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.^[1] 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^[2][3][4] and the conjecture describes when the multiplicity is precisely one.

Gan–Gross–Prasad conjecture

Field	Representation theory
Conjectured by	Gan Wee Teck Benedict Gross Dipendra Prasad
Conjectured in	2012

Motivation

A motivating example is the following classical branching problem in the theory of compact Lie groups. Let ${\displaystyle \pi }$ be an irreducible finite dimensional representation of the compact unitary group ${\displaystyle U(n)}$, and consider its restriction to the naturally embedded subgroup ${\displaystyle U(n-1)}$. It is known that this restriction is multiplicity-free, but one may ask precisely which irreducible representations of ${\displaystyle U(n-1)}$ occur in the restriction.

By the Cartan–Weyl theory of highest weights, there is a classification of the irreducible representations of ${\displaystyle U(n)}$ via their highest weights which are in natural bijection with sequences of integers ${\displaystyle {\underline {a}}=(a_{1}\geq a_{2}\geq \cdots \geq a_{n})}$. Now suppose that ${\displaystyle \pi }$ has highest weight ${\displaystyle {\underline {a}}}$. Then an irreducible representation ${\displaystyle \tau }$ of ${\displaystyle U(n-1)}$ with highest weight ${\displaystyle {\underline {b}}}$ occurs in the restriction of ${\displaystyle \pi }$ to ${\displaystyle U(n-1)}$ (viewed as a subgroup of ${\displaystyle U(n)}$) if and only if ${\displaystyle {\underline {a}}}$ and ${\displaystyle {\underline {b}}}$ are interlacing, i.e. ${\displaystyle a_{1}\geq b_{1}\geq a_{2}\geq b_{2}\geq \cdots \geq b_{n-1}\geq a_{n}}$.^[5]

The Gan–Gross–Prasad conjecture then considers the analogous restriction problem for other classical groups.^[6]

Statement

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

Setup

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

Let ${\displaystyle \pi =\pi _{1}\boxtimes \pi _{2}}$ be an irreducible smooth representation of ${\displaystyle G}$ and let ${\displaystyle \nu }$ be either the trivial representation (the "Bessel case") or the Weil representation (the "Fourier–Jacobi case"). Let ${\displaystyle \varphi =\varphi _{1}\times \varphi _{2}}$ be a generic L-parameter for ${\displaystyle G=G(V)\times G(W)}$, and let ${\displaystyle \Pi _{\varphi }}$ be the associated Vogan L-packet.

Local Gan–Gross–Prasad conjecture

If ${\displaystyle \varphi }$ is a local L-parameter for ${\displaystyle G}$, then

${\displaystyle \sum _{{\text{relevant }}\pi \in \Pi _{\varphi }}\dim \operatorname {Hom} _{H}(\pi \otimes {\overline {\nu }},\mathbb {C} )=1.}$

Letting ${\displaystyle \eta _{\mathrm {GP} }}$ be the "distinguished character" defined in terms of the Langlands–Deligne local constant, then furthermore

${\displaystyle \operatorname {Hom} _{H}(\pi (\varphi ,\eta )\otimes {\overline {\nu }},\mathbb {C} )\neq 0{\text{ if and only if }}\eta =\eta _{\mathrm {GP} }.}$

Global Gan–Gross–Prasad conjecture

For a quadratic field extension ${\displaystyle E/F}$, let ${\displaystyle L_{E}(s,\pi _{1}\times \pi _{2}):=L_{E}(s,\pi _{1}\boxtimes \pi _{2},\mathrm {std} _{n}\boxtimes \mathrm {std} _{n-1})}$ where ${\displaystyle L_{E}}$ 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:

1. The period interval ${\displaystyle P_{H}}$ is nonzero when restricted to ${\displaystyle \pi }$.
2. For all places ${\displaystyle v}$, the local Hom space ${\displaystyle \operatorname {Hom} _{H(F_{v})}(\pi _{v},\nu _{v})\neq 0}$ and ${\displaystyle L_{E}(1/2,\pi _{1}\times \pi _{2})\neq 0}$.

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.^{[7][8][9][10]} 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.^[11]

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.^[12]

Hongyu He proved the Gan-Gross-Prasad conjectures for discrete series representations of the real unitary group U(p,q).^[13]

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.^[14][15][16]

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.^[17]

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.^[18][19]

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.^[20]

References

1. Gan, Wee Teck; Gross, Benedict H.; Prasad, Dipendra (2012), "Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups", Astérisque, 346: 1–109, ISBN 978-2-85629-348-5, MR 3202556
2. Aizenbud, Avraham; Gourevitch, Dmitry; Rallis, Stephen; Schiffmann, Gérard (2010), "Multiplicity-one theorems", Annals of Mathematics, 172 (2): 1407–1434, arXiv:0709.4215, doi:10.4007/annals.2010.172.1413, MR 2680495
3. Sun, Binyong (2012), "Multiplicity-one theorems for Fourier–Jacobi models", American Journal of Mathematics, 134 (6): 1655–1678, arXiv:0903.1417, doi:10.1353/ajm.2012.0044
4. Sun, Binyong; Zhu, Chen-Bo (2012), "Multiplicity-one theorems: the Archimedean case", Annals of Mathematics, 175 (1): 23–44, arXiv:0903.1413, doi:10.4007/annals.2012.175.1.2, MR 2874638
5. Weyl, Hermann (1950), The Theory of Groups and Quantum Mechanics, Dover Publications
6. Gan, Wee Teck (2014), "Recent progress on the Gross-Prasad conjecture", Acta Mathematica Vietnamica, 39 (1): 11–33, doi:10.1007/s40306-014-0047-2, ISSN 2315-4144, S2CID 256378802
7. Waldspurger, Jean-Loup (2012), "Une Formule intégrale reliée à la conjecture locale de Gross-Prasad.", Compositio Mathematica, 146 (5): 1180–1290, arXiv:0902.1875, doi:10.1112/S0010437X10004744
8. Waldspurger, Jean-Loup (2012), "Une Formule intégrale reliée à la conjecture locale de Gross-Prasad, 2ème partie: extension aux représentations tempérées.", Astérisque, 347: 171–311
9. Waldspurger, Jean-Loup (2012), "La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes spéciaux orthogonaux.", Astérisque, 347: 103–166
10. Waldspurger, Jean-Loup (2012), "Calcul d'une valeur d'un facteur epsilon par une formule intégrale.", Astérisque, 347
11. Moeglin, Colette; Waldspurger, Jean-Loup (2012), "La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général", Astérisque, 347
12. Beuzart-Plessis, Raphaël (2012), "La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires", PhD Thesis
13. He, Hongyu (2017), "On the Gan-Gross-Prasad conjectures for U(p,q)", Inventiones Mathematicae, 209 (3): 837–884, arXiv:1508.02032, doi:10.1007/s00222-017-0720-x
14. Ginzburg, David; Jiang, Dihua; Rallis, Stephen (2004), "On the nonvanishing of the central value of the Rankin–Selberg L-functions.", Journal of the American Mathematical Society, 17 (3): 679–722, doi:10.1090/S0894-0347-04-00455-2
15. Ginzburg, David; Jiang, Dihua; Rallis, Stephen (2005), "On the nonvanishing of the central value of the Rankin–Selberg L-functions, II.", Automorphic Representations, L-functions and Applications: Progress and Prospects, Berlin: Ohio State Univ. Math. Res. Inst. Publ. 11, de Gruyter: 157–191, doi:10.1515/9783110892703.157, ISBN 978-3-11-017939-2
16. Ginzburg, David; Jiang, Dihua; Rallis, Stephen (2009), "Models for certain residual representations of unitary groups. Automorphic forms and L-functions I.", Global Aspects, Providence, RI: Contemp. Math., 488, Amer. Math. Soc.: 125–146
17. Zhang, Wei (2014), "Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups.", Annals of Mathematics, 180 (3): 971–1049, arXiv:0903.1413, doi:10.4007/annals.2012.175.1.2, MR 2874638
18. Liu, Yifeng (2014), "Relative trace formulae toward Bessel and Fourier–Jacobi periods of unitary groups.", Manuscripta Mathematica, 145 (1–2): 1–69, arXiv:1012.4538, doi:10.1007/s00229-014-0666-x
19. Xue, Hang (2014), "The Gan–Gross–Prasad conjecture for U(n) × U(n).", Advances in Mathematics, 262: 1130–1191, doi:10.1016/j.aim.2014.06.010, MR 3228451
20. Jiang, Dihua; Zhang, Lei (2020), "Arthur parameters and cuspidal automorphic modules of classical groups.", Annals of Mathematics, 191 (3): 739–827, arXiv:1508.03205, doi:10.4007/annals.2020.191.3.2