On Cuspidal Spectrum of Classical Groups - Cornell Math
On Cuspidal Spectrum of Classical Groups Dihua Jiang University of Minnesota
Simons Symposia on Geometric Aspects of the Trace Formula April 10-16, 2016
Square-Integrable Automorphic Forms I
G a reductive algebraic group defined over a number field F .
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A is the ring of adeles of F .
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XG := G(F )\G(A)1 , where G(A)1 := ∩χ∈X ∗ (G) ker |χ|A .
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L2 (XG ) denotes the space of functions: φ : XG → C such that Z |φ(g)|2 dg < ∞. G(Q)\G(A)1
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A2 (G) is the set of equivalence classes of irreducible unitary representations of G(A) occurring in the discrete spectrum L2disc (XG ).
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Acusp (G) is the subset of A2 (G) consisting of those automorphic representations of G(A) occurring in the cuspidal spectrum L2cusp (XG ).
Theory of Endoscopic Classification Theorem (Arthur, Mok, Kaletha-Minguez-Shin-White) Let G∗ be an F -quasisplit classical group and G be a pure inner form of G∗ over F . For any π ∈ Acusp (G), there is a global Arthur e 2 (G∗ ), which is G-relevant, such that parameter ψ ∈ Ψ e ψ (G) π∈Π e ψ (G) is the global Arthur packet of G associated to ψ. where Π I
We may form the global Arthur-Vogan packet as union of the e ψ (G) over all the pure inner forms G global Arthur packets Π of G∗ : e ψ [G∗ ] := ∪G Π e ψ (G). Π
e 2 (G): Examples Global Arthur Parameters Ψ I I
G∗ = SO∗2n+1 , F -split, and (G∗ )∨ = Sp2n (C). e 2 (G∗ ) (global Arthur parameters) is written as Each ψ ∈ Ψ a formal sum of simple Arthur parameters: ψ = ψ1 ψ2 · · · ψr
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where ψi = (τi , bi ), with τi ∈ Acusp (GLai ); ai , bi ≥ 1; and P r i=1 ai bi = 2n. If i 6= j, either τi ∼ 6 τj or bi 6= bj , with the parity condition = e 2 (SO∗ that ai · bi is even and ψi ∈ Ψ ). Pr ai bi +1 Endoscopy Structure: 2n = i=1 ai · bi , SO∗a1 ·b1 +1 × · · · e ψ (·) Π ⊗ ··· 1
× SO∗ar ·br +1 =⇒ SO∗2n+1 e ψ (·) e ψ (·) ⊗ Π =⇒ Π r
e 2 (G): Examples Global Arthur Parameters Ψ I I
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G∗ = Sp∗2n , F -split, and (G∗ )∨ = SO2n+1 (C). e 2 (G∗ ) is written as a formal sum of simple Arthur Each ψ ∈ Ψ parameters: ψ = ψ1 ψ2 · · · ψr where ψi = (τi , bi ), with τQ i ∈ Acusp (GLai ); ai , bi ≥ 1; P r r bi a b = 2n + 1; and i=1 i i i=1 ωτi = 1. If i 6= j, either τi ∼ 6 τj or bi 6= bj , with the parity: = 1 2
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e 2 (SO∗a b ); If ai · bi is even, then ψi ∈ Ψ i i e 2 (Sp∗a b −1 ). If ai · bi is is odd, then ψi ∈ Ψ i i
Endoscopy Structure: 2n + 1 =
Pr
i=1 ai
· bi ,
Q ∗ SO∗2li × =⇒ Sp∗2n aj bj =2lj +1 Sp2lj e ψ (·) ⊗ ⊗a b =2l +1 Π e ψ (·) =⇒ Π e ψ (·) ⊗ai bi =2li Π i j j j j Q
ai bi =2li
e 2 (G): Examples Global Arthur Parameters Ψ I
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e 2 (G∗ ) is generic if A parameter ψ = ψ1 ψ2 · · · ψr ∈ Ψ b1 = · · · = br = 1. e 2 (G∗ ) are: Generic global Arthur parameters φ ∈ Φ φ = (τ1 , 1) (τ2 , 1) · · · (τr , 1) with τi ∈ Acusp (GLai ) that τi ∼ 6 τj if i 6= j. They are of = either symplectic or orthogonal type, depending on G∗ .
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The pure inner forms of G∗ = SO∗m are G = SOm (V, q) for non-deg. quad. spaces (V, q) over F with the same dimension and discriminant.
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If G is a pure inner form of G∗ , then L G = L G∗ . e 2 (G∗ ), the endoscopic classification may define the For φ ∈ Φ e φ (G∗ ) and also define the global global Arthur packet Π e φ (G), which is non-empty if φ is G-relevant. Arthur packet Π
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Endoscopic Classification and Langlands Functoriality
A(GLNG ) πψ ↑ e 2 (G∗ )G Ψ ψ . e ψ (G) A2 (G) ∩ Π
& ⇐⇒
e ψ (G∗ ) ∩ A2 (G∗ ) Π
Problems Based on Endoscopic Classification
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e ψ (G) ∩ Acusp (G) = ∅? A Simple Question: Π e ψ (G) ∩ Acusp (G) 6= ∅, call ψ cuspidal. If Π
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What can one say about the cuspidal ψ?
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Write ψ = (τ1 , b1 ) · · · (τr , br ). How to bound these integers b1 , · · · , br if ψ is cuspidal?
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This leads to a Ramanujan type bound for Acusp (G).
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For π ∈ Acusp (G), how to determine which (τ, b) occurs in the global Arthur parameter ψ of π?
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This leads to the (τ, b)-theory that characterizes the (τ, b) factor of π in terms of basic invariants of π.
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If ψ is cuspidal, how to construct explicit modules for the e ψ (G) ∩ Acusp (G)? members in Π
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This leads to the theory of twisted automorphic descents and endoscopy correspondences via integral transforms.
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Remarks I
If G∗ is F -quasisplit, the automorphic descent of Ginzburg-Rallis-Soudry constructs a generic member in e φ (G∗ ) ∩ Acusp (G∗ ) for each generic global Arthur parameter Π ψ = φ.
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If G is a pure inner form of G∗ , all members in the set e φ (G) ∩ Acusp (G) can be constructed by using the twisted Π automorphic descent developed in my work with Lei Zhang, assuming the extended Arthur-Burger-Sarnak principle.
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A different approach is taken up also joint with Baiying Liu on this issue.
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The idea is to consider Fourier coefficients of automorphic representations associated to nilpotent orbits, which leads to the information on the automorphic wave-front set.
Fourier Coefficients and Nilpotent Adjoint Orbits I
G∗ is an F -quasi-split classical group and g∗ is the Lie algebra.
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Let NG∗ be the dimension for the defining embedding G∗ → GL(NG∗ ).
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Over algebraic closure F of F , all the nilpotent elements in g∗ (F ) form a conic algebraic variety, called the nilcone N (g∗ ).
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Under the adjoint action of G∗ , N (g∗ ) decomposes into finitely many adjoint G∗ -orbits O, which are parameterized by the corresponding partitions of N = NG∗ of type G∗ .
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Over F , each F -orbit reduces to an F -stable adjoint G∗ (F )-orbits Ost , and hence the F -stable adjoint orbits in the nilcone N (g∗ ) are also parameterized by the corresponding partitions of an integer N = NG∗ of type G∗ .
Fourier Coefficients and Nilpotent Adjoint Orbits I
For X ∈ N (g∗ ), use sl2 -triple (over F ) to define a unipotent subgroup VX and a character ψX .
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Let {X, H, Y } be an sl2 -triple (over F ). Under the adjoint action of ad(H), g∗ = g−r ⊕ · · · ⊕ g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ⊕ · · · ⊕ gr .
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Ad(G∗ )(Y ) ∩ g−2 and Ad(G∗ )(X) ∩ g2 are Zariski dense in g−2 and g2 , respectively.
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Take VX to be the unipotent subgroup of G∗ such that the Lie algebra of VX is equal to ⊕i≥2 gi .
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Let ψF be a non-trivial additive character of F \A. The character ψX of VX (F ) or VX (A) is defined by ψX (v) = ψF (tr(Y log(v))).
Fourier Coefficients and Nilpotent Adjoint Orbits I
The Fourier coefficient of ϕ ∈ π ∈ A2 (G∗ ) is defined by Z F ψX (ϕ)(g) := ϕ(vg)ψX (v)−1 dv. VX (F )\VX (A)
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Since ϕ is automorphic, the nonvanishing of F ψX (ϕ) depends only on the G∗ (F )-adjoint orbit OX of X.
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The set n(ϕ) := {X ∈ N (g) | F ψX (ϕ) 6= 0} is stable under the G∗ (F )-adjoint action.
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Denoted by p(ϕ) the set of partitions p of NG∗ of type G∗ corresponding to the F -stable orbits Opst that have non-empty intersection with n(ϕ).
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pm (ϕ) is the set of all maximal partitions in p(ϕ), according to the partial ordering of partitions.
Maximal Fourier Coefficients of Automorphic Forms I
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For π ∈ A2 (G), denote by pm (π) the set of maximal members among pm (ϕ) for all ϕ ∈ π. We would like to know: I I
How to determine pm (π) in terms of other invariants of π? What can one say about π based on the structure of pm (π)?
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Folklore Conjecture: For any irreducible automorphic representation π of G, the set pm (π) is singleton.
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Write p = [p1 p2 · · · pr ] ∈ pm (π) with p1 ≥ p2 ≥ · · · ≥ pr .
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What can we say about the largest part p1 if π is cuspidal?
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This problem can be formulated to have close relation with the extended Arthur-Burger-Sarnak principle, which is important to the theory of twisted automorphic descent.
Maximal Fourier Coefficients of Automorphic Forms I
Examples: G = GLn , the G(F )-stable orbits in N (g) are parameterized by partitions of n.
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Theorem (Piatetski-Shapiro; Shalika): If π ∈ A2 (GLn ) is cuspidal, pm (π) = {[n]}. This says that any irreducible cuspidal automorphic representation has a nonzero Whittaker-Fourier coefficient.
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What happens if π ∈ A2 (GLn ) is not cuspidal?
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Moeglin-Waldspurger Theorem: Any π ∈ Adisc (GLn ) has form ∆(τ, b) (Speh residue with cuspidal support τ ⊗b ), where τ ∈ Acusp (GLa ) and n = ab.
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If π = ∆(τ, b), then pm (π) = {[ab ]} (Ginzburg (2006), J.-Liu (2013) gives a complete global proof).
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In particular, the Folklore Conjecture is verified for all π ∈ A2 (GLn )!
Maximal Fourier Coefficients and Arthur Parameters I I I
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How to understand this in terms of Arthur parametrization? ∆(τ, b) has the Arthur parameter ψ = (τ, b). The partition attached to ψ is pψ := [ba ] and pm (∆(τ, b)) = {[ab ]}. η([ba ]) = [ab ] is given by the Barbasch-Vogan duality η from GL∨ n to GLn . In this case, it is just the transpose. Take an Arthur parameter for GLn : for τi ∈ Acusp (GLai ), ψ = (τ1 , b1 ) (τ2 , b2 ) · · · (τr , br ).
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The partition attached to ψ is pψ = [ba11 ba22 · · · bar r ].
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The Arthur representation is an isobaric sum πψ = ∆(τ1 , b1 ) ∆(τ2 , b2 ) · · · ∆(τr , br ).
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Conjecture: pm (πψ ) = {ηgl∨n ,gln (pψ )}.
Maximal Fourier Coefficients and Arthur Parameters I
e 2 (G∗ ), where ψi = (τi , bi ) For ψ = ψ1 ψ2 · · · ψr ∈ Ψ with τi ∈ Acusp (GLai ) and bi ≥ 1, pψ = [ba11 · · · bar r ] is the partition of N(G∗ )∨ attached to (ψ, (G∗ )∨ ) and η(pψ ) is the Barbasch-Vogan duality of pψ from (G∗ )∨ to G∗ .
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Conjecture (J.-2014): e ψ (G∗ ) ∩ A2 (G∗ ), any partition p ∈ pm (π) has (1) For every π ∈ Π the property that p ≤ η(pψ ). e ψ (G) ∩ A2 (G) for (2) There exists at least one member π ∈ Π ∗ some pure inner form G of G that have the property: η(pψ ) ∈ pm (π).
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Remark: For a pure inner form G of G∗ , assume that the global Arthur parameter ψ is G-relevant and the Barbasch-Vogan duality η(pψ ) is a G-relevant partition of NG = NG∗ of type G∗ . The definition of Fourier coefficients also work.
Examples of the Barbasch-Vogan duality I
G = SO2n+1 and 2n = ab; Take ψ = (τ, b) for τ ∈ Acusp (GLa ), and ( 2`, if τ is orthogonal, b= 2` + 1, if τ is symplectic.
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pψ = [ba ] is the partition of 2n of type (ψ, Sp2n (C)).
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The Barbasch-Vogan duality is given as follows: b−2 [(a + 1)a (a − 1)1] if b = 2l and a is even; η(pψ ) = [ab 1] if b = 2l and a is odd; b−1 [(a + 1)a ] if b = 2l + 1.
Examples of the Barbasch-Vogan duality I
e 2 (G). Take G = Sp2n and ψ = (τ, 2b + 1) ri=2 (τi , 1) ∈ Ψ
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pψ = [(2b + 1)a (1)2m+1−a ] with 2m + 1 = (2n + 1) − 2ab.
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When a ≤ 2m and a is even, η(pψ ) =η([(2b + 1)a (1)2m+1−a ]) = [(2b + 1)a (1)2m−a ]t =[(a)2b+1 ] + [(2m − a)] = [(2m)(a)2b ].
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When a ≤ 2m and a is odd, η(pψ ) =η([(2b + 1)a (1)2m+1−a ]) =([(2b + 1)a (1)2m−a ]Sp2n )t =[(2b + 1)a−1 (2b)(2)(1)2m−a−1 ]t =[(a − 1)2b+1 ] + [(1)2b ] + [(1)2 ] + [(2m − 1 − a)] =[(2m)(a + 1)(a)2b−2 (a − 1)].
Remarks on the Conjecture I
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It is true when G = GLn and ψ is an Arthur parameter for the discrete spectrum. e 2 (G∗ ) is generic, i.e. b1 = · · · = br = 1, the partition If φ ∈ Φ pφ = [1N(G∗ )∨ ].
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The Barbasch-Vogan duality of pφ is η([1N(G∗ )∨ ]) = [NG∗ ]G∗ .
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It is clear that the partition η([1N(G∗ )∨ ]) is G-relevant only if G = G∗ is quasi-split. In this case, it the regular partition.
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The conjecture claims that any generic global Arthur packet contains a generic member for quasi-split G∗ , and hence implies the global Shahidi conjecture on genericity of tempered packets.
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This special case can be proved by the Arthur-Langlands transfer from G to GLNG and the Ginzburg-Rallis-Soudry descent.
Remarks on the Conjecture I
The conjecture is known for various cases of Sp2n (J.-Liu).
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The conjecture provides an upper bound partition for e ψ (G) ∩ A2 (G), with a given Arthur parameter ψ. π∈Π
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We are to obtain a lower bound partition, with a given ψ, for e ψ (G) ∩ A2 (G). π∈Π
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It is very interesting, but harder problem to determine pm (π), e ψ (G) ∩ A2 (G)! with a given ψ, for general members π ∈ Π
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The theory of singular automorphic forms of Howe and Li provides a lower bound partition for π ∈ Acusp (G).
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It is not hard to find that the lower bound partition provided by the theory of Howe and Li may not be the best for all π ∈ Acusp (G).
Singular Partitions and Rank in the Sense of Howe I
G = Sp2n . Take Pn = Mn Un to be the Siegel parabolic of Sp2n , with Mn ∼ = GLn and the elements of Un are of form In X u(X) = . 0 In
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By the Pontryagin duality, one has Un (F\ )\Un (A) ∼ = Sym2 (F n ).
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For a fixed nontrivial additive character ψF of F \A, a T ∈ Sym2 (F n ) corresponds to the character ψT by ψT (u(X)) := ψF (tr(T wX)), with w anti-diagonal, and the action of GLn on Sym2 (F n ) is induced from its adjoint action on Un .
Singular Partitions and Rank in the Sense of Howe I
For an automorphic form ϕ on Sp2n (A), the T - or ψT -Fourier coefficient is defined by Z F ψT (ϕ)(g) := ϕ(u(X)g)ψT−1 (u(X))du(X). Um (F )\Un (A)
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An automorphic form ϕ on Sp2n (A) is called non-singular if ϕ has a nonzero ψT -Fourier coefficient with a non-singular T .
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ϕ is called singular if ϕ has the property that if a nonzero ψT -Fourier coefficient F ψT (ϕ) is nonzero, then det(T ) = 0.
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Howe shows in 1981 that if an automorphic form ϕ on Sp2n (A) is singular, then ϕ can be expressed as a linear combination of certain theta functions.
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Jianshu Li shows in 1989 that any cuspidal automorphic form of Sp2n (A) is non-singular.
Singular Partitions and Rank in the Sense of Howe I
For a split SOm defined by a non-deg. quad. space (V, q) over + F of dim m with the Witt index [ m 2 ], let X be a maximal − totally isotropic subspace of V with dim [ m 2 ] and let X be + the dual to X : V = X − + V0 + X + with V0 the orth. complement of X − + X + (dim V0 ≤ 1).
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The generalized flag {0} ⊂ X + ⊂ V defines a maximal parabolic subgroup PX + , whose Levi part MX + is isomorphic to GL[ m2 ] and whose unipotent radical NX + is abelian if m is even; and is a two-step unipotent subgroup with its center ZX + if m is odd. When m is even, we set ZX + = NX + .
Singular Partitions and Rank in the Sense of Howe I
m )\ZX + (A) ∼ By the Pontryagin duality, ZX + (F\ = ∧2 (F [ 2 ] ).
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For any T ∈ ∧2 (F [ 2 ] ),
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ψT (z(X)) := ψF (tr(T wX)). I
For an automorphic form ϕ on G(A), the T or ψT Fourier coefficient is defined by Z ψT F (ϕ)(g) := ϕ(z(X)g)ψT−1 (z(X))dz(X). ZX + (F )\ZX + (A)
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An automorphic form ϕ on G(A) is called non-singular if ϕ m has a non-zero ψT -Fourier coefficient with T ∈ ∧2 (F [ 2 ] ) of maximal rank.
Singular Partitions and Rank in the Sense of Howe I
Denote by pns the partition corresponding to the non-singular Fourier coefficients.
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For Sp2n , pns = [2n ], which is a special partition for Sp2n .
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For SO2n+1 , one has pns
( [22e 1] = [22e 13 ]
if n = 2e; if n = 2e + 1.
This is not a special partition of SO2n+1 . I
For SO2n , one has pns
( [22e ] = [22e 12 ]
if n = 2e; if n = 2e + 1.
This is a special partition of SO2n .
Singular Partitions and Lower Bound Partitions I
J.-Liu-Savin show 2015 that for any automorphic representation π, the set pm (π) contains only special partitions.
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For SO2n+1 , pns 6∈ pm (π) for any π ∈ Acusp (SO2n+1 ).
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Any p ∈ pm (π) as π runs in Acusp (SO2n+1 ) must be bigger than or equal to the following partition ( [322e−2 12 ] if n = 2e; SO2n+1 pns = [322e−2 14 ] if n = 2e + 1.
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Gn denotes the G -expansion of the partition p , i.e., the pns n ns smallest special partition which is bigger than or equal to pns .
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n is a lower Proposition: For a split classical group Gn , pG ns m bound for partitions in the set p (π) as π runs in Acusp (Gn ).
Singular Partitions and Small Cuspidal Representations I
n is sharp? It is natural to ask whether the lower bound pG ns
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For n = 2e even, and F to be totally real, the Ikeda lifting gives π ∈ Acusp (Sp4e ) with the global Arthur parameter (τ, 2e) (1, 1), where τ ∈ Acusp (GL2 ) is self-dual and has the trivial central character.
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4e . First, for any p ∈ pm (π), one has p ≤ [22e ] = pSp ns
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4e = [22e ] ≤ p. Then, one also has pSp ns
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Proposition: If F is totally real, the Ikeda construction π of 4e } = {[22e ]}, and hence the Sp4e has pm (π) = {pSp ns Sp4e = p = [22e ] is the sharp lower non-singular partition pns ns bound for Acusp (Sp4e ).
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Note that the construction of Ikeda is not known when F is not totally real or n = 2e + 1 is odd.
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What happens in such situations?
Singular Partitions and Small Cuspidal Representations I
J.-Liu (2015): If F is totally imaginary and n ≥ 5, the global e (τ,n)(,1) (Sp2n ), with τ ∈ Acusp (GL2 ) Arthur packet Π self-dual, has no cuspidal members, where = 1 if n = 2e; and = ωτ if n = 2e + 1.
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In this case, the construction of the Ikeda lifting is impossible! Theorem (J.-Liu 2015):
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For any π ∈ Acusp (Sp2n ), pm (π) = [2n ] if and only if π is hypercuspidal in the sense of I. Piatetski-Shapiro. e 2 (Sp4e ) with τ ∈ Acusp (GL2 ), any For ψ = (τ, 2e) (1, 1) ∈ Ψ e cuspidal π ∈ Πψ (Sp4e ) has that pm (π) = [22e ]. e 2 (Sp4e+2 ) with For ψ = (τ, 2e + 1) (ωτ , 1) ∈ Ψ e ψ (Sp4e+2 ) has that τ ∈ Acusp (GL2 ), any cuspidal π ∈ Π m 2e+1 p (π) = [2 ]. e 2 (Sp6e+2 ) with τ ∈ Acusp (GL3 ), any For ψ = (τ, 2e + 1) ∈ Ψ e ψ (Sp6e+2 ) has that pm (π) = [23e+1 ]. cuspidal π ∈ Π
Cuspidality of Global Arthur Packets Theorem (J.-Liu 2015) Assume that F is totally imaginary, and Part (1) of the Conjecture e 2 (Sp2n ) with τi ∈ Acusp (GLa ) for holds. For ψ = ri=1 (τi , biP ) in Ψ i Q r i = 1, 2, · · · , r, 2n + 1 = i=1 ai bi , and ri=1 ωτbii = 1, there exist (1)
(2)
constants Na ≥ Na,b ≥ Na,b , depending on (a1 , · · · , ar ) and (b1 , · · · , br ), such that if 2n > N0 for N0 to be one of the (1) (2) e ψ (Sp2n ) ∩ Acusp (Sp2n ) is empty. Na , Na,b , Na,b , the set Π Remarks: ( P P Pr ( ri=1 ai )2 + 2( ri=1 ai ) if i=1 ai is even; 1 N = Pr a 2 ( i=1 ai ) − 1 otherwise. (1)
(2)
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Na,b , Na,b are sharper bound, but not easy to be defined.
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Conjecture: Na,b is sharp in the sense that if 2n = Na,b , e ψ (Sp2n ) ∩ Acusp (Sp2n ) is not empty then the set Π
(2)
(2)
Cuspidality of Global Arthur Packets I
Example (1): Take ψ = (τ1 , 1) (τ2 , 8) with τ1 ∈ Acusp (GL5 ) and τ2 ∈ Acusp (GL2 ). We have (1)
(2)
Na = 48 > Na,b = 24 > Na,b = 16. I
Example (2): Take ψ = (1, b1 ) (τ, b2 ) with b1 ≥ 1 odd, τ ∈ Acusp (GL2 ) of symplectic type and b2 even. Then (1)
(2)
Na = Na,b = Na,b = 8 e ψ (Sp∗) contains no cuspidal and the global Arthur packet Π members except that (b1 , b2 ) = (1, 2), (1, 4), (3, 2), or (5, 2). I
Mœglin (2008, 2011) also provide criterion for the cuspidality of global Arthur packets, which is in a different nature from what we developed here.
(τ, b)-theory and the Ramanujan Type Bound
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This is to bound b for (τ, b) to occur in π ∈ Acusp (Sp2n ).
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For π ∈ A2 (Sp2n ), if (τ, b) occurs in π, then b ≤ 2n (sharp!).
Theorem (Kudla-Rallis (1994)) For τ = χ with χ2 = 1, and π ∈ Acusp (Sp2n ), if (χ, b) occurs in π, then b ≤ 2[ n2 ] + 1. I
This bound can also be deduced from my Conjecture.
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If (τ, b) occurs in π ∈ Acusp (Sp2n ), then b ≤ 2[ n2 ] + 1.
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Question: Is 2[ n2 ] + 1 sharp upper bound for b?
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When n is even, Piatetski-Shapiro and Rallis (1987) constructed a π(χ,n+1) ∈ Acusp (Sp2n ) with a global Arthur parameter (χ, n + 1), and hence this bound 2[ n2 ] + 1 is sharp!
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Question: When n is odd, how to construct a π(χ,n) in Acusp (Sp2n ) with a global Arthur parameter (χ, n)?
(τ, b)-theory and the Ramanujan Type Bound I
For π ∈ Acusp (Sp2n ), one reads the Satake parameters of π at a unramified local place v from Sp
(Fv )
2n IndB(F v)
χ1 |·|α1 ⊗ χ2 |·|α2 ⊗ · · · ⊗ χn |·|αn ,
where B is the standard Borel of Sp2n , and χi s are unitary.
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For θ ∈ R≥0 , we say that π has R(θ) if at each unramified local place v, one has 0 ≤ αi ≤ θ for all i.
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When n is even, π(χ,n+1) has R( n2 ).
Theorem For any number field F , if n is even, every π ∈ Acusp (Sp2n ) has R( n2 ), which is sharp. I
The proof uses the Ramanujan bound for τ ∈ Acusp (GL2 ) given by Kim-Sarnak (2003) and by Blomer-Brumley (2011).
(τ, b)-theory and the Ramanujan Type Bound I
Theorem For any number field F , if n is odd, every π ∈ Acusp (Sp2n ) has 7 R( 64 + n−1 2 ). I
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The proof is the same. In this case, it is a hard problem to find a sharp bound even assuming the Ramanujan conjecture.
Theorem (J.-Liu (2015)) If F is totally imaginary, and if n ≥ 5 is odd, then every π in Acusp (Sp2n ) has R( n−1 2 ). I
It is not known if the bound is sharp.
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One expects that the above discussion should be extended to other classical groups.
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My work in progress joint with Lei Zhang and Baiying Liu is to figure out the small cuspidal members in the global Arthur e ψ (G) for general global Arthur parameters ψ. packets Π
Simons Symposia on Geometric Aspects of the Trace Formula April 10-16, 2016
Square-Integrable Automorphic Forms I
G a reductive algebraic group defined over a number field F .
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A is the ring of adeles of F .
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XG := G(F )\G(A)1 , where G(A)1 := ∩χ∈X ∗ (G) ker |χ|A .
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L2 (XG ) denotes the space of functions: φ : XG → C such that Z |φ(g)|2 dg < ∞. G(Q)\G(A)1
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A2 (G) is the set of equivalence classes of irreducible unitary representations of G(A) occurring in the discrete spectrum L2disc (XG ).
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Acusp (G) is the subset of A2 (G) consisting of those automorphic representations of G(A) occurring in the cuspidal spectrum L2cusp (XG ).
Theory of Endoscopic Classification Theorem (Arthur, Mok, Kaletha-Minguez-Shin-White) Let G∗ be an F -quasisplit classical group and G be a pure inner form of G∗ over F . For any π ∈ Acusp (G), there is a global Arthur e 2 (G∗ ), which is G-relevant, such that parameter ψ ∈ Ψ e ψ (G) π∈Π e ψ (G) is the global Arthur packet of G associated to ψ. where Π I
We may form the global Arthur-Vogan packet as union of the e ψ (G) over all the pure inner forms G global Arthur packets Π of G∗ : e ψ [G∗ ] := ∪G Π e ψ (G). Π
e 2 (G): Examples Global Arthur Parameters Ψ I I
G∗ = SO∗2n+1 , F -split, and (G∗ )∨ = Sp2n (C). e 2 (G∗ ) (global Arthur parameters) is written as Each ψ ∈ Ψ a formal sum of simple Arthur parameters: ψ = ψ1 ψ2 · · · ψr
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where ψi = (τi , bi ), with τi ∈ Acusp (GLai ); ai , bi ≥ 1; and P r i=1 ai bi = 2n. If i 6= j, either τi ∼ 6 τj or bi 6= bj , with the parity condition = e 2 (SO∗ that ai · bi is even and ψi ∈ Ψ ). Pr ai bi +1 Endoscopy Structure: 2n = i=1 ai · bi , SO∗a1 ·b1 +1 × · · · e ψ (·) Π ⊗ ··· 1
× SO∗ar ·br +1 =⇒ SO∗2n+1 e ψ (·) e ψ (·) ⊗ Π =⇒ Π r
e 2 (G): Examples Global Arthur Parameters Ψ I I
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G∗ = Sp∗2n , F -split, and (G∗ )∨ = SO2n+1 (C). e 2 (G∗ ) is written as a formal sum of simple Arthur Each ψ ∈ Ψ parameters: ψ = ψ1 ψ2 · · · ψr where ψi = (τi , bi ), with τQ i ∈ Acusp (GLai ); ai , bi ≥ 1; P r r bi a b = 2n + 1; and i=1 i i i=1 ωτi = 1. If i 6= j, either τi ∼ 6 τj or bi 6= bj , with the parity: = 1 2
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e 2 (SO∗a b ); If ai · bi is even, then ψi ∈ Ψ i i e 2 (Sp∗a b −1 ). If ai · bi is is odd, then ψi ∈ Ψ i i
Endoscopy Structure: 2n + 1 =
Pr
i=1 ai
· bi ,
Q ∗ SO∗2li × =⇒ Sp∗2n aj bj =2lj +1 Sp2lj e ψ (·) ⊗ ⊗a b =2l +1 Π e ψ (·) =⇒ Π e ψ (·) ⊗ai bi =2li Π i j j j j Q
ai bi =2li
e 2 (G): Examples Global Arthur Parameters Ψ I
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e 2 (G∗ ) is generic if A parameter ψ = ψ1 ψ2 · · · ψr ∈ Ψ b1 = · · · = br = 1. e 2 (G∗ ) are: Generic global Arthur parameters φ ∈ Φ φ = (τ1 , 1) (τ2 , 1) · · · (τr , 1) with τi ∈ Acusp (GLai ) that τi ∼ 6 τj if i 6= j. They are of = either symplectic or orthogonal type, depending on G∗ .
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The pure inner forms of G∗ = SO∗m are G = SOm (V, q) for non-deg. quad. spaces (V, q) over F with the same dimension and discriminant.
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If G is a pure inner form of G∗ , then L G = L G∗ . e 2 (G∗ ), the endoscopic classification may define the For φ ∈ Φ e φ (G∗ ) and also define the global global Arthur packet Π e φ (G), which is non-empty if φ is G-relevant. Arthur packet Π
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Endoscopic Classification and Langlands Functoriality
A(GLNG ) πψ ↑ e 2 (G∗ )G Ψ ψ . e ψ (G) A2 (G) ∩ Π
& ⇐⇒
e ψ (G∗ ) ∩ A2 (G∗ ) Π
Problems Based on Endoscopic Classification
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e ψ (G) ∩ Acusp (G) = ∅? A Simple Question: Π e ψ (G) ∩ Acusp (G) 6= ∅, call ψ cuspidal. If Π
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What can one say about the cuspidal ψ?
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Write ψ = (τ1 , b1 ) · · · (τr , br ). How to bound these integers b1 , · · · , br if ψ is cuspidal?
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This leads to a Ramanujan type bound for Acusp (G).
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For π ∈ Acusp (G), how to determine which (τ, b) occurs in the global Arthur parameter ψ of π?
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This leads to the (τ, b)-theory that characterizes the (τ, b) factor of π in terms of basic invariants of π.
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If ψ is cuspidal, how to construct explicit modules for the e ψ (G) ∩ Acusp (G)? members in Π
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This leads to the theory of twisted automorphic descents and endoscopy correspondences via integral transforms.
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Remarks I
If G∗ is F -quasisplit, the automorphic descent of Ginzburg-Rallis-Soudry constructs a generic member in e φ (G∗ ) ∩ Acusp (G∗ ) for each generic global Arthur parameter Π ψ = φ.
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If G is a pure inner form of G∗ , all members in the set e φ (G) ∩ Acusp (G) can be constructed by using the twisted Π automorphic descent developed in my work with Lei Zhang, assuming the extended Arthur-Burger-Sarnak principle.
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A different approach is taken up also joint with Baiying Liu on this issue.
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The idea is to consider Fourier coefficients of automorphic representations associated to nilpotent orbits, which leads to the information on the automorphic wave-front set.
Fourier Coefficients and Nilpotent Adjoint Orbits I
G∗ is an F -quasi-split classical group and g∗ is the Lie algebra.
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Let NG∗ be the dimension for the defining embedding G∗ → GL(NG∗ ).
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Over algebraic closure F of F , all the nilpotent elements in g∗ (F ) form a conic algebraic variety, called the nilcone N (g∗ ).
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Under the adjoint action of G∗ , N (g∗ ) decomposes into finitely many adjoint G∗ -orbits O, which are parameterized by the corresponding partitions of N = NG∗ of type G∗ .
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Over F , each F -orbit reduces to an F -stable adjoint G∗ (F )-orbits Ost , and hence the F -stable adjoint orbits in the nilcone N (g∗ ) are also parameterized by the corresponding partitions of an integer N = NG∗ of type G∗ .
Fourier Coefficients and Nilpotent Adjoint Orbits I
For X ∈ N (g∗ ), use sl2 -triple (over F ) to define a unipotent subgroup VX and a character ψX .
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Let {X, H, Y } be an sl2 -triple (over F ). Under the adjoint action of ad(H), g∗ = g−r ⊕ · · · ⊕ g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ⊕ · · · ⊕ gr .
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Ad(G∗ )(Y ) ∩ g−2 and Ad(G∗ )(X) ∩ g2 are Zariski dense in g−2 and g2 , respectively.
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Take VX to be the unipotent subgroup of G∗ such that the Lie algebra of VX is equal to ⊕i≥2 gi .
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Let ψF be a non-trivial additive character of F \A. The character ψX of VX (F ) or VX (A) is defined by ψX (v) = ψF (tr(Y log(v))).
Fourier Coefficients and Nilpotent Adjoint Orbits I
The Fourier coefficient of ϕ ∈ π ∈ A2 (G∗ ) is defined by Z F ψX (ϕ)(g) := ϕ(vg)ψX (v)−1 dv. VX (F )\VX (A)
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Since ϕ is automorphic, the nonvanishing of F ψX (ϕ) depends only on the G∗ (F )-adjoint orbit OX of X.
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The set n(ϕ) := {X ∈ N (g) | F ψX (ϕ) 6= 0} is stable under the G∗ (F )-adjoint action.
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Denoted by p(ϕ) the set of partitions p of NG∗ of type G∗ corresponding to the F -stable orbits Opst that have non-empty intersection with n(ϕ).
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pm (ϕ) is the set of all maximal partitions in p(ϕ), according to the partial ordering of partitions.
Maximal Fourier Coefficients of Automorphic Forms I
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For π ∈ A2 (G), denote by pm (π) the set of maximal members among pm (ϕ) for all ϕ ∈ π. We would like to know: I I
How to determine pm (π) in terms of other invariants of π? What can one say about π based on the structure of pm (π)?
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Folklore Conjecture: For any irreducible automorphic representation π of G, the set pm (π) is singleton.
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Write p = [p1 p2 · · · pr ] ∈ pm (π) with p1 ≥ p2 ≥ · · · ≥ pr .
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What can we say about the largest part p1 if π is cuspidal?
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This problem can be formulated to have close relation with the extended Arthur-Burger-Sarnak principle, which is important to the theory of twisted automorphic descent.
Maximal Fourier Coefficients of Automorphic Forms I
Examples: G = GLn , the G(F )-stable orbits in N (g) are parameterized by partitions of n.
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Theorem (Piatetski-Shapiro; Shalika): If π ∈ A2 (GLn ) is cuspidal, pm (π) = {[n]}. This says that any irreducible cuspidal automorphic representation has a nonzero Whittaker-Fourier coefficient.
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What happens if π ∈ A2 (GLn ) is not cuspidal?
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Moeglin-Waldspurger Theorem: Any π ∈ Adisc (GLn ) has form ∆(τ, b) (Speh residue with cuspidal support τ ⊗b ), where τ ∈ Acusp (GLa ) and n = ab.
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If π = ∆(τ, b), then pm (π) = {[ab ]} (Ginzburg (2006), J.-Liu (2013) gives a complete global proof).
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In particular, the Folklore Conjecture is verified for all π ∈ A2 (GLn )!
Maximal Fourier Coefficients and Arthur Parameters I I I
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How to understand this in terms of Arthur parametrization? ∆(τ, b) has the Arthur parameter ψ = (τ, b). The partition attached to ψ is pψ := [ba ] and pm (∆(τ, b)) = {[ab ]}. η([ba ]) = [ab ] is given by the Barbasch-Vogan duality η from GL∨ n to GLn . In this case, it is just the transpose. Take an Arthur parameter for GLn : for τi ∈ Acusp (GLai ), ψ = (τ1 , b1 ) (τ2 , b2 ) · · · (τr , br ).
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The partition attached to ψ is pψ = [ba11 ba22 · · · bar r ].
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The Arthur representation is an isobaric sum πψ = ∆(τ1 , b1 ) ∆(τ2 , b2 ) · · · ∆(τr , br ).
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Conjecture: pm (πψ ) = {ηgl∨n ,gln (pψ )}.
Maximal Fourier Coefficients and Arthur Parameters I
e 2 (G∗ ), where ψi = (τi , bi ) For ψ = ψ1 ψ2 · · · ψr ∈ Ψ with τi ∈ Acusp (GLai ) and bi ≥ 1, pψ = [ba11 · · · bar r ] is the partition of N(G∗ )∨ attached to (ψ, (G∗ )∨ ) and η(pψ ) is the Barbasch-Vogan duality of pψ from (G∗ )∨ to G∗ .
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Conjecture (J.-2014): e ψ (G∗ ) ∩ A2 (G∗ ), any partition p ∈ pm (π) has (1) For every π ∈ Π the property that p ≤ η(pψ ). e ψ (G) ∩ A2 (G) for (2) There exists at least one member π ∈ Π ∗ some pure inner form G of G that have the property: η(pψ ) ∈ pm (π).
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Remark: For a pure inner form G of G∗ , assume that the global Arthur parameter ψ is G-relevant and the Barbasch-Vogan duality η(pψ ) is a G-relevant partition of NG = NG∗ of type G∗ . The definition of Fourier coefficients also work.
Examples of the Barbasch-Vogan duality I
G = SO2n+1 and 2n = ab; Take ψ = (τ, b) for τ ∈ Acusp (GLa ), and ( 2`, if τ is orthogonal, b= 2` + 1, if τ is symplectic.
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pψ = [ba ] is the partition of 2n of type (ψ, Sp2n (C)).
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The Barbasch-Vogan duality is given as follows: b−2 [(a + 1)a (a − 1)1] if b = 2l and a is even; η(pψ ) = [ab 1] if b = 2l and a is odd; b−1 [(a + 1)a ] if b = 2l + 1.
Examples of the Barbasch-Vogan duality I
e 2 (G). Take G = Sp2n and ψ = (τ, 2b + 1) ri=2 (τi , 1) ∈ Ψ
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pψ = [(2b + 1)a (1)2m+1−a ] with 2m + 1 = (2n + 1) − 2ab.
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When a ≤ 2m and a is even, η(pψ ) =η([(2b + 1)a (1)2m+1−a ]) = [(2b + 1)a (1)2m−a ]t =[(a)2b+1 ] + [(2m − a)] = [(2m)(a)2b ].
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When a ≤ 2m and a is odd, η(pψ ) =η([(2b + 1)a (1)2m+1−a ]) =([(2b + 1)a (1)2m−a ]Sp2n )t =[(2b + 1)a−1 (2b)(2)(1)2m−a−1 ]t =[(a − 1)2b+1 ] + [(1)2b ] + [(1)2 ] + [(2m − 1 − a)] =[(2m)(a + 1)(a)2b−2 (a − 1)].
Remarks on the Conjecture I
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It is true when G = GLn and ψ is an Arthur parameter for the discrete spectrum. e 2 (G∗ ) is generic, i.e. b1 = · · · = br = 1, the partition If φ ∈ Φ pφ = [1N(G∗ )∨ ].
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The Barbasch-Vogan duality of pφ is η([1N(G∗ )∨ ]) = [NG∗ ]G∗ .
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It is clear that the partition η([1N(G∗ )∨ ]) is G-relevant only if G = G∗ is quasi-split. In this case, it the regular partition.
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The conjecture claims that any generic global Arthur packet contains a generic member for quasi-split G∗ , and hence implies the global Shahidi conjecture on genericity of tempered packets.
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This special case can be proved by the Arthur-Langlands transfer from G to GLNG and the Ginzburg-Rallis-Soudry descent.
Remarks on the Conjecture I
The conjecture is known for various cases of Sp2n (J.-Liu).
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The conjecture provides an upper bound partition for e ψ (G) ∩ A2 (G), with a given Arthur parameter ψ. π∈Π
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We are to obtain a lower bound partition, with a given ψ, for e ψ (G) ∩ A2 (G). π∈Π
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It is very interesting, but harder problem to determine pm (π), e ψ (G) ∩ A2 (G)! with a given ψ, for general members π ∈ Π
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The theory of singular automorphic forms of Howe and Li provides a lower bound partition for π ∈ Acusp (G).
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It is not hard to find that the lower bound partition provided by the theory of Howe and Li may not be the best for all π ∈ Acusp (G).
Singular Partitions and Rank in the Sense of Howe I
G = Sp2n . Take Pn = Mn Un to be the Siegel parabolic of Sp2n , with Mn ∼ = GLn and the elements of Un are of form In X u(X) = . 0 In
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By the Pontryagin duality, one has Un (F\ )\Un (A) ∼ = Sym2 (F n ).
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For a fixed nontrivial additive character ψF of F \A, a T ∈ Sym2 (F n ) corresponds to the character ψT by ψT (u(X)) := ψF (tr(T wX)), with w anti-diagonal, and the action of GLn on Sym2 (F n ) is induced from its adjoint action on Un .
Singular Partitions and Rank in the Sense of Howe I
For an automorphic form ϕ on Sp2n (A), the T - or ψT -Fourier coefficient is defined by Z F ψT (ϕ)(g) := ϕ(u(X)g)ψT−1 (u(X))du(X). Um (F )\Un (A)
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An automorphic form ϕ on Sp2n (A) is called non-singular if ϕ has a nonzero ψT -Fourier coefficient with a non-singular T .
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ϕ is called singular if ϕ has the property that if a nonzero ψT -Fourier coefficient F ψT (ϕ) is nonzero, then det(T ) = 0.
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Howe shows in 1981 that if an automorphic form ϕ on Sp2n (A) is singular, then ϕ can be expressed as a linear combination of certain theta functions.
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Jianshu Li shows in 1989 that any cuspidal automorphic form of Sp2n (A) is non-singular.
Singular Partitions and Rank in the Sense of Howe I
For a split SOm defined by a non-deg. quad. space (V, q) over + F of dim m with the Witt index [ m 2 ], let X be a maximal − totally isotropic subspace of V with dim [ m 2 ] and let X be + the dual to X : V = X − + V0 + X + with V0 the orth. complement of X − + X + (dim V0 ≤ 1).
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The generalized flag {0} ⊂ X + ⊂ V defines a maximal parabolic subgroup PX + , whose Levi part MX + is isomorphic to GL[ m2 ] and whose unipotent radical NX + is abelian if m is even; and is a two-step unipotent subgroup with its center ZX + if m is odd. When m is even, we set ZX + = NX + .
Singular Partitions and Rank in the Sense of Howe I
m )\ZX + (A) ∼ By the Pontryagin duality, ZX + (F\ = ∧2 (F [ 2 ] ).
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For any T ∈ ∧2 (F [ 2 ] ),
m
ψT (z(X)) := ψF (tr(T wX)). I
For an automorphic form ϕ on G(A), the T or ψT Fourier coefficient is defined by Z ψT F (ϕ)(g) := ϕ(z(X)g)ψT−1 (z(X))dz(X). ZX + (F )\ZX + (A)
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An automorphic form ϕ on G(A) is called non-singular if ϕ m has a non-zero ψT -Fourier coefficient with T ∈ ∧2 (F [ 2 ] ) of maximal rank.
Singular Partitions and Rank in the Sense of Howe I
Denote by pns the partition corresponding to the non-singular Fourier coefficients.
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For Sp2n , pns = [2n ], which is a special partition for Sp2n .
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For SO2n+1 , one has pns
( [22e 1] = [22e 13 ]
if n = 2e; if n = 2e + 1.
This is not a special partition of SO2n+1 . I
For SO2n , one has pns
( [22e ] = [22e 12 ]
if n = 2e; if n = 2e + 1.
This is a special partition of SO2n .
Singular Partitions and Lower Bound Partitions I
J.-Liu-Savin show 2015 that for any automorphic representation π, the set pm (π) contains only special partitions.
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For SO2n+1 , pns 6∈ pm (π) for any π ∈ Acusp (SO2n+1 ).
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Any p ∈ pm (π) as π runs in Acusp (SO2n+1 ) must be bigger than or equal to the following partition ( [322e−2 12 ] if n = 2e; SO2n+1 pns = [322e−2 14 ] if n = 2e + 1.
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Gn denotes the G -expansion of the partition p , i.e., the pns n ns smallest special partition which is bigger than or equal to pns .
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n is a lower Proposition: For a split classical group Gn , pG ns m bound for partitions in the set p (π) as π runs in Acusp (Gn ).
Singular Partitions and Small Cuspidal Representations I
n is sharp? It is natural to ask whether the lower bound pG ns
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For n = 2e even, and F to be totally real, the Ikeda lifting gives π ∈ Acusp (Sp4e ) with the global Arthur parameter (τ, 2e) (1, 1), where τ ∈ Acusp (GL2 ) is self-dual and has the trivial central character.
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4e . First, for any p ∈ pm (π), one has p ≤ [22e ] = pSp ns
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4e = [22e ] ≤ p. Then, one also has pSp ns
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Proposition: If F is totally real, the Ikeda construction π of 4e } = {[22e ]}, and hence the Sp4e has pm (π) = {pSp ns Sp4e = p = [22e ] is the sharp lower non-singular partition pns ns bound for Acusp (Sp4e ).
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Note that the construction of Ikeda is not known when F is not totally real or n = 2e + 1 is odd.
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What happens in such situations?
Singular Partitions and Small Cuspidal Representations I
J.-Liu (2015): If F is totally imaginary and n ≥ 5, the global e (τ,n)(,1) (Sp2n ), with τ ∈ Acusp (GL2 ) Arthur packet Π self-dual, has no cuspidal members, where = 1 if n = 2e; and = ωτ if n = 2e + 1.
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In this case, the construction of the Ikeda lifting is impossible! Theorem (J.-Liu 2015):
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For any π ∈ Acusp (Sp2n ), pm (π) = [2n ] if and only if π is hypercuspidal in the sense of I. Piatetski-Shapiro. e 2 (Sp4e ) with τ ∈ Acusp (GL2 ), any For ψ = (τ, 2e) (1, 1) ∈ Ψ e cuspidal π ∈ Πψ (Sp4e ) has that pm (π) = [22e ]. e 2 (Sp4e+2 ) with For ψ = (τ, 2e + 1) (ωτ , 1) ∈ Ψ e ψ (Sp4e+2 ) has that τ ∈ Acusp (GL2 ), any cuspidal π ∈ Π m 2e+1 p (π) = [2 ]. e 2 (Sp6e+2 ) with τ ∈ Acusp (GL3 ), any For ψ = (τ, 2e + 1) ∈ Ψ e ψ (Sp6e+2 ) has that pm (π) = [23e+1 ]. cuspidal π ∈ Π
Cuspidality of Global Arthur Packets Theorem (J.-Liu 2015) Assume that F is totally imaginary, and Part (1) of the Conjecture e 2 (Sp2n ) with τi ∈ Acusp (GLa ) for holds. For ψ = ri=1 (τi , biP ) in Ψ i Q r i = 1, 2, · · · , r, 2n + 1 = i=1 ai bi , and ri=1 ωτbii = 1, there exist (1)
(2)
constants Na ≥ Na,b ≥ Na,b , depending on (a1 , · · · , ar ) and (b1 , · · · , br ), such that if 2n > N0 for N0 to be one of the (1) (2) e ψ (Sp2n ) ∩ Acusp (Sp2n ) is empty. Na , Na,b , Na,b , the set Π Remarks: ( P P Pr ( ri=1 ai )2 + 2( ri=1 ai ) if i=1 ai is even; 1 N = Pr a 2 ( i=1 ai ) − 1 otherwise. (1)
(2)
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Na,b , Na,b are sharper bound, but not easy to be defined.
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Conjecture: Na,b is sharp in the sense that if 2n = Na,b , e ψ (Sp2n ) ∩ Acusp (Sp2n ) is not empty then the set Π
(2)
(2)
Cuspidality of Global Arthur Packets I
Example (1): Take ψ = (τ1 , 1) (τ2 , 8) with τ1 ∈ Acusp (GL5 ) and τ2 ∈ Acusp (GL2 ). We have (1)
(2)
Na = 48 > Na,b = 24 > Na,b = 16. I
Example (2): Take ψ = (1, b1 ) (τ, b2 ) with b1 ≥ 1 odd, τ ∈ Acusp (GL2 ) of symplectic type and b2 even. Then (1)
(2)
Na = Na,b = Na,b = 8 e ψ (Sp∗) contains no cuspidal and the global Arthur packet Π members except that (b1 , b2 ) = (1, 2), (1, 4), (3, 2), or (5, 2). I
Mœglin (2008, 2011) also provide criterion for the cuspidality of global Arthur packets, which is in a different nature from what we developed here.
(τ, b)-theory and the Ramanujan Type Bound
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This is to bound b for (τ, b) to occur in π ∈ Acusp (Sp2n ).
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For π ∈ A2 (Sp2n ), if (τ, b) occurs in π, then b ≤ 2n (sharp!).
Theorem (Kudla-Rallis (1994)) For τ = χ with χ2 = 1, and π ∈ Acusp (Sp2n ), if (χ, b) occurs in π, then b ≤ 2[ n2 ] + 1. I
This bound can also be deduced from my Conjecture.
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If (τ, b) occurs in π ∈ Acusp (Sp2n ), then b ≤ 2[ n2 ] + 1.
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Question: Is 2[ n2 ] + 1 sharp upper bound for b?
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When n is even, Piatetski-Shapiro and Rallis (1987) constructed a π(χ,n+1) ∈ Acusp (Sp2n ) with a global Arthur parameter (χ, n + 1), and hence this bound 2[ n2 ] + 1 is sharp!
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Question: When n is odd, how to construct a π(χ,n) in Acusp (Sp2n ) with a global Arthur parameter (χ, n)?
(τ, b)-theory and the Ramanujan Type Bound I
For π ∈ Acusp (Sp2n ), one reads the Satake parameters of π at a unramified local place v from Sp
(Fv )
2n IndB(F v)
χ1 |·|α1 ⊗ χ2 |·|α2 ⊗ · · · ⊗ χn |·|αn ,
where B is the standard Borel of Sp2n , and χi s are unitary.
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For θ ∈ R≥0 , we say that π has R(θ) if at each unramified local place v, one has 0 ≤ αi ≤ θ for all i.
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When n is even, π(χ,n+1) has R( n2 ).
Theorem For any number field F , if n is even, every π ∈ Acusp (Sp2n ) has R( n2 ), which is sharp. I
The proof uses the Ramanujan bound for τ ∈ Acusp (GL2 ) given by Kim-Sarnak (2003) and by Blomer-Brumley (2011).
(τ, b)-theory and the Ramanujan Type Bound I
Theorem For any number field F , if n is odd, every π ∈ Acusp (Sp2n ) has 7 R( 64 + n−1 2 ). I
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The proof is the same. In this case, it is a hard problem to find a sharp bound even assuming the Ramanujan conjecture.
Theorem (J.-Liu (2015)) If F is totally imaginary, and if n ≥ 5 is odd, then every π in Acusp (Sp2n ) has R( n−1 2 ). I
It is not known if the bound is sharp.
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One expects that the above discussion should be extended to other classical groups.
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My work in progress joint with Lei Zhang and Baiying Liu is to figure out the small cuspidal members in the global Arthur e ψ (G) for general global Arthur parameters ψ. packets Π
