It is well known that Howe's correspondence has been proved for reductive dual pair over local p-adic fields with p≠2. In this thesis, we examine Howe's correspondence for the dual pair (U(1), U(1,1)) over dyadic fields, i.e. p-adic fields with p = 2. We first give a classification of unitary irreducible admissible representations of U(1, 1), using minimal nondegenerate representations introduced by Allen Moy and some related Hecke algebras. This classification helps us to identify those representations occurring in the correspondence....[ Read more ]

It is well known that Howe's correspondence has been proved for reductive dual pair over local p-adic fields with p≠2. In this thesis, we examine Howe's correspondence for the dual pair (U(1), U(1,1)) over dyadic fields, i.e. p-adic fields with p = 2. We first give a classification of unitary irreducible admissible representations of U(1, 1), using minimal nondegenerate representations introduced by Allen Moy and some related Hecke algebras. This classification helps us to identify those representations occurring in the correspondence.

In Chapter 1, we classify unitary irreducible admissible representations of U(1, 1). Since principal series are well known in the literature, we mainly focus our interest on supercuspidals. Some Hecke algebras are involved in the discussion. We classify all the minimal nondegenerate representations, including some new types which will not occur when p is odd. Except for a finite number of supercuspidal representations, we give construction and parametrization of all the other representations.

Chapter 2 and Chapter 3 are devoted to construction of the dual pair. We utilize Schrödinger model and lattice model of Weil representation in our discussion. For each character of U(1), we show that the corresponding eigenspace is nonzero, and we give a detailed description of those eigenfunctions in each eigenspace. In particular, we calculated the "largest subgroup" (in some sense defined in the chapter) which have nonzero fixed eigenfunctions.

In Chapter 4, we use the above facts to calculate the minimal nondegenerate representations of U(1, 1) occurring in the dual pair correspondence. Furthermore, under our classification of Charter 1, some eigenspaces as U(1,1) representations are identified. The conclusions are summarized in theorems of this chapter, which is the main contribution of this thesis.