On Cuspidal Spectrum of Classical Groups

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1 On Cuspidal Spectrum of Classical Groups Dihua Jiang University of Minnesota Simons Symposia on Geometric Aspects of the Trace Formula April 10-16, 2016

2 Square-Integrable Automorphic Forms G a reductive algebraic group defined over a number field F. A is the ring of adeles of F. X G := G(F )\G(A) 1, where G(A) 1 := χ X (G) ker χ A. L 2 (X G ) denotes the space of functions: φ : X G C such that G(Q)\G(A) 1 φ(g) 2 dg <. A 2 (G) is the set of equivalence classes of irreducible unitary representations of G(A) occurring in the discrete spectrum L 2 disc (X G). A cusp (G) is the subset of A 2 (G) consisting of those automorphic representations of G(A) occurring in the cuspidal spectrum L 2 cusp(x G ).

3 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 A cusp (G), there is a global Arthur parameter ψ Ψ 2 (G ), which is G-relevant, such that π Π ψ (G) where Π ψ (G) is the global Arthur packet of G associated to ψ. We may form the global Arthur-Vogan packet as union of the global Arthur packets Π ψ (G) over all the pure inner forms G of G : Π ψ [G ] := G Πψ (G).

4 Global Arthur Parameters Ψ 2 (G): Examples G = SO 2n+1, F -split, and (G ) = Sp 2n (C). Each ψ Ψ 2 (G ) (global Arthur parameters) is written as a formal sum of simple Arthur parameters: ψ = ψ 1 ψ 2 ψ r where ψ i = (τ i, b i ), with τ i A cusp (GL ai ); a i, b i 1; and r i=1 a ib i = 2n. If i j, either τ i = τj or b i b j, with the parity condition that a i b i is even and ψ i Ψ 2 (SO a i b i +1 ). Endoscopy Structure: 2n = r i=1 a i b i, SO a 1 b 1 +1 SO a r b r+1 = SO 2n+1 Π ψ1 ( ) Πψr ( ) = Π ψ ( )

5 Global Arthur Parameters Ψ 2 (G): Examples G = Sp 2n, F -split, and (G ) = SO 2n+1 (C). Each ψ Ψ 2 (G ) is written as a formal sum of simple Arthur parameters: ψ = ψ 1 ψ 2 ψ r where ψ i = (τ i, b i ), with τ i A cusp (GL ai ); a i, b i 1; r i=1 a ib i = 2n + 1; and r i=1 ωb i τ i = 1. If i j, either τ i = τj or b i b j, with the parity: 1 If a i b i is even, then ψ i Ψ 2 (SO a ib i ); 2 If a i b i is is odd, then ψ i Ψ 2 (Sp a ib i 1). Endoscopy Structure: 2n + 1 = r i=1 a i b i, a i b i =2l i SO 2l i a j b j =2l j +1 Sp 2l j = Sp 2n ai b i =2l i Πψi ( ) aj b j =2l j +1 Π ψj ( ) = Π ψ ( )

6 Global Arthur Parameters Ψ 2 (G): Examples A parameter ψ = ψ 1 ψ 2 ψ r Ψ 2 (G ) is generic if b 1 = = b r = 1. Generic global Arthur parameters φ Φ 2 (G ) are: φ = (τ 1, 1) (τ 2, 1) (τ r, 1) with τ i A cusp (GL ai ) that τ i = τj if i j. They are of either symplectic or orthogonal type, depending on G. The pure inner forms of G = SO m are G = SO m (V, q) for non-deg. quad. spaces (V, q) over F with the same dimension and discriminant. If G is a pure inner form of G, then L G = L G. For φ Φ 2 (G ), the endoscopic classification may define the global Arthur packet Π φ (G ) and also define the global Arthur packet Π φ (G), which is non-empty if φ is G-relevant.

7 Endoscopic Classification and Langlands Functoriality A(GL NG ) π ψ Ψ 2 (G ) G ψ A 2 (G) Π ψ (G) Πψ (G ) A 2 (G )

8 Problems Based on Endoscopic Classification A Simple Question: Π ψ (G) A cusp (G) =? If Π ψ (G) A cusp (G), call ψ cuspidal. What can one say about the cuspidal ψ? Write ψ = (τ 1, b 1 ) (τ r, b r ). How to bound these integers b 1,, b r if ψ is cuspidal? This leads to a Ramanujan type bound for A cusp (G). For π A cusp (G), how to determine which (τ, b) occurs in the global Arthur parameter ψ of π? This leads to the (τ, b)-theory that characterizes the (τ, b) factor of π in terms of basic invariants of π. If ψ is cuspidal, how to construct explicit modules for the members in Π ψ (G) A cusp (G)? This leads to the theory of twisted automorphic descents and endoscopy correspondences via integral transforms.

9 Remarks If G is F -quasisplit, the automorphic descent of Ginzburg-Rallis-Soudry constructs a generic member in Π φ (G ) A cusp (G ) for each generic global Arthur parameter ψ = φ. If G is a pure inner form of G, all members in the set Π φ (G) A cusp (G) can be constructed by using the twisted automorphic descent developed in my work with Lei Zhang, assuming the extended Arthur-Burger-Sarnak principle. A different approach is taken up also joint with Baiying Liu on this issue. 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.

10 Fourier Coefficients and Nilpotent Adjoint Orbits G is an F -quasi-split classical group and g is the Lie algebra. Let N G be the dimension for the defining embedding G GL(N G ). Over algebraic closure F of F, all the nilpotent elements in g (F ) form a conic algebraic variety, called the nilcone N (g ). 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 = N G of type G. Over F, each F -orbit reduces to an F -stable adjoint G (F )-orbits O st, and hence the F -stable adjoint orbits in the nilcone N (g ) are also parameterized by the corresponding partitions of an integer N = N G of type G.

11 Fourier Coefficients and Nilpotent Adjoint Orbits For X N (g ), use sl 2 -triple (over F ) to define a unipotent subgroup V X and a character ψ X. Let {X, H, Y } be an sl 2 -triple (over F ). Under the adjoint action of ad(h), g = g r g 2 g 1 g 0 g 1 g 2 g r. Ad(G )(Y ) g 2 and Ad(G )(X) g 2 are Zariski dense in g 2 and g 2, respectively. Take V X to be the unipotent subgroup of G such that the Lie algebra of V X is equal to i 2 g i. Let ψ F be a non-trivial additive character of F \A. The character ψ X of V X (F ) or V X (A) is defined by ψ X (v) = ψ F (tr(y log(v))).

12 Fourier Coefficients and Nilpotent Adjoint Orbits The Fourier coefficient of ϕ π A 2 (G ) is defined by F ψ X (ϕ)(g) := ϕ(vg)ψ X (v) 1 dv. V X (F )\V X (A) Since ϕ is automorphic, the nonvanishing of F ψ X (ϕ) depends only on the G (F )-adjoint orbit O X of X. The set n(ϕ) := {X N (g) F ψ X (ϕ) 0} is stable under the G (F )-adjoint action. Denoted by p(ϕ) the set of partitions p of N G of type G corresponding to the F -stable orbits Op st that have non-empty intersection with n(ϕ). p m (ϕ) is the set of all maximal partitions in p(ϕ), according to the partial ordering of partitions.

13 Maximal Fourier Coefficients of Automorphic Forms For π A 2 (G), denote by p m (π) the set of maximal members among p m (ϕ) for all ϕ π. We would like to know: How to determine p m (π) in terms of other invariants of π? What can one say about π based on the structure of p m (π)? Folklore Conjecture: For any irreducible automorphic representation π of G, the set p m (π) is singleton. Write p = [p 1 p 2 p r ] p m (π) with p 1 p 2 p r. What can we say about the largest part p 1 if π is cuspidal? 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.

14 Maximal Fourier Coefficients of Automorphic Forms Examples: G = GL n, the G(F )-stable orbits in N (g) are parameterized by partitions of n. Theorem (Piatetski-Shapiro; Shalika): If π A 2 (GL n ) is cuspidal, p m (π) = {[n]}. This says that any irreducible cuspidal automorphic representation has a nonzero Whittaker-Fourier coefficient. What happens if π A 2 (GL n ) is not cuspidal? Moeglin-Waldspurger Theorem: Any π A disc (GL n ) has form (τ, b) (Speh residue with cuspidal support τ b ), where τ A cusp (GL a ) and n = ab. If π = (τ, b), then p m (π) = {[a b ]} (Ginzburg (2006), J.-Liu (2013) gives a complete global proof). In particular, the Folklore Conjecture is verified for all π A 2 (GL n )!

15 Maximal Fourier Coefficients and Arthur Parameters How to understand this in terms of Arthur parametrization? (τ, b) has the Arthur parameter ψ = (τ, b). The partition attached to ψ is p ψ := [b a ] and p m ( (τ, b)) = {[a b ]}. η([b a ]) = [a b ] is given by the Barbasch-Vogan duality η from GL n to GL n. In this case, it is just the transpose. Take an Arthur parameter for GL n : for τ i A cusp (GL ai ), ψ = (τ 1, b 1 ) (τ 2, b 2 ) (τ r, b r ). The partition attached to ψ is p ψ = [b a 1 1 ba 2 2 bar r ]. The Arthur representation is an isobaric sum π ψ = (τ 1, b 1 ) (τ 2, b 2 ) (τ r, b r ). Conjecture: p m (π ψ ) = {η gl n,gl n (p ψ )}.

16 Maximal Fourier Coefficients and Arthur Parameters For ψ = ψ 1 ψ 2 ψ r Ψ 2 (G ), where ψ i = (τ i, b i ) with τ i A cusp (GL ai ) and b i 1, p ψ = [b a 1 1 bar r ] is the partition of N (G ) attached to (ψ, (G ) ) and η(p ψ ) is the Barbasch-Vogan duality of p ψ from (G ) to G. Conjecture (J.-2014): (1) For every π Π ψ (G ) A 2 (G ), any partition p p m (π) has the property that p η(p ψ ). (2) There exists at least one member π Π ψ (G) A 2 (G) for some pure inner form G of G that have the property: η(p ψ ) p m (π). 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(p ψ ) is a G-relevant partition of N G = N G of type G. The definition of Fourier coefficients also work.

17 Examples of the Barbasch-Vogan duality G = SO 2n+1 and 2n = ab; Take ψ = (τ, b) for τ A cusp (GL a ), and { 2l, if τ is orthogonal, b = 2l + 1, if τ is symplectic. p ψ = [b a ] is the partition of 2n of type (ψ, Sp 2n (C)). The Barbasch-Vogan duality is given as follows: [(a + 1)a b 2 (a 1)1] if b = 2l and a is even; η(p ψ ) = [a b 1] if b = 2l and a is odd; [(a + 1)a b 1 ] if b = 2l + 1.

18 Examples of the Barbasch-Vogan duality Take G = Sp 2n and ψ = (τ, 2b + 1) r i=2 (τ i, 1) Ψ 2 (G). p ψ = [(2b + 1) a (1) 2m+1 a ] with 2m + 1 = (2n + 1) 2ab. When a 2m and a is even, η(p ψ ) =η([(2b + 1) a (1) 2m+1 a ]) = [(2b + 1) a (1) 2m a ] t =[(a) 2b+1 ] + [(2m a)] = [(2m)(a) 2b ]. When a 2m and a is odd, η(p ψ ) =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)].

19 Remarks on the Conjecture It is true when G = GL n and ψ is an Arthur parameter for the discrete spectrum. If φ Φ 2 (G ) is generic, i.e. b 1 = = b r = 1, the partition p φ = [1 N (G ) ]. The Barbasch-Vogan duality of p φ is η([1 N (G ) ]) = [N G ] G. It is clear that the partition η([1 N (G ) ]) is G-relevant only if G = G is quasi-split. In this case, it the regular partition. 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. This special case can be proved by the Arthur-Langlands transfer from G to GL NG and the Ginzburg-Rallis-Soudry descent.

20 Remarks on the Conjecture The conjecture is known for various cases of Sp 2n (J.-Liu). The conjecture provides an upper bound partition for π Π ψ (G) A 2 (G), with a given Arthur parameter ψ. We are to obtain a lower bound partition, with a given ψ, for π Π ψ (G) A 2 (G). It is very interesting, but harder problem to determine p m (π), with a given ψ, for general members π Π ψ (G) A 2 (G)! The theory of singular automorphic forms of Howe and Li provides a lower bound partition for π A cusp (G). 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 A cusp (G).

21 Singular Partitions and Rank in the Sense of Howe G = Sp 2n. Take P n = M n U n to be the Siegel parabolic of Sp 2n, with M n = GLn and the elements of U n are of form ( ) In X u(x) =. 0 I n By the Pontryagin duality, one has U n (F )\U n (A) = Sym 2 (F n ). For a fixed nontrivial additive character ψ F of F \A, a T Sym 2 (F n ) corresponds to the character ψ T by ψ T (u(x)) := ψ F (tr(t wx)), with w anti-diagonal, and the action of GL n on Sym 2 (F n ) is induced from its adjoint action on U n.

22 Singular Partitions and Rank in the Sense of Howe For an automorphic form ϕ on Sp 2n (A), the T - or ψ T -Fourier coefficient is defined by F ψ T (ϕ)(g) := ϕ(u(x)g)ψ 1 T (u(x))du(x). U m(f )\U n(a) An automorphic form ϕ on Sp 2n (A) is called non-singular if ϕ has a nonzero ψ T -Fourier coefficient with a non-singular T. ϕ is called singular if ϕ has the property that if a nonzero ψ T -Fourier coefficient F ψ T (ϕ) is nonzero, then det(t ) = 0. Howe shows in 1981 that if an automorphic form ϕ on Sp 2n (A) is singular, then ϕ can be expressed as a linear combination of certain theta functions. Jianshu Li shows in 1989 that any cuspidal automorphic form of Sp 2n (A) is non-singular.

23 Singular Partitions and Rank in the Sense of Howe For a split SO m 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 + V 0 + X + with V 0 the orth. complement of X + X + (dim V 0 1). The generalized flag {0} X + V defines a maximal parabolic subgroup P X +, whose Levi part M X + is isomorphic to GL [ m 2 ] and whose unipotent radical N X + is abelian if m is even; and is a two-step unipotent subgroup with its center Z X + if m is odd. When m is even, we set Z X + = N X +.

24 Singular Partitions and Rank in the Sense of Howe By the Pontryagin duality, For any T 2 (F [ m 2 ] ), Z X +(F )\Z X +(A) = 2 (F [ m 2 ] ). ψ T (z(x)) := ψ F (tr(t wx)). For an automorphic form ϕ on G(A), the T or ψ T Fourier coefficient is defined by F ψ T (ϕ)(g) := ϕ(z(x)g)ψ 1 T (z(x))dz(x). Z X + (F )\Z X + (A) An automorphic form ϕ on G(A) is called non-singular if ϕ has a non-zero ψ T -Fourier coefficient with T 2 (F [ m 2 ] ) of maximal rank.

25 Singular Partitions and Rank in the Sense of Howe Denote by p ns the partition corresponding to the non-singular Fourier coefficients. For Sp 2n, p ns = [2 n ], which is a special partition for Sp 2n. For SO 2n+1, one has p ns = { [2 2e 1] if n = 2e; [2 2e 1 3 ] if n = 2e + 1. This is not a special partition of SO 2n+1. For SO 2n, one has p ns = { [2 2e ] if n = 2e; [2 2e 1 2 ] if n = 2e + 1. This is a special partition of SO 2n.

26 Singular Partitions and Lower Bound Partitions J.-Liu-Savin show 2015 that for any automorphic representation π, the set p m (π) contains only special partitions. For SO 2n+1, p ns p m (π) for any π A cusp (SO 2n+1 ). Any p p m (π) as π runs in A cusp (SO 2n+1 ) must be bigger than or equal to the following partition { [32 2e ] if n = 2e; p Gn ns p SO 2n+1 ns = [32 2e ] if n = 2e + 1. denotes the G n -expansion of the partition p ns, i.e., the smallest special partition which is bigger than or equal to p ns. Proposition: For a split classical group G n, p Gn is a lower ns bound for partitions in the set p m (π) as π runs in A cusp (G n ).

27 Singular Partitions and Small Cuspidal Representations It is natural to ask whether the lower bound p Gn ns is sharp? For n = 2e even, and F to be totally real, the Ikeda lifting gives π A cusp (Sp 4e ) with the global Arthur parameter (τ, 2e) (1, 1), where τ A cusp (GL 2 ) is self-dual and has the trivial central character. First, for any p p m (π), one has p [2 2e ] = p Sp 4e. ns Then, one also has p Sp 4e = [2 2e ] p. ns Proposition: If F is totally real, the Ikeda construction π of Sp 4e has p m (π) = {p Sp 4e} = {[2 2e ]}, and hence the ns non-singular partition p Sp 4e = p ns ns = [2 2e ] is the sharp lower bound for A cusp (Sp 4e ). Note that the construction of Ikeda is not known when F is not totally real or n = 2e + 1 is odd. What happens in such situations?

28 Singular Partitions and Small Cuspidal Representations J.-Liu (2015): If F is totally imaginary and n 5, the global Arthur packet Π (τ,n) (ɛ,1) (Sp 2n ), with τ A cusp (GL 2 ) self-dual, has no cuspidal members, where ɛ = 1 if n = 2e; and ɛ = ω τ if n = 2e + 1. In this case, the construction of the Ikeda lifting is impossible! Theorem (J.-Liu 2015): 1 For any π A cusp (Sp 2n ), p m (π) = [2 n ] if and only if π is hypercuspidal in the sense of I. Piatetski-Shapiro. 2 For ψ = (τ, 2e) (1, 1) Ψ 2 (Sp 4e ) with τ A cusp (GL 2 ), any cuspidal π Π ψ (Sp 4e ) has that p m (π) = [2 2e ]. 3 For ψ = (τ, 2e + 1) (ω τ, 1) Ψ 2 (Sp 4e+2 ) with τ A cusp (GL 2 ), any cuspidal π Π ψ (Sp 4e+2 ) has that p m (π) = [2 2e+1 ]. 4 For ψ = (τ, 2e + 1) Ψ 2 (Sp 6e+2 ) with τ A cusp (GL 3 ), any cuspidal π Π ψ (Sp 6e+2 ) has that p m (π) = [2 3e+1 ].

29 Cuspidality of Global Arthur Packets Theorem (J.-Liu 2015) Assume that F is totally imaginary, and Part (1) of the Conjecture holds. For ψ = r i=1 (τ i, b i ) in Ψ 2 (Sp 2n ) with τ i A cusp (GL ai ) for i = 1, 2,, r, 2n + 1 = r i=1 a ib i, and r i=1 ωb i τ i = 1, there exist constants N a N (1) a,b N (2) a,b, depending on (a 1,, a r ) and (b 1,, b r ), such that if 2n > N 0 for N 0 to be one of the N a, N (1) a,b, N (2) a,b, the set Π ψ (Sp 2n ) A cusp (Sp 2n ) is empty. Remarks: { ( r 1 i=1 N a = a i) 2 + 2( r i=1 a i) if r i=1 a iis even; ( r i=1 a i) 2 1 otherwise. 2 N (1) a,b, N (2) a,b are sharper bound, but not easy to be defined. 3 Conjecture: N (2) (2) a,b is sharp in the sense that if 2n = N a,b, then the set Π ψ (Sp 2n ) A cusp (Sp 2n ) is not empty

30 Cuspidality of Global Arthur Packets Example (1): Take ψ = (τ 1, 1) (τ 2, 8) with τ 1 A cusp (GL 5 ) and τ 2 A cusp (GL 2 ). We have N a = 48 > N (1) a,b = 24 > N (2) a,b = 16. Example (2): Take ψ = (1, b 1 ) (τ, b 2 ) with b 1 1 odd, τ A cusp (GL 2 ) of symplectic type and b 2 even. Then N a = N (1) a,b = N (2) a,b = 8 and the global Arthur packet Π ψ (Sp ) contains no cuspidal members except that (b 1, b 2 ) = (1, 2), (1, 4), (3, 2), or (5, 2). Mœglin (2008, 2011) also provide criterion for the cuspidality of global Arthur packets, which is in a different nature from what we developed here.

31 (τ, b)-theory and the Ramanujan Type Bound This is to bound b for (τ, b) to occur in π A cusp (Sp 2n ). For π A 2 (Sp 2n ), if (τ, b) occurs in π, then b 2n (sharp!). Theorem (Kudla-Rallis (1994)) For τ = χ with χ 2 = 1, and π A cusp (Sp 2n ), if (χ, b) occurs in π, then b 2[ n 2 ] + 1. This bound can also be deduced from my Conjecture. If (τ, b) occurs in π A cusp (Sp 2n ), then b 2[ n 2 ] + 1. Question: Is 2[ n 2 ] + 1 sharp upper bound for b? When n is even, Piatetski-Shapiro and Rallis (1987) constructed a π (χ,n+1) A cusp (Sp 2n ) with a global Arthur parameter (χ, n + 1), and hence this bound 2[ n 2 ] + 1 is sharp! Question: When n is odd, how to construct a π (χ,n) in A cusp (Sp 2n ) with a global Arthur parameter (χ, n)?

32 (τ, b)-theory and the Ramanujan Type Bound For π A cusp (Sp 2n ), one reads the Satake parameters of π at a unramified local place v from Ind Sp 2n (Fv) B(F v) χ 1 α 1 χ 2 α 2 χ n αn, where B is the standard Borel of Sp 2n, and χ i s are unitary. For θ R 0, we say that π has R(θ) if at each unramified local place v, one has 0 α i θ for all i. When n is even, π (χ,n+1) has R( n 2 ). Theorem For any number field F, if n is even, every π A cusp (Sp 2n ) has ), which is sharp. R( n 2 The proof uses the Ramanujan bound for τ A cusp (GL 2 ) given by Kim-Sarnak (2003) and by Blomer-Brumley (2011).

33 (τ, b)-theory and the Ramanujan Type Bound Theorem For any number field F, if n is odd, every π A cusp (Sp 2n ) has R( n 1 2 ). 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 A cusp (Sp 2n ) has R( n 1 2 ). It is not known if the bound is sharp. One expects that the above discussion should be extended to other classical groups. My work in progress joint with Lei Zhang and Baiying Liu is to figure out the small cuspidal members in the global Arthur packets Π ψ (G) for general global Arthur parameters ψ.

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