Postdoctoral Fellow
Department of Mathematics
The Weizmann Institute of Science
Herzl 234, Rehovot 7610001
Israel
Email: yuanqing[dot]cai[at]weizmann[dot]ac[dot]il
Office: Goldsmith Building, Room 212
I recently received my PhD from Boston College, under the supervision of Solomon Friedberg.
I'm interested in number theory
and
representation theory.
In particular, my interests include:
Functoriality in the Langlands program
Automorphic forms on covering groups
Rankin-Selberg integrals of L-functions
Combinatorial aspects of representation theory
Weyl group multiple Dirichlet series
Publications and Preprints
(with
S. Friedberg,
D. Ginzburg
and
E. Kaplan)
Doubling Constructions and Tensor Product L-functions: the Linear Case, preprint.
On the arxiv, here.
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).
(with
S. Friedberg,
D. Ginzburg
and
E. Kaplan)
Doubling Constructions for Covering Groups and Tensor Product L-Functions, preprint.
Research announcement on the arxiv, here.
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.
Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition.
Preprint here.
Also on the arxiv, here.
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.
Fourier Coefficients for Theta Representations on Covers of General Linear Groups.
Preprint here.
Also on the arxiv, here.
Video presentation.
To appear in Trans. Amer. Math. Soc.
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.