Global aspects

In these lectures, I will talk about the theory of global theta lifts. Given a cuspidal automorphic form, its theta lift is defined as an integral against the theta kernel. In this way, one can construct an automorphic representation of one group out of another in a fairly explicit and systematic way. The major difficulty is the issue of non-vanishing of the theta lift. A systematic method of showing the nonvanishing is the Rallis inner product formula. The Rallis inner product formula, combined with the Siegel-Weil formula, expresses the inner product of theta lifts in terms of the L-function of the given automorphic representation. I will talk about this theory in detail.