# Math 263C: Automorphic Forms and Representations of Metaplectic Groups and Howe Duality

This class will meet 11:30-12:20 MWF in Building 200, room 30.

• Lecture Notes. (In progress.) Please report typos.
• Also recommended: Piatetski-Shapiro's paper Work of Waldspurger in Springer LNM 1041 (available on-line through the Stanford libraries).

Metaplectic groups are central extensions of groups such as $GL(r)$ or $Sp(2r)$ over a local field the or adele ring of a number field. Thus if the ground field contains the group $\mu_n$ of $n$-th roots of unity, there is a central $1\rightarrow \mu_n\rightarrow \widetilde{GL}(r,F)\rightarrow GL(r,F)\rightarrow 1.$ When $n=2$, the automorphic theory is the natural setting for the theory of modular forms of half-integral weight.

A feature of this theory is the oscillator representation of the metaplectic double cover of $Sp(2r)$. Coming out of the Stone-von-Neumann theorem classifying the representations of the Heisenberg group, the properties of the oscillator representation explain theta correspondences, and important phenomena in automorphic forms. For example, restricting the oscillator representation to a subgroup of $Sp(2r)$ of the form $O(k)\times Sp(2l)$ where $kl=r$ gives relations between automorphic forms on orthogonal and symplectic groups that explain many important phenomena in number theory.

We will consider the oscillator representations of the metaplectic double cover of $Sp(2r)$ and its role in Howe duality, dual reductive pairs, and examples of the theta correspondences such as the Shimura correspondence and work of Waldspurger. We will consider both the local theory and certain topics in the global theory such as the work of Waldspurger.

If time allows, we will consider some aspects of the $n>2$ situation.