Let (G, G ′ ) be a finite reductive dual pair of a symplectic group and an orthogonal group. The Howe correspondence establishes a correspondence between a subset of irreducible characters of G and a subset of irreducible characters of G ′ . The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a semisimple element s in the connected component (G * ) 0 of the dual group of G and the set of irreducible unipotent characters of the centralizer of s in G * . In this paper, we prove the commutativity (up to a twist of the sign character) between these two correspondences under some restriction on the characteristic of the finite field.

In this paper we give a complete description of the Howe correspondence of unipotent characters for a finite reductive dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization when the characteristic of the base field is not equal to 2. That is, the conjecture by Aubert-Michel-Rouquier is proved.

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In this paper, we prove that the depths of irreducible admissible representations are preserved by the local theta correspondence for any type I reductive dual pairs over a nonarchimedean local field.

In this paper, we construct a splitting of the metaplectic cover of the reductive dual pairs of orthogonal and symplectic groups or the reductive dual pairs of unitary groups over a nonarchimedean local field with respect to a generalized lattice model of the Weil representation. We also prove a result concerning the splitting that we construct and the theta dichotomy for unitary group. The splitting plays a very crucial role in the study of theta correspondence for p-adic and finite reductive dual pairs. 0. Introduction. Let F be a p-adic field with odd residual characteristic. Let D be F itself or a quadratic extension of F. Let O D be the ring of integers, p D be the maximal ideal in O D , f D be the (finite) residue field and be a prime element in D. Let V (resp. V) be a finite-dimensional nondegenerate-hermitian (resp.-hermitian) space over D where , are 1 or −1 and = −1. Let U (V) (resp. U (V)) denote the group of isometries of V (resp. V). We can define a skew-symmetric F-bilinear form on W := V ⊗ D V. Then the pair of two groups (U (V), U (V)) forms a reductive dual pair in the symplectic group Sp(W). In particular, we have embeddings ι V : U (V) → Sp(W) and ι V : U (V) → Sp(W). Let (M [g], S) be a model of the Weil (projective) representation of Sp(W) with respect to a fixed nontrivial character ψ of F. Then there is a twococycle c : Sp(W) × Sp(W) → C × associated to (M [g], S) given by M [g] • M [g ] = c(g, g)M [gg ]. This two-cocycle c(g, g) determines an extension 1 −→ C × −→ Sp(W) −→ Sp(W) −→ 1. The group Sp(W) is called the metaplectic cover of Sp(W). The projective representation (M [g], S) of Sp(W) can be lifted as an ordinary representation (ω(g), S) of Sp(W).