Local aspects

In the course we shall introduce basic concepts around the local theta correspondence in the non-archimedean case. After introducing the Heisenberg group over a non-archimedean local field, we will define the corresponding Heisenberg-Weil representation and introduce the related metaplectic group, a cover of a symplectic group, and its oscillator representation. On the other hand, there is a concept of a dual reductive pair, consisting of mutually commuting reductive subgroups inside this symplectic group. Roughly, the oscillator representation restricts to a representation of the groups in a dual reductive pair, and is small enough that its isotypic quotients decompose as a tensor product of representations of the two groups. This enables one to relate the irreducible, smooth, complex representations of one group in a dual reductive pair to a similar representation of the other group. This is, the so-called, theta correspondence. In the course we aim to describe this correspondence more precisely, state the basic results,  and  present some of the usual tools for the study. If time permits, we shall also explain the most recent developments.