My principal area of research has been the local and global aspects of the
theory of Automorphic Representations. Starting with my thesis work at Harvard
in 1989, I have been involved in understanding a basic question in representation
theory of groups over local and global fields: how do representations of these
groups decompose when restricted to subgroups. These are called branching
laws, and play a large role in the theory of automorphic forms. Since one is
typically dealing with infinite dimensional representations, this is a subtle question
specially when the subgroup is noncompact. Each case where multiplicity one
is known, has been of considerable importance to the subject, and has played
a important role in explicit construction of L-functions: examples of Whittaker
models, and spherical representations are two cases in point.
The local Langlands program, now proven in many cases, such as for GL(n)
by Harris and Taylor, and also by Henniart, and then by J. arthur for classical groups,
relates representation theory of classical groups over local fields to Galois representations,
and offers the possibility that some of these questions about branching laws may have
an answer in terms of Galois representations, and indeed my work together with Benedict Gross,
and then with Wee Teck Gan, points to the importance of these Galois representations in
several questions about branching laws.
By now a large class of examples are studied and understood, such as restriction of
representations of SO(n + 1) to SO(n), or of U(n + 1) to U(n), in which multiplicity one
theorems have been proved by various authors. My work with Benedict Gross
and Wee Teck Gan, makes precise predictions about these branching laws using
what are called epsilon factors (of symplectic representations of the Weil-Deligne group),
which are certain Gauss sums taking values in ±1. The conjectures seem to imply that the
internal structure of what is called an L-packet is dictated by these epsilon factors, just as the set of L-packets
themselves are parametrized by L-functions.
Two papers with Gan and Gross in Asterisque 346 (2012) extended the framework of
my earlier work with Gross to all classical, and metaplectic groups.
There has been a lot of attention drawn by these conjectures in the recent
past, and a number of theorems proven along the way: the most spectacular
progress was made by Waldspurger who in a series of papers –one jointly with
Moeglin– totalling more than 500 pages proved what has come to
be known as the Gan-Gross-Prasad conjectures in the local non-Archimedean
case for the orthogonal groups. (The conjectures for the case of Unitary groups in both the Archimedean
and non-Archimedean cases have since been settled by R. Beuzart-Plessis.)
There are some global theorems proved by Ginzburg, Jiang, Lei Zhang. Jacquet and Rallis
developed a program using the relative trace formula approach
to the global Gan-Gross-Prasad conjectures which has been very successfully used by Wei Zhang,
with recent further progress (2020) Raphaël Beuzart-Plessis, Yifeng Liu, Wei Zhang, Xinwen Zhu.
These conjectures have found further interest as they seem to be one large
class of examples as the higher dimensional analogue of the Gross-Zagier the-
orem, one of the most striking theorems in Arithmetic Geometry from the 80’s.
This line of thought is being actively pursued by Shou-Wu Zhang, Wei Zhang
and their collaborators. They have recently obtained a relative trace formula
proof of the Gross-Zagier theorem as a first step in their program.
The global questions about branching laws concern period integrals of auto-
morphic functions on sub-arithmetic variety, such as the integral on H(Q)\H(A) of
an automorphic function on G(Q)\G(A).
These period integrals (more specifically, the relationship of these period in-
tegrals to central L-values) have played an important role in the recent work of
Soundararajan and Holowinsky in what is called QUE, Quantum Unique Ergodic-
ity conjecture, which asserts that for modular forms f_i of weight k_i on the upper
half plane, the measures y^{k_i} |f_i |^2 dxdy/y^{2}
tend to the invariant measure on the upper half plane, and amounts to a weak
sub-convexity estimate for triple product L-value at the center.