Gan–Gross–Prasad conjecture


In mathematics, the Gan–Gross–Prasad conjecture is a restriction problem in the representation theory of real or p-adic Lie groups posed by Gan Wee Teck, Benedict Gross, and Dipendra Prasad. The problem originated from a conjecture of Gross and Prasad for special orthogonal groups but was later generalized to include all four classical groups. In the cases considered, it is known that the multiplicity of the restrictions is at most one
and the conjecture describes when the multiplicity is precisely one.

Motivation

A motivating example is the following classical branching problem in the theory of compact Lie groups. Let be an irreducible finite dimensional representation of the compact unitary group, and consider its restriction to the naturally embedded subgroup. It is known that this restriction is multiplicity-free, but one may ask precisely which irreducible representations of occur in the restriction.
By the Cartan–Weyl theory of highest weights, there is a classification of the irreducible representations of via their highest weights which are in natural bijection with sequences of integers.
Now suppose that has highest weight. Then an irreducible representation of with highest weight occurs in the restriction of to if and only if and are interlacing, i.e..
The Gan–Gross–Prasad conjecture then considers the analogous restriction problem for other classical groups.

Statement

The conjecture has slightly different forms for the different classical groups. The formulation for general unitary groups is as follows.

Setup

Let be a finite-dimensional vector space over a field not of characteristic equipped with a non-degenerate sesquilinear form that is -symmetric or the Weil representation.
Let be a generic L-parameter for, and let be the associated Vogan L-packet.

Local Gan–Gross–Prasad conjecture

If is a local L-parameter for, then
Letting be the "distinguished character" defined in terms of the Langlands–Deligne local constant, then furthermore

Global Gan–Gross–Prasad conjecture

For a quadratic field extension, let where is the global L-function obtained as the product of local L-factors given by the local Langlands conjectures.
The following are equivalent:
  1. The period interval is nonzero when restricted to.
  2. For all places, the local Hom space and.

    Current status

Local Gan–Gross–Prasad conjecture

In a series of four papers between 2010 and 2012, Jean-Loup Waldspurger proved the local Gan–Gross–Prasad conjecture for tempered representations of orthogonal groups over p-adic fields. In 2012, Colette Moeglin and Waldspurger then proved the local Gan–Gross–Prasad conjecture for generic non-tempered representations of orthogonal groups over p-adic fields.
In his 2013 thesis, Raphaël Beuzart-Plessis proved the local Gan–Gross–Prasad conjecture for the tempered representations of unitary groups in the p-adic Hermitian case under the same hypotheses needed to establish the local Langlands conjecture.
Hongyu He proved the Gan-Gross-Prasad conjectures for discrete series representations of the real unitary group U.

Global Gan–Gross–Prasad conjecture

In a series of papers between 2004 and 2009, David Ginzburg, Dihua Jiang, and Stephen Rallis showed the implies direction of the global Gan–Gross–Prasad conjecture for unitary groups.
In the Bessel case of the global Gan–Gross–Prasad conjecture for unitary groups, Wei Zhang used the theory of the relative trace formula by Hervé Jacquet and the work on the fundamental lemma by Zhiwei Yun to prove that the conjecture is true subject to certain local conditions in 2014.
In the Fourier–Jacobi case of the global Gan–Gross–Prasad conjecture for unitary groups, Yifeng Liu and Hang Xue showed that the conjecture holds in the skew-Hermitian case, subject to certain local conditions.