[Paper Review] Multiplicity preservation for orthogonal-symplectic and unitary dual pair correspondences

Jian-Shu Li, Binyong Sun+1 more|ArXiv.org|Mar 8, 2009

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper proves multiplicity preservation for theta correspondences in orthogonal-symplectic and unitary dual pairs over non-archimedean local fields of characteristic zero. Using geometric invariant theory and Gelfand-Kazhdan criteria for distributions, the authors establish that the dimension of the space of $ω_{ψ}$-invariant functionals on $π \otimes \pi'$ is at most one, confirming the multiplicity one property for these dual pair correspondences.

ABSTRACT

Over a non-archimedean local field of characteristic zero, we prove the multiplicity preservation for orthogonal-symplectic dual pair correspondences and unitary dual pair correspondences.

Motivation & Objective

• To establish the multiplicity one property for theta correspondences in orthogonal-symplectic and unitary dual pairs over non-archimedean local fields of characteristic zero.
• To extend Waldspurger’s result (for odd residue characteristic) and Howe’s archimedean result to the general non-archimedean setting.
• To prove that the space of $ω_{ψ}$-invariant functionals on $π \otimes \pi'$ has dimension at most one for any genuine irreducible admissible smooth representations $π$ and $π'$.

Proposed method

• Constructs the metaplectic cover $ω_{ψ}$ of the symplectic group acting on a Heisenberg group via the oscillator representation.
• Introduces extended groups $̄{G}$, $̄{G}'$, and $̄{Sp}(\mathbf{E})$ to handle the action of $-1$ and stabilize orbits under the group action.
• Uses the Gelfand-Kazhdan criterion for distributions on totally disconnected groups to reduce the multiplicity problem to invariance of generalized functions under a larger group.
• Applies a geometric result showing that $G$-orbits in $\mathbf{E}'$ are stable under $\breve{G}$, which implies invariance of bi-$\mathbf{G}$-invariant generalized functions under $\{\pm 1\} \ltimes (\breve{\mathbf{G}} \times \breve{\mathbf{G}})$.
• Leverages the fact that the contragredient of the oscillator representation is isomorphic to its dual, allowing symmetry in the Hom space to conclude the bound.
• Applies the Gelfand-Kazhdan criterion to the group $\mathbf{J} = \mathbf{G} \ltimes \mathbf{H}$ to show that $\dim \mathrm{Hom}_{\mathbf{G}}(\Pi, \mathbb{C}) \cdot \dim \mathrm{Hom}_{\mathbf{G}}(\Pi^\vee, \mathbb{C}) \leq 1$ for irreducible representations $\Pi$ of $\mathbf{J}$.

Experimental results

Research questions

• RQ1 Does the theta correspondence between orthogonal-symplectic and unitary dual pairs preserve multiplicity one over non-archimedean local fields of characteristic zero?
• RQ2 Can the multiplicity one property be established uniformly across all classical groups in the dual pair setting?
• RQ3 How does the action of the extended group $\breve{G}$ affect the stability of orbits in the dual pair construction?

Key findings

• The dimension of the space of $\widetilde{G} \times \widetilde{G}'$-invariant functionals on $\omega_{\psi} \otimes \pi \otimes \pi'$ is at most one for any genuine irreducible admissible smooth representations $\pi$ and $\pi'$.
• The result confirms the multiplicity one conjecture for orthogonal-symplectic and unitary dual pair correspondences in the non-archimedean setting with characteristic zero.
• The proof relies on the invariance of bi-$\mathbf{G}$-invariant generalized functions under the action of $\{\pm 1\} \ltimes (\breve{\mathbf{G}} \times \breve{\mathbf{G}})$, established via geometric orbit stability.
• The orbit stability result follows from the existence of an element in $\breve{G} \setminus G$ that negates a given $x \in \tilde{\mathfrak{g}}$ and preserves a flag $F$, implying $\breve{G}$-stability of $G$-orbits.
• The Gelfand-Kazhdan criterion is applied to the group $\mathbf{J} = \mathbf{G} \ltimes \mathbf{H}$ to bound the product of dimensions of Hom spaces to at most one.
• The final conclusion $\dim \mathrm{Hom}_{\widetilde{G} \times \widetilde{G}'}(\omega_{\psi}, \pi \otimes \pi') \leq 1$ is derived by symmetry and duality, using the fact that $\omega_{\psi}^\vee \cong \omega_{\psi}$ and $\dim \mathrm{Hom}_{\mathbf{G}}(\omega_{\psi} \otimes \pi^\vee \otimes \pi^{\prime\vee}, \mathbb{C}) = \dim \mathrm{Hom}_{\mathbf{G}}(\omega_{\psi}^\vee \otimes \pi \otimes \pi^{\prime}, \mathbb{C})$.