This is the first in a series of papers on the work promised in the research announcement below. In this preprint, we treat symplectic and even orthogonal groups. Subsequent papers (in preparation) will treat odd orthogonal and general spin groups, the metaplectic covering version of these integrals, and applications to functoriality coming from combining this work with the converse theorem (and independent of the trace formula).
In the 1980's, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models.
This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.
In this work, we give a generalization of the doubling method. We present a family of integrals representing tensor product L-functions of classical groups with general linear groups.
Our construction is uniform over all classical groups and their non-linear coverings, and is applicable to all cuspidal representations.
These integrals remove the main obstruction to proving the existence of endoscopic lifts for all automorphic representations without using the trace formula.
Braidless Weights, Minimal Representatives and the Weyl Group Multiple Dirichlet Series.
For a semisimple Lie algebra admitting a good enumeration, we prove a parameterization for the elements in its Weyl group.
As an application, we give coordinate-free comparison between the crystal graph description (when it is known) and the Lie-theoretic description of the Weyl group multiple Dirichlet series in the stable range.
Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition.
To appear in Manuscripta Math.
For type A root systems, we generalize a lemma of Casselman-Shalika. We use it to calculate the unipotent orbits attached to degenerate Eisenstein series on general linear groups. This confirms a conjecture of David Ginzburg.
In this paper we determine the unipotent orbits attached to the theta representations on covers of general linear groups. In some special cases, we show that the theta representations support certain (degenerate) types of unique functionals.