In this note, assuming the nonvanishing result of explicit theta correspondence for the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$, we show that, for metapletic–orthogonal dual pair over $\mathbb {R}$ and the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$, the theta correspondence is compatible with tempered condition by directly estimating the matrix coefficients, without using the classification theorem.

In an earlier paper of Wee Teck Gan and Gordan Savin, the local Langlands correspondence for metaplectic groups over a nonarchimedean local field of characteristic zero was established. In this paper, we formulate and prove a local intertwining relation for metaplectic groups assuming the local intertwining relation for non-quasi-split odd special orthogonal groups.

Using theta correspondence, we classify the irreducible representations of Mp2n in terms of the irreducible representations of SO2n+1 and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger for n=1.

In this paper, we construct explicitly endoscopic representations of $\widetilde{\mathrm{Sp}}_{2n}$, the metaplectic cover of a symplectic group of rank $n$. We do this in the automorphic case, and also in the local case, over a $p$-adic field.

AMS 2000 Mathematics subject classification: Primary 11F70