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 formula_1 be an irreducible finite dimensional representation of the compact unitary group formula_2, and consider its restriction to the naturally embedded subgroup formula_3. It is known that this restriction is multiplicity-free, but one may ask precisely which irreducible representations of formula_3 occur in the restriction. By the Cartan–Weyl theory of highest weights, there is a classification of the irreducible representations of formula_2 via their highest weights which are in natural bijection with sequences of integers formula_6. Now suppose that formula_1 has highest weight formula_8. Then an irreducible representation formula_9 of formula_3 with highest  weight formula_11 occurs in the restriction of formula_1 to formula_3 (viewed as a subgroup of formula_2) if and only if formula_8 and formula_11 are interlacing, i.e. formula_17. 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 unitary groups is as follows. Setup. Let formula_18 be a finite-dimensional vector space over a field formula_19 not of characteristic formula_20 equipped with a non-degenerate sesquilinear form that is formula_21-Hermitian (i.e. formula_22 if the form is Hermitian and formula_23 if  the form is skew-Hermitian). Let formula_24 be a non-degenerate subspace of formula_18 such that formula_26 and formula_27 is of dimension formula_28. Then let formula_29, where formula_30 is the unitary group preserving the form on formula_18, and let formula_32 be the diagonal subgroup of formula_33. Let formula_34 be an irreducible smooth representation of formula_33 and let formula_36 be either the trivial representation (the "Bessel case") or the Weil representation (the "Fourier–Jacobi case"). Let formula_37 be a generic L-parameter for formula_29, and let formula_39 be the associated Vogan L-packet. Local Gan–Gross–Prasad conjecture. If formula_40 is  a local L-parameter for formula_33, then Letting formula_43 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 formula_45, let formula_46 where formula_47 is the global L-function obtained as the product of local L-factors given by the local Langlands conjectures. The conjecture states that the following are equivalent: 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 special 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 special 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(p,q). Global Gan–Gross–Prasad conjecture. In a series of papers between 2004  and 2009, David Ginzburg, Dihua Jiang, and Stephen Rallis showed the (1) implies (2) direction of the global Gan–Gross–Prasad conjecture for all quasisplit classical 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. In the Bessel case of the global Gan–Gross–Prasad conjecture for special orthogonal groups and unitary groups, Dihua Jiang and Lei Zhang used the theory of twisted automorphic descents to prove that (1) implies (2) in its full generality, i.e. for any irreducible cuspidal automorphic representation with a generic global Arthur parameter, and that (2) implies (1) subject to a certain global assumption.