Using theta correspondence, we obtain a classification of the irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover, we show that our classifications coincide with the local Langlands correspondence established by Arthur [Art13] and formulated precisely by Atobe-Gan [AG17b] for quasi-split even orthogonal groups.
NotationLet F be a non-archimedean local field of characteristic 0 and residue characteristic p. Let W F be the Weil group of F . Let | • | F be the normalized absolute value of F . We fix a non-trivial additive character ψ of F , and for c ∈ F × , we define an additive character ψ c of F byNote that any non-trivial additive character of F is of the form ψ c for some c ∈ F × . Let (•, •) F be the quadratic Hilbert symbol of F .If G is the F -rational points of a linear algebraic group, we let Irr(G) be the set of equivalence classes of irreducible smooth representations of G and Irr temp (G) be the subset of Irr(G) consisting of tempered representations of G. Let π be a representation of G, the contragredient representation of π is denote by π ∨ .