Number Theory Seminar @ University of Melbourne

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.

The theory of automorphic forms sometimes requires working with covers of reductive groups, rather than with linear algebraic groups. For example, the Jacobi theta function naturally lives on the metaplectic group, a double cover of SL2, which is not linear. This motivates the study of representations of covers of p-adic reductive groups.

For such covering groups one would like to prove a local Langlands correspondence (LLC). In the case of linear groups, the LLC was established for the principal Bernstein block by combining the Borel-Casselman Theorem, which reduces the problem to the representation theory of affine Hecke algebras, with the work of Kazhdan-Lusztig on the representation theory of these algebras.

To extend this result to covering groups one must prove an analog of the Borel-Casselman Theorem. For tame covers this was accomplished by Savin, but the non-tame case has remained open. In this talk, I will describe recent joint work with Shuichiro Takeda in which we prove an analog of the Borel-Casselman Theorem for non-tame double covers.