the descent under moment maps. For Harish–Chandra modules, associated cycles satisfy

$\mathrm{AC}(\Theta(\Pi)) \preceq \vartheta(\mathrm{AC}(\Pi))$

and equality holds in stable range.

This controls finer invariants in the character expansions and nilpotent geometry of representations, vital for wavefront set calculations and transfer of models.

5. Algebraic, Analytic, Modular, and C*-Algebraic Frameworks

Algebraic vs. Analytic Correspondence

Howe’s “automatic continuity” question—whether the algebraic (Harish–Chandra module) and smooth (Casselman–Wallach) theta correspondences coincide—has an affirmative answer except for quaternionic type I dual pairs (Bao et al., 2016, Sun et al., 2020). The essential ingredient is the equality of first occurrence indices (via conservation relations), which implies that algebraic methods remain valid in smooth representation theory.

Modular Setting

The modular local theta correspondence is established for coefficient fields of positive characteristic (Trias, 15 Jul 2025). When $\ell$ is large and "banal," correspondence is bijective and mirrors the classical case; for small or "bad" $\ell$ (e.g., dividing certain pro-orders), failures occur, highlighting arithmetic subtleties absent over $\mathbb{C}$ .

C*-algebraic and Functorial Perspective

Recent work (Goffeng et al., 10 Dec 2024, 2207.13484) recasts the local theta correspondence as a continuous functor between categories of representations of group C*-algebras, realized via the oscillator bimodule and Rieffel induction. In equal rank and stable range cases, this induces a strong Morita equivalence, preserves distribution characters, formal degrees, and is continuous in the Fell topology. This viewpoint integrates representation theory with noncommutative geometry and offers new analytical invariants alongside classical harmonic analysis.

6. Applications and Further Developments

Unitary Representation Theory and Small Representations

Theta lifts classify and construct unitarizable representations in Archimedean and non-Archimedean contexts (Zhu, 11 Nov 2025), including small and minimal representations, via successive lifts from compact or minimal groups. In stable ranges, explicit Rallis inner product formulas and the conservation relations yield decisive information about preservation of unitarity.

Arithmetic and Geometric Langlands Aspects

In the geometric setting (Farang-Hariri, 2015), the theory is formulated at the Iwahori level: the theta correspondence is realized as a bimodule over affine Hecke algebras and matches the geometric Langlands bimodule via convolution kernels in equivariant K-theory. This connects theta functoriality with Arthur-Langlands packets and nilpotent orbital data.

Extensions, Generalizations, and Open Questions

The framework extends to almost unramified representations (Liu, 2021), model transitions for periods (Liu, 2016), and explicit parameter-level formulas for regular supercuspidals (Zhang, 2018). The strong Functoriality and compatibility with Galois distinction, as well as the geometric and categorical enhancements, point toward further unification, deeper links with automorphic forms, and conjectural "global" versions rooted in global–local compatibility of the theta correspondence.

Summary Table: Core Properties of Local Theta Correspondence

Property	Statement/Formula	Reference
Multiplicity One	$\dim\Hom(\omega, \pi \otimes \pi') \le 1$	(Zhu, 11 Nov 2025, Sun et al., 2020)
Conservation Relation	$n(\pi)+n(\pi\otimes \det)=\dim V$	(Sun et al., 2012, Zhu, 11 Nov 2025)
Langlands Parameter Lift	$\varphi_\sigma=(\varphi_\pi \otimes \chi_V\chi_W) \ominus \chi_WS_\ell$	(Atobe et al., 2016, Bakic et al., 2022)
Nilpotent Model Transfer	$\Wh_\gamma (\pi) \cong \Wh_{\gamma'}(\theta(\pi^{\vee}))$	(Zhu, 11 Nov 2025, Zhu, 2018)
Algebraic=Smooth	$\Theta_{\mathrm{alg}}(\pi) \cong \Theta_{\mathrm{sm}}(\pi)$	(Bao et al., 2016, Sun et al., 2020)
Modular Case	Bijection for large $\ell$ ; failure for small $\ell$	(Trias, 15 Jul 2025)
C*-algebraic Functor	Theta = Morita equivalence between spectra of reduced C*-algebras	(2207.13484, Goffeng et al., 10 Dec 2024)

The local theta correspondence, thus, serves as a central bridge between harmonic analysis, arithmetic representation theory, global automorphic forms, and emerging geometric/noncommutative frameworks. It synthesizes analytic, algebraic, categorical, and invariant-theoretic perspectives, enabling explicit constructions, classification, and functoriality in the representation theory of classical groups.