View source for Gan–Gross–Prasad conjecture

{{Infobox mathematical statement
| name = Gan–Gross–Prasad conjecture
| image = 
| caption = 
| field = [[Representation theory]]
| conjectured by = [[Gan Wee Teck]]<br>[[Benedict Gross]]<br>[[Dipendra Prasad]]
| conjecture date = 2012
| first proof by = 
| first proof date = 
| open problem = 
| known cases =
| implied by = 
| generalizations =
| consequences =
}}

In [[mathematics]], the '''Gan–Gross–Prasad conjecture''' is a [[restricted representation|restriction]] problem in [[representation of a Lie group|the representation theory]] of [[real number|real]] or [[p-adic number|{{nowrap|''p''-adic}}]] [[Lie group]]s posed by [[Gan Wee Teck]], [[Benedict Gross]], and [[Dipendra Prasad]].<ref>{{Citation | last1=Gan | first1=Wee Teck |authorlink1=Gan Wee Teck| last2=Gross | first2=Benedict H. |authorlink2=Benedict Gross | last3=Prasad | first3=Dipendra | authorlink3= Dipendra Prasad |title=Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups | mr=3202556 | year=2012 | journal=[[Astérisque]] | isbn=978-2-85629-348-5 | volume=346 | pages=1–109}}</ref> The problem originated from a [[conjecture]] of Gross and Prasad for [[orthogonal group|special orthogonal groups]] but was later generalized to include all four [[classical group]]s. In the cases considered, it is known that the [[Multiplicity-one theorem|multiplicity of the restrictions is at most one]]<ref>{{Citation | last1=Aizenbud| first1=Avraham | last2=Gourevitch | first2=Dmitry | last3=Rallis | first3=Stephen |authorlink3= Stephen Rallis| last4= Schiffmann | first4=Gérard | title=Multiplicity-one theorems | year=2010 | journal=[[Annals of Mathematics]] | volume=172 | issue=2 | pages=1407–1434 | mr=2680495 | arxiv=0709.4215 | doi=10.4007/annals.2010.172.1413 }}</ref><ref>{{Citation | last=Sun | first=Binyong | authorlink =Sun Binyong | title=Multiplicity-one theorems for Fourier–Jacobi models | year=2012 | journal=[[American Journal of Mathematics]] | doi=10.1353/ajm.2012.0044 | volume=134 | issue=6 | pages=1655–1678| arxiv=0903.1417 }}</ref><ref>{{Citation | last1=Sun | first1=Binyong | authorlink1=Sun Binyong| last2=Zhu |first2=Chen-Bo | authorlink2=Chen-Bo Zhu |title=Multiplicity-one theorems: the Archimedean case | year=2012 | journal=Annals of Mathematics | mr=2874638 | volume=175 | issue=1 | pages=23–44 | doi=10.4007/annals.2012.175.1.2| doi-access=free | arxiv=0903.1413 }}</ref>
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 group#Compact Lie groups|compact Lie groups]]. Let <math>\pi</math> be an [[irreducible representation|irreducible]] finite-dimensional representation of the compact [[unitary group]] <math>U(n)</math>, and consider its restriction to the naturally embedded [[subgroup]] <math>U(n-1)</math>. It is known that this restriction is multiplicity-free, but one may ask precisely which irreducible representations of <math>U(n-1)</math> occur in the restriction.

By the [[Theorem of the highest weight|Cartan–Weyl theory of highest weights]], there is a classification of the irreducible representations of <math>U(n)</math> via their [[Weight (representation_theory)#Highest weight|highest weights]] which are in natural [[bijection]] with sequences of [[integer]]s <math>\underline{a} = (a_1 \geq a_2 \geq \cdots \geq a_n)</math>.
Now suppose that <math>\pi</math> has highest weight <math>\underline{a}</math>. Then an irreducible representation <math>\tau</math> of <math>U(n-1)</math> with highest weight <math>\underline{b}</math> occurs in the restriction of <math>\pi</math> to <math>U(n-1)</math> (viewed as a subgroup of <math>U(n)</math>) if and only if <math>\underline{a}</math> and <math>\underline{b}</math> are interlacing, i.e. <math>a_1 \geq b_1 \geq a_2 \geq b_2 \geq \cdots \geq b_{n-1} \geq a_n</math>.<ref>{{citation|first=Hermann|last=Weyl|authorlink=Hermann Weyl|title=The Theory of Groups and Quantum Mechanics|publisher=Dover Publications|year=1950}}</ref>

The Gan–Gross–Prasad conjecture then considers the analogous restriction problem for other classical groups.<ref>{{Citation | last=Gan | first=Wee Teck |authorlink=Gan Wee Teck | title=Recent progress on the Gross-Prasad conjecture | year=2014 | journal=Acta Mathematica Vietnamica | doi=10.1007/s40306-014-0047-2 | issn=2315-4144 | volume=39 | issue=1 | pages=11–33| s2cid=256378802 }}</ref>

==Statement==
The conjecture has slightly different forms for the different classical groups. The formulation for [[Unitary group#Indefinite forms|unitary group]]s is as follows.

===Setup===
Let <math>V</math> be a finite-dimensional [[vector space]] over a [[field (mathematics)|field]] <math>k</math> not of [[characteristic (algebra)|characteristic]] <math>2</math> equipped with a non-degenerate [[sesquilinear form]] that is <math>\varepsilon</math>-Hermitian (i.e. <math>\varepsilon = 1</math> if the form is Hermitian and <math>\varepsilon = -1</math> if the form is skew-Hermitian). Let <math>W</math> be a non-degenerate [[linear subspace|subspace]] of <math>V</math> such that <math>V = W \oplus W^\perp</math> and <math>W^\perp</math> is of dimension <math>(\varepsilon + 1)/2</math>. Then let <math>G = G(V) \times G(W)</math>, where <math>G(V)</math> is the unitary group preserving the form on <math>V</math>, and let <math>H = \Delta G(W)</math> be the [[diagonal subgroup]] of <math>G</math>.

Let <math>\pi = \pi_1 \boxtimes \pi_2</math> be an irreducible smooth representation of <math>G</math> and let <math>\nu</math> be either the [[trivial representation]] (the "Bessel case") or the [[Metaplectic group|Weil representation]] (the "Fourier–Jacobi case").
Let <math>\varphi = \varphi_1 \times \varphi_2</math> be a generic [[Local Langlands conjectures#Local Langlands conjectures for other groups|L-parameter]] for <math>G = G(V) \times G(W)</math>, and let <math>\Pi_\varphi</math> be the associated Vogan L-packet.

===Local Gan–Gross–Prasad conjecture===
If <math>\varphi</math> is a local L-parameter for <math>G</math>, then
:<math>\sum_{\text{relevant } \pi \in \Pi_\varphi} \dim \operatorname{Hom}_H (\pi \otimes \overline{\nu}, \mathbb{C}) = 1.</math>

Letting <math>\eta_{\mathrm{GP}}</math> be the "distinguished character" defined in terms of the [[Langlands–Deligne local constant]], then furthermore
:<math>\operatorname{Hom}_H (\pi(\varphi, \eta) \otimes \overline{\nu}, \mathbb{C}) \neq 0 \text{ if and only if } \eta = \eta_{\mathrm{GP}}.</math>

===Global Gan–Gross–Prasad conjecture===
For a quadratic field extension <math>E/F</math>, let <math>L_E(s, \pi_1 \times \pi_2) := L_E(s, \pi_1 \boxtimes \pi_2, \mathrm{std}_n \boxtimes \mathrm{std}_{n-1})</math> where <math>L_E</math> 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:
# The period interval <math>P_H</math> is nonzero when restricted to <math>\pi</math>.
# For all places <math>v</math>, the local Hom space <math>\operatorname{Hom}_{H(F_v)}(\pi_v, \nu_v) \neq 0</math> and <math>L_E(1/2, \pi_1 \times \pi_2) \neq 0</math>.

==Current status==
===Local Gan–Gross–Prasad conjecture===
In a series of four papers between 2010 and 2012, [[Jean-Loup Waldspurger]] [[mathematical proof|proved]] the local Gan–Gross–Prasad conjecture for [[tempered representation]]s of special orthogonal groups over [[p-adic number|{{nowrap|''p''-adic}} fields]].<ref>{{Citation | last=Waldspurger | first=Jean-Loup | authorlink = Jean-Loup Waldspurger | title=Une Formule intégrale reliée à la conjecture locale de Gross-Prasad. | year=2012 | journal=[[Compositio Mathematica]]| volume=146 | issue=5 | pages=1180–1290| doi=10.1112/S0010437X10004744 | arxiv=0902.1875 }}</ref><ref>{{Citation | last=Waldspurger | first=Jean-Loup | authorlink = Jean-Loup Waldspurger| title=Une Formule intégrale reliée à la conjecture locale de Gross-Prasad, 2ème partie: extension aux représentations tempérées. | year=2012 | journal=Astérisque | volume=347 | pages=171–311}}</ref><ref>{{Citation | last=Waldspurger | first=Jean-Loup | authorlink = Jean-Loup Waldspurger| title=La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes spéciaux orthogonaux. | year=2012 | journal=Astérisque | volume=347 | pages=103–166}}</ref><ref>{{Citation | last=Waldspurger | first=Jean-Loup | authorlink = Jean-Loup Waldspurger| title=Calcul d'une valeur d'un facteur epsilon par une formule intégrale. | year=2012 | journal=Astérisque | volume=347}}</ref> In 2012, [[Colette Moeglin]] and Waldspurger then proved the local Gan–Gross–Prasad conjecture for generic non-tempered representations of special orthogonal groups over {{nowrap|''p''-adic}} fields.<ref>{{Citation | last1=Moeglin | first1=Colette | authorlink1=Colette Moeglin |last2=Waldspurger | first2=Jean-Loup | authorlink2 = Jean-Loup Waldspurger | title=La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général | year=2012 | journal=Astérisque | volume=347}}</ref>

In his 2013 thesis, Raphaël Beuzart-Plessis proved the local Gan–Gross–Prasad conjecture for the tempered representations of unitary groups in the {{nowrap|''p''-adic}} Hermitian case under the same hypotheses needed to establish the [[local Langlands conjecture]].<ref>{{Citation | last=Beuzart-Plessis| first=Raphaël | title=La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires | year=2012 | journal=PhD Thesis}}</ref>

Hongyu He proved the Gan-Gross-Prasad conjectures for discrete series representations of the real unitary group U(p,q).<ref>{{Citation | last=He | first= Hongyu | title= On the Gan-Gross-Prasad conjectures for U(p,q) | year=2017 | journal=[[Inventiones Mathematicae]] | volume=209 | number=3| pages=837–884 | doi=10.1007/s00222-017-0720-x| arxiv=1508.02032 }}</ref>

===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.<ref>{{Citation | last1=Ginzburg | first1=David |authorlink1= David Ginzburg | last2=Jiang | first2=Dihua |authorlink2=Dihua Jiang | last3=Rallis | first3=Stephen |authorlink3=Stephen Rallis| title=On the nonvanishing of the central value of the Rankin–Selberg L-functions. | year=2004 | journal=[[Journal of the American Mathematical Society]] | volume=17 | issue=3 | pages=679–722 | doi=10.1090/S0894-0347-04-00455-2  | doi-access = free}}</ref><ref>{{Citation | last1=Ginzburg | first1=David |authorlink1= David Ginzburg | last2=Jiang | first2=Dihua |authorlink2=Dihua Jiang | last3=Rallis | first3=Stephen |authorlink3=Stephen Rallis| title=On the nonvanishing of the central value of the Rankin–Selberg L-functions, II. | year=2005 | journal=Automorphic Representations, L-functions and Applications: Progress and Prospects | publisher=Ohio State Univ. Math. Res. Inst. Publ. 11, de Gruyter | place=Berlin | pages=157–191| doi=10.1515/9783110892703.157 | isbn=978-3-11-017939-2 }}</ref><ref>{{Citation | last1=Ginzburg | first1=David |authorlink1= David Ginzburg | last2=Jiang | first2=Dihua |authorlink2=Dihua Jiang | last3=Rallis | first3=Stephen |authorlink3=Stephen Rallis | title=Models for certain residual representations of unitary groups. Automorphic forms and L-functions I. | year=2009 | journal=Global Aspects | publisher=Contemp. Math., 488, Amer. Math. Soc. | place=Providence, RI | pages=125–146}}</ref>

In the Bessel case of the global Gan–Gross–Prasad conjecture for unitary groups, [[Wei Zhang (mathematician) | Wei Zhang]] used the theory of the [[Arthur–Selberg trace formula|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.<ref>{{Citation | last=Zhang | first=Wei |authorlink= Wei Zhang (mathematician)| title=Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups. | year=2014 | journal=[[Annals of Mathematics]] | mr=2874638 | volume=180 | issue=3 | pages=971–1049 | doi=10.4007/annals.2012.175.1.2| doi-access=free | arxiv=0903.1413 }}</ref>

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.<ref>{{Citation | last=Liu | first=Yifeng |authorlink=Yifeng Liu | title=Relative trace formulae toward Bessel and Fourier–Jacobi periods of unitary groups. | year=2014 | journal=Manuscripta Mathematica | volume=145 | issue=1–2 | pages=1–69 | doi=10.1007/s00229-014-0666-x| arxiv=1012.4538 }}</ref><ref>{{Citation | last=Xue | first=Hang| title=The Gan–Gross–Prasad conjecture for U(n) × U(n). | year=2014 | journal=[[Advances in Mathematics]] | volume=262 | pages=1130–1191 | doi=10.1016/j.aim.2014.06.010 | mr=3228451| doi-access= }}</ref>

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.<ref>{{Citation | last1=Jiang | first1=Dihua | authorlink1=Dihua Jiang |last2=Zhang | first2=Lei | title=Arthur parameters and cuspidal automorphic modules of classical groups. | year=2020 | journal=[[Annals of Mathematics]] | volume=191 | issue=3 | pages=739–827 | doi=10.4007/annals.2020.191.3.2 | arxiv=1508.03205 }}</ref>

==References==
{{Reflist|colwidth=30em}}

{{DEFAULTSORT:Gan-Gross-Prasad conjecture}}
[[Category:Automorphic forms]]
[[Category:Conjectures]]
[[Category:Number theory]]
[[Category:Langlands program]]
[[Category:Representation theory of Lie groups]]
[[Category:Representation theory of groups]]
[[Category:Zeta and L-functions]]