On Wave Front Sets of Global Arthur Packets of Classical Groups: Upper Bound

We prove a conjecture of the first-named author ([J14]) on the upper bound Fourier coefficients of automorphic forms in Arthur packets of all classical groups over any number field. This conjecture generalizes the global version of the local tempered $L$-packet conjecture of F. Shahidi ([Sh90] and [Sh10]).


Introduction
In the classical theory of automorphic forms, Fourier coefficients encode abundant arithmetic information of automorphic forms.In the modern theory of automorphic forms, i.e. the theory of automorphic representations of reductive algebraic groups defined over a number field k (or a global field), Fourier coefficients bridge the connection from harmonic analysis to number theory via automorphic forms.When the reductive group is GL n , the general linear group, by a classical theorem of I. Piatetski-Shapiro ( [PS79]) and J. Shalika ([S74]), every cuspidal automorphic representation of GL n (A), where A is the ring of adeles of k, has a non-zero Whittaker-Fourier coefficient.This fundamental result has been indispensable in the theory, especially the theory of automorphic L-functions.The theorem of Piatetski-Shapiro and Shalika has been extended to the discrete spectrum of GL n (A) in [JL13] and to the isobaric sum automorphic spectrum of GL n (A) in [LX21].
In general, due to the nature of the discrete spectrum of squareintegrable automorphic forms on reductive algebraic groups G, one has to consider more general version of Fourier coefficients, i.e.Fourier coefficients of automorphic forms associated to nilpotent orbits in the Lie algebra g of G.Such general Fourier coefficients of automorphic forms, including Bessel-Fourier coefficients and Fourier-Jacobi coefficients have been widely used in theory of automorphic L-functions via integral representation method (see [GPSR97], [GJRS11], [JZ14] and [JZ20], for instance), in automorphic descent method of Ginzburg, Rallis and Soudry to produce special cases of explicit Langlands functorial transfers ( [GRS11]), and in the Gan-Gross-Prasad conjecture on vanishing of the central value of certain automorphic L-functions of symplectic type ( [GJR04], [JZ20], and [GGP12]).More recent applications of such general Fourier coefficients to explicit constructions of endoscopy transfers for classical groups can be found in [J14] (and also in [G12] for split classical groups).
In this paper, we consider following classical groups defined over k, G n = Sp 2n , SO 2n+1 , O α 2n , quasi-split, and U n , quasi-split or inner forms.We follow the formulation in [GGS17] for the definition of generalized Whittaker-Fourier coefficients of automorphic forms associated to nilpotent orbits, see Section 2 for details.It is well-known that nilpotent orbits of the quasi-split classical group G n are parameterized by symplectic or orthogonal partitions and certain quadratic forms when G n = Sp 2n , SO 2n+1 , O α 2n , by relevant partitions when G n = U n (see [CM93], [N11] and [W01], for instance).For any irreducible automorphic representation π of G n (A), let n(π) be the set of nilpotent orbits providing nonzero generalized Whittaker-Fourier coefficients for π, which is called the wave front set of π, as in [JLS16] for instance.Let n m (π) be the subset that consists of maximal elements in n(π) under the dominance ordering of nilpotent orbits, and denote by p m (π) the set of the partitions of type G n corresponding to nilpotent orbits in n m (π).
It is an interesting problem to determine the structure of the set n m (π) and equivalently the set p m (π) for any given irreducible automorphic representation π of G n (A), by means of other invariants of π.When π occurs in the discrete spectrum of square integrable automorphic functions on G n (A), the global Arthur parameter attached to π ([Ar13], [Mok15], [KMSW14]) is clearly a fundamental invariant for π.An important conjecture made in [J14], which is the natural generalization of the global version of the local tempered L-packet conjecture of F. Shahidi ([Sh90] and [Sh10]), asserts an intrinsic relation between the structure of the global Arthur parameter of π to the structure of the set p m (π).It is well-known that the conjecture of Shahidi and its global version (see [JL16a,Section 3] for discussion and proof) has played a fundamental role in the understanding of the local and global Arthur packets for generic Arthur parameters, according the endoscopic classification of J. Arthur ([Ar13] and also [Mok15] and [KMSW14]).It is well expected that the conjecture made in [J14] for general global Arthur parameters will be important to the understanding of the structure of general global Arthur packets.
To state the conjecture of [J14], for simplicity, we briefly recall the endoscopic classification of the discrete spectrum for G n (A) from [Ar13] for G n = Sp 2n , SO 2n+1 , O α 2n .The set of global Arthur parameters for the discrete spectrum of G n is denoted, as in [Ar13], by Ψ 2 (G n ), the elements of which are of the form (1.1) where ψ i are pairwise different simple global Arthur parameters of orthogonal type (when G n = Sp 2n , O α 2n ) or symplectic type (when G n = SO 2n+1 ), and have the form ψ i = (τ i , b i ).The notations are explained in order.Let A cusp (GL a i ) be the set of equivalence classes of irreducible cuspidal automorphic representations of GL a i (A).We have when G n = O α 2n , following [Ar13, Section 1.4].More precisely, for each 1 ≤ i ≤ r, ψ i = (τ i , b i ) satisfies the following conditions: if τ i is of symplectic type (i.e., L(s, τ i , ∧ 2 ) has a pole at s = 1), then b i is even (when where Π ψ (ǫ ψ ) denotes the subset of Π ψ consisting of members which occur in the discrete spectrum, and m ψ is the discrete multiplicity of Π, which is either 1 or 2.
As in [J14], one may call Π ψ (ǫ ψ ) the automorphic L 2 -packet attached to ψ.For π ∈ Π ψ (ǫ ψ ), the structure of the global Arthur parameter ψ deduces constraints on the structure of p m (π), which is given by the following conjecture of the first-named author.
Conjecture 4.2 in [J14] consists of the upper-bound conjecture (Conjecture 1.2) and the sharpness conjecture ([J14, Conjecture 4.2, Part (3)], i.e., there exists π ∈ Π ψ (ǫ ψ ) such that η g ∨ ,g (p(ψ)) ∈ p m (π)).It is clear that if the global Arthur parameter ψ is generic, then [J14, Conjecture 4.2] asserts that the corresponding global Arthur packet Π ψ (ǫ ψ ) contains a automorphic member that is generic, i.e. has a nonzero Whittaker-Fourier coefficient.This is the global version of the local tempered L-packet conjecture of Shahidi ([Sh90]) and was proved in [JL16a, Section 3] by using automorphic descent of D. Ginzgurg, S. Rallis, and D. Soudry ( [GRS11]).The goal of this paper is to prove Conjecture 1.2 for general global Arthur parameters.The sharpness conjecture is of global in nature and will be fully considered in future projects.
In [JL16], using the method of local descent, we partially prove Conjecture 1.2 for G n = Sp 2n , namely, for any π ∈ Π ψ (ǫ ψ ), if a partition p ∈ p m (π), then p ≤ L η g ∨ ,g (p(ψ)), under the lexicographical order.We refer to [J14, Section 4] for more discussion on this conjecture and related topics.
In order to prove Conjecture 1.2, we study the structure of the unramified local components π v of π and of the set p m (π v ) which is defined similarly as p m (π).Our discussion reduces the general situation to a special case of strongly negative unramified unitary representations of G n (see Section 3 for details).In such a special situation, the structure of the wave front set (Theorem 3.6) can be deduced as a special case from [Oka21, Theorem 1.5].
To be more precise, first, for the Arthur parameter ψ = ⊞ r i=1 (τ i , b i ), by [JL16, Proposition 6.1] (see Proposition 4.1), there exist infinitely many finite places v such that G n (k v ) is split, τ i,v 's all have trivial central characters, and hence π v is the unramified component of an induced representation of the following form is a special family of strongly negative representations which have Arthur parameters of the form ⊕ s j=1 1 W ′ F ⊗ S 2n j +1 (see Section 3.2 for details), with W ′ F being the Weil-Deligne group and It is known that the wave front set of σ (hence of π v ) is bounded above by the induced orbits once we know the leading orbits for the wave front set of σ sn .On the other hand, Okada ([Oka21, Theorem 1.5] computed the leading orbits in the wave front set of those unramified representations whose Arthur parameters are trivial when restricting to the Weil-Deligne group. We remark that for non-quasi-split even orthogonal groups, once the Arthur classification being carried out (see [CZ21a,CZ21b] for recent progress in this direction), Conjecture 1.2 can be proved by similar arguments.
In the last part of this paper, we study the wave front set of the unramified unitary dual for split classical groups G n = Sp 2n , SO 2n+1 , O 2n .Under a conjecture on the wave front set of negative representations (Conjecture 8.1), we are able to determine the set p m (π) for general unramified unitary representations (Theorem 8.2).This provides a reduction towards understanding the wave front set of the whole unramified unitary dual, which has its own interests.
The structure of this paper is as follows.In Section 2, we recall certain twisted Jacquet modules and Fourier coefficients associated to nilpotent orbits, following the formulation in [GGS17].The structure of unramified unitary dual of G n (F v ) was determined by D. Barbasch in [Bar10] and by G. Muic and M. Tadic in [MT11] with different approaches.In Section 3, we recall from [MT11] the results on unramified unitary dual.In Section 4, we determine, for any given global Arthur parameter ψ ∈ Ψ 2 (G n ), the unramified components π v of any π ∈ Π ψ (ǫ ψ ) in terms of the classification data in [MT11], and prove Theorem 1.3 by means of Theorem 4.2 which is about certain properties of p ∈ p m (π v ).Theorem 4.2 is technical and will be proved in Sections 5, 6, and 7, for G n = Sp 2n , SO 2n+1 , O α 2n , respectively.In Section 8, we determine the leading orbits in the wave front set of general unramified unitary representations assuming Conjecture 8.1 for split classical groups G n = Sp 2n , SO 2n+1 , O 2n (Theorem 8.2).

Fourier coefficients associated to nilpotent orbits
In this section, we recall certain twisted Jacquet modules and Fourier coefficients associated to nilpotent orbits, following the formulation of R. Gomez, D. Gourevitch and S. Sahi in [GGS17].
Let G be a reductive group defined over a field F of characteristic zero, and g be the Lie algebra of G = G(F ).Given any semi-simple element s ∈ g, under the adjoint action, g is decomposed into a direct sum of eigenspaces g s i corresponding to eigenvalues i.The element s is called rational semi-simple if all its eigenvalues are in Q.Given a nilpotent element u and a semi-simple element s in g, the pair (s, u) is called a Whittaker pair if s is a rational semi-simple element, and u ∈ g s −2 .The element s in a Whittaker pair (s, u) is called a neutral element for u if there is a nilpotent element v ∈ g such that (v, s, u) is an sl 2 -triple.A Whittaker pair (s, u) with s being a neutral element is called a neutral pair.
Given any Whittaker pair (s, u), define an anti-symmetric form ω u on g × g by here κ is the Killing form on g.For any rational number r ∈ Q, let g s ≥r = ⊕ r ′ ≥r g s r ′ .Let u s = g s ≥1 and let n s,u be the radical of For any X ∈ g, let g X be the centralizer of X in g.By [GGS17, Lemma 3.2.6],one has n s,u = g s ≥2 + g s 1 ∩ g u .Note that if the Whittaker pair (s, u) comes from an sl 2 -triple (v, s, u), then n s,u = g s ≥2 .We denote by N s,u = exp(n s,u ) the corresponding unipotent subgroup of G.
When F = k v is a non-Archimedean local field, we take ψ : F → C × to be a fixed non-trivial additive character and define a character of N s,u by Let σ be an irreducible admissible representation of G(F ).The twisted Jacquet module of π associated to a Whittaker pair (s, u) is defined to be σ Ns,u,ψu .Let n(σ) be the set of nilpotent orbits O ⊂ g such that the twisted Jacquet module σ Ns,u,ψu is non-zero for some neutral pair (s, u) with u ∈ O.
When F = k is a number field, let A be the ring of adeles, and let ψ : F \A → C × be a fixed non-trivial additive character.Extend the killing form κ to g(A) × g(A).Define a character of N s,u (A) by It is clear from the definition that the character ψ u (n) is trivial when restricted to the discrete subgroup N s,u (F ), and hence can be viewed as a function on [N s,u ] := N s,u (F )\N s,u (A).Let π be an irreducible automorphic representation of G(A).For any φ ∈ π, the degenerate Whittaker-Fourier coefficient of φ attached to a Whittaker pair (s, u) is defined to be If (s, u) is a neutral pair, then F s,u (φ) is also called a generalized Whittaker-Fourier coefficient of φ.Define which is called the Fourier coefficient of π.The wave-front set n(π) of π is defined to be the set of nilpotent orbits O such that F s,u (π) is non-zero for some neutral pair (s, u) with u ∈ O.
Note that if σ Ns,u,ψu , or F s,u (π), is non-zero for some neutral pair (s, u) with u ∈ O, then it is non-zero for any such neutral pair (s, u), since the non-vanishing property of such Whittaker models or Fourier coefficients does not depend on the choices of representatives of O.Moreover, we let n m (σ) and n m (π) be the set of maximal elements in the wave front sets n(σ) and n(π), respectively, under the natural ordering of nilpotent orbits (i.e., In this paper, we mainly consider classical groups G n = Sp 2n , SO 2n+1 , O α 2n , quasi-split, and U n , quasi-split or inner forms, and study the sets p(π), or p(σ), partitions corresponding to the nilpotent orbits in the wave front sets, for any irreducible automorphic representation π of G n (A), which occurs in the discrete spectrum of G n (A) as displayed in Theorem 1.1, and for some unramified local components σ = π v .

Unramified Unitary Dual of split classical groups
We take F = k v to be a non-Archimedean local field of k.In this section, we recall the classification of the unramified unitary dual of the split classical groups G n = Sp 2n , SO 2n+1 , O 2n over F , which was obtained by D. Barbasch in [Bar10] and by G. Muic and M. Tadic in [MT11], using different methods.
We recall the unramified unitary dual of split classical groups from the work of Muic and Tadic in [MT11].The classification in [MT11] starts from classifying two special families of irreducible unramified representations of G n (F ) that are called strongly negative and negative, respectively.We refer to [M06] for definitions of strongly negative and negative representations, respectively, and for more related discussion on those two families of unramified representations.In the following, we recall from [MT11] the classification of these two families in terms of Jordan blocks.Their classification also provides explicit constructions of the two families of unramified representations.
A pair (χ, m), where χ is an unramified unitary character of F * and m ∈ Z >0 , is called a Jordan block.When G n = Sp 2n , O 2n , define Jord sn (n) to be the collection of all sets Jord of the following form: (3.1) {(λ 0 , 2n 1 + 1), . . ., (λ 0 , 2n k + 1), (1 GL 1 , 2m 1 + 1), . . ., (1 GL 1 , 2m l + 1)} where λ 0 is the unique non-trivial unramified unitary character of F * of order 2, given by the local Hilbert symbol (δ, •) F * , with δ being a non-square unit in O F ; k is even, and l is odd when G n = Sp 2n and is even when G n = O 2n .There are also following constraints: When G n = SO 2n+1 , define Jord sn (n) to be the collection of all sets Jord of the following form: both k and l are even and k i=1 (2n i ) + l j=1 (2m j ) = 2n.For each Jord ∈ Jord sn (n), we can associate a representation σ(Jord), which is the unique irreducible unramified subquotient of the following induced representation.When G n = Sp 2n , it is given by (3.5) Theorem 3.1 (Theorem 5-8, [MT11]).Assume that n > 0. The map Jord → σ(Jord) defines a one-to-one correspondence between the set Jord sn (n) to the set of all irreducible strongly negative unramified representations of G n (F ).
The inverse of the map in Theorem 3.1 is denoted by σ → Jord(σ).Based on the classification in Theorem 3.1, irreducible negative unramified representations can be constructed from irreducible strongly negative unramified representations of smaller rank groups as follows.
Theorem 3.2 (Thereom 5-10, [MT11]).For any sequence of pairs (χ 1 , n 1 ), . . ., (χ t , n t ) with χ i being unramified unitary characters of F * and n i ∈ Z ≥1 , for 1 ≤ i ≤ t, and for a strongly negative representation σ sn of G n ′ (F ) with t i=1 n i + n ′ = n, the unique irreducible unramified subquotient of the following induced representation is negative and it is a subrepresentation.
Conversely, any irreducible negative unramified representation σ neg of G n (F ) can be obtained from the above construction.The data (χ 1 , n 1 ), . . ., (χ t , n t ) and σ sn are unique, up to permutations and taking inverses of χ i 's.
For any irreducible negative unramified representation σ neg with data in Theorem 3.2, we define [M07], any irreducible negative representation is unitary.In particular, we have the following Corollary 3.3.Any irreducible negative unramified representation of G n (F ) is unitary.
To describe the general unramified unitary dual, we need to recall the following definition.
Let M u,unr (n) be the subset of M unr (n) consisting of pairs (e, σ neg ), which satisfy the following conditions: (1) Write elements in e(χ, m) as follows: with k, l ∈ Z ≥0 .They satisfy the following conditions: Theorem 3.5 (Theorem 5-14, [MT11]).The map defines a one-to-one correspondence between the set M u,unr (n) and the set of equivalence classes of all irreducible unramified unitary representations of G n (F ).
In Sections 4-7, we will mainly consider the following type of unramified unitary representations: Type (I): An irreducible unramified unitary representations of G n (F ) is called of Type (I) if it is of the following form: where σ neg is the unique irreducible negative unramified subrepresentation of the following induced representation with σ sn being the unique strongly negative unramified constituent of the following induced representation: Next, we recall the following theorem, which is a special case of [Oka21, Theorem 1.5].We remark that the spherical representations considered in [Oka21, Theorem 1.5] are those with Arthur parameters that are trivial on the Weil-Deligne group (see [Oka21, Page 5] for the setting), while general unramified representations have Arthur parameters that are trivial on the subgroup I F × SL 2 (C), where I F is the inertia subgroup of the Weil group W F .Theorem 3.6 (Theorem 1.5, [Oka21]).Let σ sn is a strongly negative unramified representation of G n .If G n = Sp 2n , O 2n and if the Jordan Block of σ sn is of the form then the maximal partitions of the wave-front set of σ sn is given by then the maximal partitions of the wave-front set of σ sn is given by

Arthur parameters and unramified local components
In this section, in terms of the classification of the unramified unitary dual of G n , we study the structure of the unramified local components We write F = k v and first consider the case of G n = Sp 2n , O α 2n .Rewrite the global Arthur parameter ψ as follows: Let J 1 be the subset of J such that ω τ j,v = 1, and The local unramified Arthur parameter ψ v has the following structures: • For i ∈ I, where 0 ≤ β i q < 1 2 , for 1 ≤ q ≤ a i , and χ i q 's are unramified unitary characters of where 0 ≤ β j q < 1 2 , for 1 ≤ q ≤ a j , and χ j q 's are unramified unitary characters of F * .• For j ∈ J 2 , where 0 ≤ β j q < 1 2 , for 1 ≤ q ≤ a j , and χ j q 's are unramified unitary characters of F * .
Note that Jord 1 is a multi-set.Let Jord 2 be a set consists of different Jordan blocks with odd multiplicities in Jord 1 .Thus Jord 2 has the form of (3.1).By Theorem 3.1, there is a corresponding irreducible strongly negative unramified representation σ sn .Then we define the following Jordan blocks: Finally, we define By Theorem 3.2, corresponding to the data Jord 3 and σ sn , there is an irreducible negative unramified presentation σ neg .Let e I = {(χ i q , 2b i , β i q ), i ∈ I, 1 ≤ q ≤ a i , β i q > 0}, e J 1 = {(χ j q , 2b j + 1, β j q ), j ∈ J 1 , 1 ≤ q ≤ a j , β j q > 0}, e J 2 = {(χ j q , 2b j + 1, Then we define Since the unramified component π v is unitary, we must have that (e, σ neg ) ∈ M u,unr (n), and π v is exactly the irreducible unramified unitary representation σ of G n (F ) which corresponds to (e, σ neg ) as in Theorem 3.5.Now we consider the case of G n = SO 2n+1 .Rewrite the global Arthur parameter ψ as follows: where Let J 1 be the subset of J such that ω τ j,v = 1, and J 2 = J\J 1 , that is, for j ∈ J 2 , ω τ j,v = λ 0 .Let S 1 be the subset of S such that ω τs,v = 1, and S 2 = S\S 1 , that is, for s ∈ S 2 , ω τs,v = λ 0 .The local unramified Arthur parameter ψ v has the following structures: where 0 ≤ β i q < 1 2 , for 1 ≤ q ≤ a i , and χ i q 's are unramified unitary characters of F * .
where 0 ≤ β j q < 1 2 , for 1 ≤ q ≤ a j , and χ j q 's are unramified unitary characters of where 0 ≤ β j q < 1 2 , for 1 ≤ q ≤ a j , and χ j q 's are unramified unitary characters of F * .
where 0 ≤ β s q < 1 2 , for 1 ≤ q ≤ a s , and χ s q 's are unramified unitary characters of where 0 ≤ β s q < 1 2 , for 1 ≤ q ≤ a s , and χ s q 's are unramified unitary characters of F * .We define Jord 1 = {(λ 0 , 2b j ), j ∈ J 2 ; (λ 0 , 2b s ), s ∈ S 2 ; ( Note that Jord 1 is a multi-set.Let Jord 2 be a set consists of different Jordan blocks with odd multiplicities in Jord 1 .Thus Jord 2 has the form of (3.1).By Theorem 3.1, there is a corresponding irreducible strongly negative unramified representation σ sn .Then we define the following Jordan blocks: By Theorem 3.2, corresponding to the data Jord 3 and σ sn , there is an irreducible negative unramified presentation σ neg . Let Since the unramified component π v is unitary, we must have that (e, σ neg ) ∈ M u,unr (n), and π v is exactly the irreducible unramified unitary representation σ of G n (F ) which corresponds to (e, σ neg ) as in Theorem 3.5.

4.2.
Proof of Theorem 1.3.The following result from [JL16] is needed for the proof of Theorem 1.3.
Proposition 4.1 (Proposition 6.1, [JL16]).For any finitely many nonsquare elements α i / ∈ k * /(k * ) 2 , 1 ≤ i ≤ t, there are infinitely many finite places v such that α i ∈ (k * v ) 2 , for any 1 ≤ i ≤ t.Now we are going to prove Theorem 1.3.First we consider the cases of Assume that {τ i 1 , . . ., τ iq } is a multi-set of all the τ 's with non-trivial central characters.Since all τ i j 's are self-dual, the central characters ω τ i j 's are all quadratic characters, which are parametrized by global non-square elements.Assume that ω τ i j = χ α i j , where α i j ∈ k * /(k * ) 2 , and χ α i j is the quadratic character given by the global Hilbert symbol (•, α i j ).Note that {α i 1 , . . ., α iq } is a multi-set.By Proposition 4.1, there are infinitely many finite places v, such that α and α i j 's are all squares in k v .Therefore, for the given ψ, there are infinitely many finite places v such that G n (k v ) split and all τ i,v 's have trivial central characters.From the discussion in Section 3, for any π ∈ Π ψ (ǫ ψ ), there is a finite local place v with such a property that π v is an irreducible unramified unitary representation of Type (I) as in (3.7).
We are going to discuss the connection with the classification of D. Barbasch in [Bar10].
Assume first that G n = Sp 2n , O α 2n .If σ is an irreducible unramified unitary representation of G n (k v ) corresponding to the pair (e, σ neg ) ∈ M u,unr (n), then the orbit Ǒ corresponding to σ in [Bar10] is given by the following partition: When π v is of Type (I) as in (3.7), the orbit Ǒ corresponding to σ = π v in [Bar10] is given by the following partition: (2m i + 1))] (4.6) which turns out to be p(ψ) exactly.
Assume now that G n = SO 2n+1 .If σ is an irreducible unramified unitary representation of G n (k v ) corresponding to the pair (e, σ neg ) ∈ M u,unr (n), then the orbit Ǒ corresponding to σ in [Bar10] is given by the following partition: When π v is of Type I as in (3.7), the orbit Ǒ corresponding to σ = π v in [Bar10] is given by the following partition: which turns out to be p(ψ) exactly.
We claim that for the cases of G n = Sp 2n , SO 2n+1 , O α 2n , Theorem 1.3 can be deduced from the following theorem whose proof will be given in the next three sections.Theorem 4.2.Let σ be an irreducible unramified unitary representations of G n (k v ) of Type (I) as in (3.7).For any p ∈ p m (σ), the following bound (2m i + 1))] (4.9) holds with the partition on the left-hand side from (4.6) when G n = Sp 2n , O 2n ; and the following bound holds with the partition on the left-hand side from (4.7) when G n = SO 2n+1 .
For the case of G n = U n , E/k a quadratic extension, by similar arguments, for any π ∈ Π ψ (ǫ ψ ), there is a finite local place v such that , split, and π v is unramified.Then, Theorem 1.3 is simply implied by the classfication of the unramified unitary dual of GL n ( [Tad86]) and the result of Moeglin and Waldspurger on the wave front set of representations of GL n ([MW87, Section II.2]).Note that for G n = U n , the Barbasch-Vogan-Spaltenstein duality is just the transpose of partitions.We omit the details here.
This completes the proof of Theorem 1.3.
Remark 4.3.We expect that the method of proving Theorem 1.3 in this paper also applies to the inner forms of even orthogonal groups, once the full Arthur classification of the discrete spectrum being carried out (see [CZ21a,CZ21b] for recent progress in this direction).The same method can also be applied to the metaplectic double cover of symplectic groups, whose proof will appear elsewhere.Note that for the metaplectic double cover of symplectic groups, the notion of Barbasch-Vogan-Spaltenstein duality has been defined in [BMSZ20].

Proof of Theorem 4.2, G n = Sp 2n
First, we recall the following general lemma which can be deduced from the argument in [MW87, Section II.1.3].
Lemma 5.1 (Section II.1.3,[MW87]).Let G be a reductive group defined over a non-Archimedean local field F , and Q = MN be a parabolic subgroup of G. Let δ be an irreducible admissible representation of M.
where q and g are the Lie algebras of Q and G, respectively.For induced nilpotent orbits, see [CM93, Chapter 7].Now we prove Theorem 4.2 for the case that G n = Sp 2n .By the assumption of Theorem 4.2, σ is of Type (I) and is of the following form: where σ neg is the unique irreducible negative unramified subrepresentation of the following induced representation with σ sn being the unique strongly negative unramified constituent of the following induced representation: As representations of general linear groups, it is known that for any given character χ and any integer k.By Lemma 5.1, we have By Theorem 3.6, Lemma 5.1, and by [CM93, Theorem 7.3.3]on formula for induced nilpotent orbits, for any p ∈ p m (σ), we have where 2k = ( l i=1 (2m i + 1)) − 1.To prove Theorem 4.2 in the case, it suffices to show the following lemma.
Lemma 5.2.The following identity Proof.Recall that and where given any On the other hand, we have .

Case (2):
To carry out the Sp 2n -collapse of p(ψ) − , we also need to list all the different odd k i 's between 2m 1 + 1 and k 1 as and k i,2 0 by (k i 0 +1, k i 0 −1), for 1 ≤ i ≤ s 0 .On the other hand, we obtain , for 1 ≤ i ≤ s 0 .Hence, we deduce that (5.1) still holds.
This completes the proof of the lemma.
The proof of Theorem 4.2 has been completed for G n = Sp 2n .

Proof of Theorem 4.2, G n = SO 2n+1
By the assumption of Theorem 4.2, σ is of Type (I) and is of the following form: where σ neg is the unique irreducible negative unramified subrepresentation of the following induced representation with σ sn being the unique strongly negative unramified constituent of the following induced representation: Also recall that l is even, and 0 As in Section 5, by Lemma 5.1, we have By Theorem 3.6, Lemma 5.1, and by [CM93, Theorem 7.3.3]on formula for induced nilpotent orbits, any p ∈ p m (σ) has the following upper bound , where 2k = l i=1 (2m i ).To prove Theorem 4.2 in this case, it suffices to show the following lemma.
Lemma 6.1.The following identity Proof.Recall that and where for any given partition On the other hand, we have .
Given any partition p of SO 2n+1 , it is known that (p is indeed an orthogonal partition.Hence we obtain that   [( Therefore, we only need to show that (6.1) where we omit the "(2m 1 − 1)1"-term if 2m 1 = 0.
We are going to rewrite the partition [ To proceed, we separate into the following cases: (1) k s ≤ 2m l ; (2) k s > 2m l .
This completes the proof of the lemma.
The proof of Theorem 4.2 has been completed for G n = SO 2n+1 .

Proof of Theorem 4.2, G n = O 2n
By the assumption of Theorem 4.2, σ is of Type (I) and is of the following form: where σ neg is the unique irreducible negative unramified subrepresentation of the following induced representation with σ sn being the unique strongly negative unramified constituent of the following induced representation: Also recall that l is even, and 0 As in Sections 5 and 6, by Lemma 5.1, we have By Theorem 3.6, Lemma 5.1, and by [CM93, Theorem 7.3.3]on formula for induced nilpotent orbits, any p ∈ p m (σ) has the following upper bound with 2k = l i=1 (2m i + 1).To prove Theorem 4.2 in this case, it suffices to show the following lemma.

Also recall that given any partition
By .
It is easy to see that is a partition of the following form where p i l with 1 ≤ i ≤ 2m 1 + 1, p i 2j with 1 ≤ i ≤ 2m l+1−2j − 2m l−2j and 1 ≤ j ≤ l−2 2 , and p k 0 with 1 ≤ k ≤ m 0 are all even; and p i 2j+1 with 1 ≤ i ≤ 2m l−2j − 2m l−2j−1 and 0 ≤ j ≤ l−2 2 are all odd; and finally Then the partition 2[( is equal to the following partition ].Following the recipe on carrying out the O 2n -collapse ([CM93, Lemma 6.3.8]),we obtain that the partition   [( is equal to the following partition can be written as Hence we obtain that This completes the proof of the lemma.
The proof of Theorem 4.2 has been completed for G n = O 2n .

On the wave front set of unramified unitary representations
In this last section, we study the wave front set of the unramified unitary representations for split classical groups G n = Sp 2n , SO 2n+1 , O 2n .Under assumptions on the leading orbits in the wave front set of negative representations, we determine the set p m (π) for general unramified unitary representations.This reduction has its own interests.
We have the following conjecture on the maximal partitions in the wave front set of negative representations.
Based on Conjecture 8.1, we obtain the explicit description on the maximal partitions in the wave-front set of general irreducible unramified unitary representations π of G n (F ).
Theorem 8.2.Assume Conjecture 8.1 is true.For any irreducible unramified unitary representation π of G n (F ), the maximal partitions in the wave-front set p(π) are given as follows: We remark that Ciubotaru, Mason-Brown, and Okada ([CMO21]) recently computed the maximal orbits in the wave front set of irreducible Iwahori-spherical representations of split reductive p-adic groups with "real infinitesimal characters", which partially proved Conjecture 8.1 and Theorem 8.2.This provides evidence for Conjecture 8.1.
By Lemma 5.1, we have By Lemma 5.1, and by [CM93, Theorem 7.3.3]on formula for induced nilpotent orbits, we obtain that Hence, by the assumption, to prove Theorem 8.2, it suffices to show the following lemma which will be proved case-by-case in the following subsections.

For any given partition
(1) When q v ≤ 2r l+k , we have (a) p u ≤ 2r l+k and (b) p u > 2r l+k .
(2) When q v > 2r l+k , we have (a) p u ≤ q v and (b) p u > q v .
when G n = Sp 2n ; 2n, when G n = SO 2n+1 or O α 2n , and the central character of ψ and the central characters of τ i 's satisfy the following constraints:i ω b i τ i = 1, when G n = Sp 2n or SO 2n+1 ; η α , Then, we prove Theorem 1.3.We first consider the cases of G n = Sp 2n , SO 2n+1 , O α 2n and leave the case of G n = U n to the end of the section.4.1.Unramified structure of Arthur parameters.For a given global Arthur parameter ψ ∈ Ψ 2 (G n ), Π ψ (ε ψ ) is the corresponding automorphic L 2 -packet.It is clear that the irreducible unramified representations, which are the local components of π ∈ Π ψ (ε ψ ), are determined by the local Arthur parameter ψ v at almost all unramified local places v of k.We fix one of the members, π ∈ Π ψ (ε ψ ), and describe the unramified local component π v at a finite local place v, where the local Arthur parameter j ) and τ s ∈ A cusp (GL 2as+1 ) are of orthogonal type for k+1 ≤ j ≤ k + l and k + l + 1 ≤ s ≤ k + l + 2t + 1.Similarly, we define I := {1, 2, . . ., k}, J := {k + 1, k + 2, . . ., k + l}, S := {k + l + 1, k + l + 2, . . ., k + l + 2t + 1} Gn when G n = SO 2n+1 .8.1.Proof of Lemma 8.3, G n = Sp 2n .By similar arguments as in the proof of the Sp 2n -case of Lemma 5.2, we only need to show that   [(