In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field F of characteristic 0. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.

Abstract. In this paper, we highlight and state precisely the local Langlands correspondence for quasi-split O 2n established by Arthur. We give two applications: Prasad's conjecture and Gross-Prasad conjecture for O n . Also, we discuss the Arthur conjecture for O 2n , and establish the Arthur multiplicity formula for O 2n .

Abstract. We prove the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case under some assumptions. This is the last case of the local Gan-Gross-Prasad conjectures which has not been established. We also prove two of Prasad's conjectures on the local theta correspondence in the almost equal rank case.

In this paper, we give an explicit computable algorithm for the Zelevinsky-Aubert duals of irreducible representations of p-adic symplectic and odd special orthogonal groups. To do this, we establish explicit formulas for certain derivatives and socles. We also give a combinatorial criterion for the irreducibility of certain parabolically induced representations. Contents1. Introduction 1 2. Notation and preliminaries 4 3. The theory of ρ-derivatives 9 4. The algorithm 14 5. The endoscopic classification 15 6. Best matching functions, the ugly and the negative case 19 7. Explicit formulas for derivatives and socles: The positive case 21 8. Explicit formulas for derivatives and socles: A non-cuspidal case 28 9. Some examples of Zelevinsky-Aubert duality 35 References 37

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