Fourier coefficients attached to small automorphic representations of ${\mathrm{SL}}_n(\mathbb{A})$
Olof Ahlén Henrik P. A. Gustafsson Axel Kleinschmidt Baiying Liu Daniel Persson
math.RT hep-th math.NT 11F70, 22E55, 11F30


equationsection


Fourier coefficients of small automorphic representations]Fourier coefficients attached to small automorphic representations of _n(𝔸)


O. Ahlén]Olof Ahlén
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

Am Mühlenberg 1

14476 Potsdam, Germany
olof.ahlen@aei.mpg.de

H. Gustafsson]Henrik P. A. Gustafsson
Department of Physics

Chalmers University of Technology

 SE-412 96 Gothenburg, Sweden
henrik.gustafsson@chalmers.se

A. Kleinschmidt]Axel Kleinschmidt
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

Am Mühlenberg 1

14476 Potsdam, Germany
Solvay Institutes 

ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium
axel.kleinschmidt@aei.mpg.de


B. Liu]Baiying Liu
Department of Mathematics

Purdue University

150 N. University St

West Lafayette, IN, 47907
liu2053@purdue.edu

D. Persson]Daniel Persson
Department of Mathematical Sciences 

Chalmers University of Technology

 SE-412 96 Gothenburg, Sweden
daniel.persson@chalmers.se

[2000]Primary 11F70, 22E55; Secondary 11F30







We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of _n(𝔸) are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro–Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory. 



[
    [
    December 30, 2023
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§ INTRODUCTION




 §.§ Background and motivation


Let F be a number field and  be the associated ring of adeles. 
Let  be a reductive algebraic group defined over F and π an (irreducible) automorphic representation of () as defined in <cit.>.

Fix a Borel subgroup B and let P ⊂ be a standard parabolic subgroup with Levi decomposition P=LU, and let ψ : U(F)\ U(𝔸)→ℂ^× be a global unitary character. 
Given any automorphic form φ∈π one can consider the following function on ():

    ℱ_U(φ, ψ; g)=_U(F)\ U(𝔸)φ(ug)ψ^-1(u)   du .

This can be viewed as a Fourier coefficient of the automorphic form φ with respect to the unipotent subgroup U. Fourier coefficients of automorphic forms carry a wealth of arithmetic and representation-theoretic information. For example, in the case of classical modular forms on the upper half-plane, Fourier coefficients are well-known to encode information about the count of rational points on elliptic curves. On the other hand, for higher rank Lie groups  their arithmetic content is not always transparent, but they always encode important representation-theoretic information. Langlands showed that the constant terms in the Fourier expansion of Eisenstein series provide a   source for automorphic L-functions <cit.>, and Shahidi extended this method (now called the Langlands-Shahidi method) to include also the non-constant Fourier coefficients <cit.>.

Theta correspondences provide realizations of Langlands functorial transfer between automorphic representations π and π' of two different groups  and '. In this context automorphic forms attached to minimal automorphic representations play a key role <cit.>. The wave front set of a minimal representation π_min of a group  is the closure of the smallest non-trivial nilpotent coadjoint orbit 𝒪_min of  <cit.>. The automorphic realizations of minimal representations are characterized by having very few non-vanishing Fourier coefficients <cit.>. Conversely, the method of descent <cit.> can be viewed as an inverse to the functorial lifting, in which an automorphic representation of a general linear group _n is transferred to a representation of a smaller classical group . Also in this case do Fourier coefficients of small representations enter in a crucial way. 

In general it is a difficult problem to obtain explicit formulas for Fourier coefficients for higher rank groups, let alone settle the question of whether an automorphic form φ can be reconstructed from only a subset of its Fourier coefficients.
For cusp forms on _n this is possible due to the Piatetski-Shapiro–Shalika formula <cit.> that allows to reconstruct φ from its Whittaker coefficients; i.e. the Fourier coefficients with respect to the unipotent radical N of the Borel subgroup B ⊂. These coefficients are sums of Eulerian Whittaker coefficients on subgroups of , and their non-archimedean parts can be obtained from the Casselman–Shalika formula <cit.> as described in <cit.>. However, even if this gives us complete control of the Fourier expansion with respect to N it does not automatically give us a way of calculating an arbitrary Fourier coefficient ℱ_U(φ, ψ; g) with respect to some other unipotent subgroup U. Such coefficients play an important role in the construction of L-functions, and also carry information about non-perturbative effects in string theory as described in section <ref>.

Expanding upon the classic results of <cit.>, Miller and Sahi proved in <cit.> that for automorphic forms φ attached to a minimal representation π_min of E_6 and E_7, any Fourier coefficient ℱ_U(φ, ψ; g) is completely determined by maximally degenerate Whittaker coefficients of the form

    _N(F)\ N(𝔸)φ(ng)ψ_α (n)^-1 dn  ,

where ψ_α is non-trivial only on the one-parameter subgroup of N corresponding to the simple root α. This result maybe viewed as a global version of the classic results of Moeglin-Waldspurger in the non-archimedean setting <cit.>, and Matumoto in the archimedean setting <cit.>. 


For the special cases of _3 and _4 the Miller–Sahi results were generalized in <cit.> (following related results in <cit.>) to automorphic forms attached to a next-to-minimal automorphic representation π_ntm. It was shown that any Fourier coefficient is completely determined by (<ref>) and coefficients of the following form

    ∫_N(F)\ N(𝔸)φ(ng)ψ_α, β (n)^-1 dn ,

where ψ_α, β is only supported on strongly orthogonal pairs of simple roots (α, β) which here reduces to that [E_α, E_β]=0 <cit.>. The main goal of the present paper is to use the techniques of <cit.>, in particular the notion of Whittaker pair, to extend the above results to all of _n. 



 §.§ Summary of results


We now summarize our main results. In the rest of this paper we will consider _n for n ≥ 5 where we have fixed a Borel subgroup with the unipotent radical N. 
Let also T be the diagonal elements of _n(F) and, for a character ψ_0 on N, let T_ψ_0 be the stabilizer of ψ_0 under the action [h.ψ_0](n) = ψ_0(h n h^-1) for h ∈ T. 

Define

    Γ_i(ψ_0) 
            (_n-i(F))_Ŷ_n-i(F)     1 ≤ i ≤ n-2 
    
            (T_ψ_0∩ T_ψ_α_n-1)  T_ψ_0    i = n-1  ,

where (_n-i(F))_Ŷ is the stabilizer of Ŷ = ^t(1, 0, 0, …, 0) ∈_(n-i)× 1(F) and consists of elements 
[ 1 ξ; 0 h ], with h ∈_n-i-1(F) and ξ∈_1 × (n-i-1)(F).
When ψ_0 = 1 we write Γ_i(1) as Γ_i.

Similarly, let (_j(F))_X̂ be the stabilizer of X̂ = (0, …, 0, 1) ∈_1× j(F) with respect to multiplication on the right, ψ_0 a character on N, and define

    Λ_j(ψ_0) 
            (_j(F))_X̂_j(F)     2 < j ≤ n 
    
            (T_ψ_0∩ T_ψ_α_1)  T_ψ_0    j = 2   ,

where, again, we denote Λ_j(1) = Λ_j.

Define also the embeddings ι, ι̂: _n-i→_n for any 0 ≤ i ≤ n-1 as

    ι(γ) =
        [ I_i   0;   0   γ ]    ι̂(γ) =
        [   γ   0;   0 I_i ]  ,

where we for brevity suppress their dependence on i. Note that for i = 0, they are just the identity maps for _n.

The following theorem expands an automorphic form φ attached to a small automorphic representation of _n in terms of highly degenerate Whittaker coefficients similar to how cusp forms on _n can be expanded in terms of Whittaker coefficients with the Piatetski-Shapiro–Shalika formula <cit.>. Expansion of non-cuspidal automorphic forms on _n in terms of Whittaker coefficients were discussed in <cit.>.


    
    Let π be a minimal or next-to-minimal irreducible automorphic representation of _n(), and let φ∈π.
    
        
  *  If π=π_min, then φ has the expansion
            
    φ(g) = _N(F)\ N(𝔸)φ(ng)   dn + ∑_i=1^n-1∑_γ∈Γ_i _N(F)\ N(𝔸)φ(n ι(γ) g) ψ^-1_α_i(n)   dn   .

        
  *  If π=π_ntm,  then φ has the expansion
            
    φ(g) = _N(F)  N()φ(vg)   dv + ∑_i=1^n-1∑_γ∈Γ_i_N(F) N()φ(v ι(γ) g) ψ^-1_α_i(v)   dv +
    
                    + ∑_j=1^n-3∑_i=j+2^n-1∑_γ_i ∈Γ_i(ψ_α_j) 
    γ_j ∈Γ_j_N(F) N()φ(v ι(γ_i) ι(γ_j) g) ψ^-1_α_j, α_i (v)   dv  .

    


Note that the Whittaker coefficients in the last sum of case <ref> have characters supported on two strongly orthogonal (or commuting) simple roots. 
As mentioned in section <ref> and further described in <cit.>, the Whittaker coefficients are sums of Eulerian Whittaker coefficients on smaller subgroups _n, whose non-archimedean parts can be computed by the Casselman–Shalika formula <cit.>. 
The more degenerate a Whittaker coefficient is the smaller the subgroup we need to consider (and on which character becomes generic). Thus, maximally degenerate Whittaker coefficients, and the ones with characters supported on two commuting simple roots become particularly simple and are, in principle, one, or a product of two, known _2 Whittaker coefficients respectively.

Next, we consider Fourier coefficients on maximal parabolic subgroups. 
Let P_m the maximal parabolic subgroup of SL_n with respect to the simple root α_m and let U = U_m be the unipotent radical and L_m be the corresponding Levi subgroup which stabilizes U_m under conjugation. 
For an element l∈ L_m(F) and a character ψ_U on U_m we obtain another character ψ_U^l by conjugation as

    ψ_U^l(u) = ψ_U(lul^-1)   .
 

Fourier coefficients ℱ_U with conjugated characters are related by l-translates of their arguments

    ℱ_U(φ, ψ_U^l; g)    = _U(F)  U()φ(ug) ψ_U^-1(lul^-1)   du 
       = _U(F)  U()φ(l^-1u'lg) ψ_U^-1(u')   du' = ℱ_U(φ, ψ_U; lg)   ,

where we have first made the variable substitution u' = lul^-1 and then used the automorphic invariance since l ∈ L_m(F). 
This means that we only need to compute the Fourier coefficients of one character per L_m(F)-orbit.

We show in section <ref> that a character can be parametrized by an element y ∈ g by (<ref>) denoted by ψ_y, which, under conjugation, satisfies ψ_y^l = ψ_l^-1 y l according to (<ref>).

In section <ref> and appendix <ref>, we describe these orbits  following <cit.> and construct standard characters ψ_y(Y_r(d)) on U_m based on anti-diagonal (n-m) × m rank r matrices Y_r(d), where d ∈ F^×/(F^×)^2 for n = 2r = 2m and d = 1 otherwise (in which case we suppress the d), and y(Y_r(d)) is defined as

    y(Y_r(d)) =
       [    0_m      0; Y_r(d)  0_n-m ] .


Let π be a minimal or next-to-minimal automorphic representations, and

    r_π = 
        
            1    if π is a minimal automorphic representation
    
            2    if π is a next-to-minimal automorphic representation.


We will show that only the characters with rank r ≤ r_π≤ 2 give non-vanishing Fourier coefficients.
Let us briefly define the characters with rank r ≤ 2 which will be used in the next theorem, postponing a more general definition to section <ref>.
The rank zero character is the trivial character ψ_y(Y_0) = 1 and the corresponding Fourier coefficient has been computed in <cit.> as reviewed in <cit.>.
The rank one character is ψ_y(Y_1) = ψ_α_m and the rank two character can be defined as follows

    ψ_y(Y_2)(u) = ψ(u_m,m+1 + u_m-1, m+2)      u ∈ U_m()   .


The following theorem, together with the known constant term, then allows us to compute any Fourier coefficient with respect to the unipotent radical of a maximal parabolic subgroups for automorphic forms attached to minimal and next-to-minimal automorphic representations in terms of Whittaker coefficients.


    
    Let π be a minimal or next-to-minimal irreducible automorphic representation of _n(), and let r_π be 1 or 2 respectively (which denotes the maximal rank of the character matrix Y_r).
    Let also, φ∈π, P_m be the maximal parabolic subgroup described above with its associated subgroups U≡ U_m and L_m, and let ψ_U be a non-trivial character on U_m with Fourier coefficient 
    
    ℱ_U(φ, ψ_U; g) = ∫_U_m(F) U_m()φ(ug) ψ_U^-1(u)   du  .
 
    Then, there exists an element l ∈ L_m(F) such that 
    
    ℱ_U(φ, ψ_U; g) = ℱ_U(φ, ψ_y(Y_r(d)); lg)

     for some standard character ψ_y(Y_r(d)) described above and in the proof.

    Additionally, all ℱ_U(φ, ψ_y(Y_r(d)); lg) for r > r_π vanish identically.
    The remaining (non-constant) coefficients can be expressed in terms of Whittaker coefficients on N as follows.
    
        
  *  If π = π_min:
            
                    ℱ_U(φ, ψ_y(Y_1); g)    = _N(F)N() φ(ng) ψ_α_m^-1(n)   dn   .    
            
        
  *  If π = π_ntm:
            
                    ℱ_U(φ, ψ_y(Y_1); g)    =
                [t]
                    _[N] φ(ng) ψ_α_m^-1(n)   dn + 
 
                + ∑_j=1^m-2 ∑_γ∈Λ_j(ψ_α_m)  _[N] φ(nι̂(γ) g) ψ_α_j, α_m^-1(n)   dn + 
 
                + ∑_i=m+2^n-1 ∑_γ∈Γ_i(ψ_α_m)   _[N] φ(n ι(γ) g) ψ_α_m, α_i^-1(n)   dn   . 
               
        
        
  *  If π = π_ntm:
            
                    ℱ_U(φ, ψ_y(Y_2); g)    = _C()_N(F)N() φ(nωcg)ψ_α_1,α_3^-1(n)  dn dc  ,    
            
            where ω is the Weyl element mapping the torus elements 
            
    (t_1, t_2, …, t_n) ↦ (t_m-1, t_m+2, t_m, t_m+1, t_1, t_2, …, t_m-2, t_m+3, t_m+4, …, t_n) ,

            and the subgroup C of U_m will be detailed in the proof in section <ref>.
    


As described in detail in section <ref>, F-rational nilpotent orbits of _n are characterized by ( p, d) where p is a partition of n and d ∈ F^×/(F^×)^k with k = ( p). 
If k = 1 we will often suppress the extra d = 1 and only write out the partition.

There we will also see that, for each orbit, there are natural choices of unipotent subgroups and characters related by conjugations with elements γ∈_n(F) and the corresponding Fourier coefficients (<ref>) are related by γ-translates of their arguments. 

The orbits may be partially ordered and the minimal and next-to-minimal orbits are described by the partitions [21^n-2] and [2^21^n-4], respectively. 
Besides the trivial partition, these are the only partitions whose associated Fourier coefficients are non-vanishing for φ in a minimal or next-to-minimal irreducible automorphic representation. 
In section <ref> we choose standard representatives for these orbits and specify the associated standard Fourier coefficients which we denote by ℱ^[211…] and ℱ^[221…].
For n≥ 5, we have that the trivial, minimal and next-to-minimal orbit all have k=1.

The following theorems express these standard Fourier coefficients associated with the two partitions above in terms of Fourier coefficients on maximal parabolic subgroups that, in turn, were written in terms of Whittaker coefficients in theorem <ref>.


     
    Let π be an irreducible automorphic representation of _n(), φ∈π and Y = Π_i=3^n X_e_i - e_2. 
    Then,
    
    ℱ^[211…](φ; g) = ∑_y ∈ Y(F) _U_1(F) U_1()φ(u y^-1 g) ψ_α_1^-1(u)   du    ,

    where U_1 is the unipotent radical of P_1 consisting of the first row of N. The Fourier coefficient ℱ^[211…] is for a particular standard choice of orbit representative detailed in the proof; all other choices are related simply by _n(F) translation.



    
    Let π be an irreducible automorphic representation of _n(), φ∈π, Y' = ∏_i=5^n X_e_i-e_4∏_i=5^n X_e_i-e_3 and ω be the Weyl element mapping the torus elements 
    
    (t_1, t_2, …, t_n) ↦ (t_1, t_3, t_4, t_2, t_5, t_6, …, t_n)  .

    Then,
    
    ℱ^[221…](φ; g) = ∑_y ∈ Y'(F) _U_2(F) U_2()φ(u y^-1ω g) ψ_y(Y_2)^-1(u)   du   ,

    where U_2 is the unipotent radical of P_2 consisting of the first two rows of N and ψ_y(Y_2) is defined in (<ref>) with m = 2. The Fourier coefficient ℱ^[2^21…] is for a particular standard choice of orbit representative detailed in the proof; all other choices are related simply by _n(F) translation.






 §.§ Applications in string theory


String theory is a quantum theory of gravity describing maps X: Σ→ M, where Σ is a Riemann surface (the string worldsheet) and M is a ten-dimensional pseudo-Riemannian manifold (spacetime). Its low-energy limit is a supersymmetric extension of Einstein's theory of gravity in 10 dimensions coupled to additional matter in the form of scalar fields Φ : M→ℂ and differential forms on spacetime M. Our main focus here will be the scalar fields. The scalar fields parametrize the space of string theory vacua, i.e. the moduli space ℳ. 

To make contact with a lower-dimensional world, one choice is to decompose spacetime into 

    M=ℝ^1,9-n× T^n ,

 where ℝ^1,9-n is the flat Minkowski space in 10-n dimensions and T^n  is an n-dimensional torus. In the limit when the size of the torus is small, the physics looks effectively (10-n)-dimensional and one says that the theory has been compactified. As the size of the torus is  increased the moduli space ℳ gets larger and larger due to an increased number of scalar fields Φ. The moduli space for this toroidal compactification is always of the form 

    ℳ=G(ℤ)\G(ℝ)/K ,

where G(ℝ) is a semi-simple Lie group in its split real form, K  its maximal compact subgroup and G(ℤ) an arithmetic subgroup. The group G(ℤ) is known as the U-duality group and is a symmetry of the full quantum string theory. The extreme case when n=0, i.e. for no compactification, the moduli space is given by 

    ℳ=_2(ℤ)\_2(ℝ)/SO_2  .

 Another extreme case is n=6, corresponding to four space-time dimensions, for which the moduli space is given by <cit.>

    ℳ= E_7(ℤ)\E_7(ℝ)/(SU_8/ℤ_2) .

Here E_7(ℝ) is the split real form and E_7(ℤ) its Chevalley group of integer points. The sequence of groups in between are obtained by successively removing nodes from the E_7 Dynkin diagram; see table <ref> for the complete list.

Constraints from U-duality and supersymmetry ensure that certain quantum corrections to Einstein's gravitational theory involve functions f : ℳ→ℂ that must be eigenfunctions of the ring of G(ℝ)-invariant differential operators. In particular they are eigenfunctions of the Laplacian on G(ℝ)/K with  specific eigenvalues. In addition, they must have  well-behaved growth properties in certain limits corresponding to `cusps' of ℳ. Such quantum corrections are therefore controlled by automorphic forms on ℳ.

It turns out that the relevant automorphic forms are very special and are precisely those attached to a minimal and next-to-minimal automorphic representation of the groups G <cit.>. The Fourier coefficients of such automorphic forms therefore have a direct physical interpretation: the constant terms encode perturbative quantum corrections, while the non-constant terms correspond to non-perturbative, instanton, effects <cit.>. For a recent book on automorphic representations and the connection with string theory, see <cit.>.

Fourier coefficients with respect to different choices of parabolic subgroups P⊂G correspond to different limits in string theory, and reveal different types of effects. The ones of main interest are certain maximal parabolic subgroups. Let P_α= L_α U_α denote the maximal parabolic whose Levi subgroup is L_α = M_α×_1, where M_α is obtained by removing the node in the Dynkin diagram of G corresponding to the simple root α. There are three types of maximal parabolics of main interest  in string theory (the numbering of nodes are according to the Bourbaki convention of the exceptional Lie algebras):



  * P_α_1: this is the perturbative, or string theory, limit where the Levi is of orthogonal type M_α_1=D_n;


  * P_α_2: this is the M-theory limit where the Levi is of type M_α_2=A_n;


  * P_α_n+1: this is the decompactification limit where the Levi is of exceptional type M_α_n+1=E_n (for n<6 these are strictly speaking not exceptional, but given by table <ref>).


Theorem <ref>, together with its counterpart in <cit.>, then provides explicit results for the Fourier coefficients of automorphic forms in all these parabolics for the cases n=2 or n=3 when the symmetry groups are _2×_3 or _5, respectively. The case of _5  will be treated in detail in section <ref>.







 §.§ Acknowledgements


We have greatly benefitted from many discussions with Dmitry Gourevitch, Joseph Hundley, Stephen D. Miller  and Siddhartha Sahi. We gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University during the program on “Automorphic forms, mock modular forms and string theory” in the fall of 2016 during which part of the research for this paper was performed. The fourth named author is partially supported by NSF grant DMS-1702218 and by a start-up fund from the Department of Mathematics at Purdue University.




§ NILPOTENT ORBITS AND FOURIER COEFFICIENTS




In this section, first, we introduce Whittaker pairs and nilpotent orbits with their associated Fourier coefficients following <cit.>, which is slightly more general and easier to use than the one given in <cit.>. 
Then we recall the parametrization of F-rational nilpotent orbits of _n in terms of partitions of n from <cit.> and a lemma for exchanging roots in Fourier integrals from <cit.>. 


As before, let F be a number field,  be the adele ring of F and fix a non-trivial additive character ψ on F. 
Let also  be a reductive group defined over F, or a central extension of finite degree, and let g be the Lie algebra of (F). For a semi-simple element s ∈ g, let g^s_i be defined as the eigenspace of s in g with eigenvalue i under the adjoint action decomposing g to a direct sum of eigenspaces over different eigenvalues. For any r ∈, we further define g^s_≥ r = ⊕_r' ≥ r g^s_r' and similarly for other inequality relations. For an element X ∈ g, we will also denote the centralizer of X in g as

    g_X = { x ∈ g  | [x,X]=0} .


Furthermore, a semi-simple element s is called rational semi-simple if all of its eigenvalues under the adjoint action on g are in . For such a rational semi-simple element s and a non-trivial nilpotent element u ∈ g^s_-2 we call the ordered pair (s, u) a Whittaker pair. If, for such a pair, s is also called a neutral element for u or (s, u) a neutral pair if the map g^s_0 → g^s_-2 : X ↦ [X,u] is surjective or, equivalently <cit.>, s ∈(ad(u)).

An sl_2-triple is an ordered triple (u, s, v) of elements in g that satisfy the standard commutation relations for sl_2, 

    [s, v] = 2v      [s, u] = - 2u      [v, u] = s ,

where u is called the nil-negative element, v is called the nil-positive element and s is a neutral element for u. 
We have, from <cit.>, that a Whittaker pair (s,u) comes from an sl_2-triple (u,s,v) if and only if s is a neutral element for u.

By the Jacobson–Morozov theorem, there exists an sl_2 triple for any nilpotent element u ∈ g. 
Moreover, the -conjugacy classes of sl_2-triples are one-to-one with the nilpotent orbits 𝒪_X = {gXg^-1| g ∈(F)} in g <cit.>.  

We will now construct the Fourier coefficient that is associated to a Whittaker pair (s, u). The pair defines a unipotent subgroup N_s and a character ψ_u on N_s as follows. Following <cit.>, let

    n_s = g^s_>1⊕g^s_1 ∩g_u   ,

which is a nilpotent subalgebra of g, and define N_s = exp( n_s) as the corresponding unipotent subgroup of . Then ψ_u, defined by

    ψ_u(n) = ψ(⟨ u, log(n)⟩)      n ∈ N_s() ,

is a character on N_s() where ⟨·, ·⟩ is the Killing form.

Note that if the Whittaker pair (s,u) comes from an 𝔰𝔩_2-triple (u,s,v), then, by sl_2 representation theory, (s) has integer eigenvalues with a graded decomposition of the Lie algebra g = ⊕_i ∈g^s_i and g_u ⊂⊕_i ≤ 0g^s_i <cit.>,
and thus, 

    𝔫_s=𝔤^s_≥ 2     (for neutral  s).


Let π be an automorphic representation of () and φ an automorphic form attached to π.
The Fourier coefficient associated with a Whittaker pair (s, u) is

    _s,u(φ)(g) = _N_s(F)  N_s()φ(ng)ψ^-1_u(n)dn,    g ∈()   ,

and let _s,u(π)={_s,u(φ) |φ∈π}. 
For convenience, we introduce the following notation for a unipotent subgroup U

    [U] = U(F)  U()   .


Consider the Fourier coefficient associated with a neutral Whittaker pair (s, u), and let (s', u') = (γ sγ^-1, γ u γ^-1) which is also neutral for any γ∈ G(F). 
Because of the invariance of the Killing form we have that ψ_u'(n') = ψ_u(γ^-1 n' γ) where n' ∈ [N_s'], and because of (<ref>) we have that N_s' = γ N_s γ^-1.
Thus, with a variable substitution n' = γ n γ^-1,

    _s',u'(φ)(g)    = _[γ N_s'γ^-1]φ(n'g) ψ^-1_u(γ^-1 n' γ)   dn 
       = _[N_s]φ(γ n γ^-1 g)  ψ_u^-1(n)   dn = _s,u(φ)(γ^-1g) ,

using the automorphic invariance of φ. 
Note the resemblance with (<ref>) where we made a conjugation keeping N_s invariant.
In particular, (<ref>) means that if _s,u vanishes identically then so do all Fourier coefficients associated to neutral Whittaker pairs (s',u') where u' ∈_u.
For an F-rational nilpotent orbit , we say that the coefficients _s,u with neutral s and u ∈ are Fourier coefficients attached to the nilpotent orbit .

We define the (global) wave-front set 𝒲ℱ(π) of an automorphic representation π of () as the set of nilpotent orbits 𝒪 such that ℱ_s,u(π) is non-zero, for some (and therefore all) neutral Whittaker pairs (s,u) with u ∈𝒪.
Note that nilpotent orbits can be partially ordered with respect to the inclusion of Zariski closures ' ≤ if '⊆.

We recall <cit.> as follows. 


Let π be an automorphic representation of (), let (s,u) be a Whittaker pair, and (h, u) a neutral Whittaker pair such that _h,u(π) is zero. 
Then, _s,u(π) is zero.


This means that if u ∈ where 𝒲ℱ(π) then, for any Whittaker pair (s, u), not necessarily neutral, the associated Fourier coefficient ℱ_s, u(φ) vanishes identically for φ∈π.

In this paper, we focus on the group _n where we parametrize a character on N_s by u ∈ g^s_-2 as 

    ψ_u(n)=ψ( (ulog(n)))      n ∈ N_s().

Then, for any l in the normalizer of N_s() in ()

    ψ_y^l(x)    = ψ_y(l x l^-1) = ψ(( y log(l x l^-1))) = ψ(( y l log(x) l^-1)) 
       = ψ((l^-1 y l log(x))) = ψ_l^-1 y l(x)   .


The nilpotent orbits of _n can be described by partitions p of n.
Let us characterize the F-rational orbits of _n following <cit.>.


Let p=[p_1 p_2 ⋯ p_r] be an ordered partition of n, with p_1 ≥ p_2 ≥…≥ p_r and let m = (p)=(p_1, p_2, …, p_r). 
For d ∈ F^×, define D(d) = (1, 1, …, 1, d) and let also J_p be the standard (lower triangular) Jordan matrix corresponding to p: J_p = (J_[p_1], J_[p_2], …, J_[p_3]), where J_[p] is a p× p matrix with non-zero elements only on the subdiagonal which are one.



  * For each d ∈ F^×, the matrix D(d)J_p is a representative of an F-rational nilpotent orbit of _n parametrized by p, and conversely, every orbit parametrized by p has a representative of this form. 
We say that the F-rational orbit represented by D(d)J_p is parametrized by (p,d).


  * The _n(F)-orbits represented by D(d)J_p and D(d')J_p' coincide if and only if p=p' and d ≡ d' in F^×/(F^×)^m.




The F-rational orbit ([322], 1) of _7 is represented by

    J_[322]=(J_[3], J_[2], J_[2])=[ 0 0 0 0 0 0 0; 1 0 0 0 0 0 0; 0 1 0 0 0 0 0; 0 0 0 0 0 0 0; 0 0 0 1 0 0 0; 0 0 0 0 0 0 0; 0 0 0 0 0 1 0 ] .




    Over F the F-rational orbits for different d become the same, meaning that they are completely characterized by partitions of n.
    There is partial ordering for partitions that agrees with the partial ordering of the F-orbits, where [p_1p_2… p_r] ≤ [q_1q_2… q_r] (possibly padded by zeroes) if <cit.>
    
    ∑_1≤ j ≤ n p_j ≤∑_1≤ j ≤ n q_j   for  1 ≤ n ≤ r   .

   
    The Zarisky topology over F is induced from that of F which means that we can use this partial ordering of partitions for the F-rational orbits as well. 
    Thus, when discussing the partial ordering of orbits or the closure of orbits we will sometimes not specify the F-rational orbit, but only the partition, that is, the _n(F)-orbit.



An automorphic representation π of _n() is called minimal if 𝒲ℱ(π) is the set of orbits in the closure of the minimal (non-trivial) orbit which is represented by the partition [21^n-2], and it is called next-to-minimal if it is instead the set of orbits in the closure of the next-to-minimal orbit [2^21^n-4]. 


We will now recall a general lemma for exchanging roots in Fourier coefficients from <cit.>. 
In <cit.>, the groups considered are quasi-split classical groups, but the lemma holds for any connected reductive group with exactly the same proof. 


Let G be a connected reductive group defined over F and let C be an F-subgroup of a maximal unipotent subgroup of G. 
Let also ψ_C be a non-trivial character on [C] = C(F)  C(), and X, Y two unipotent F-subgroups satisfying the following conditions:


  * X and Y normalize C;

  * X ∩ C and Y ∩ C are normal in X and Y, respectively, (X ∩ C)  X and (Y ∩ C)  Y are abelian;

  * X() and Y() preserve ψ_C under conjugation;

  * ψ_C is trivial on (X ∩ C)() and (Y ∩ C)();

  * [X, Y] ⊂ C;

  * there is a non-degenerate pairing 
    
    (X ∩ C)()    × (Y ∩ C)() →^×
     
            (x,y)    ↦ψ_C([x,y])

    which is multiplicative in each coordinate, and identifies [4](Y ∩ C)(F)  Y(F) with the dual of
X(F)(X ∩ C)()  X(),
and
(X ∩ C)(F)  X(F) with the dual of
Y(F)(Y ∩ C)()  Y().


Let B =CY and D=CX, and extend ψ_C trivially to characters of [B]=B(F) B() and [D]=D(F) D(),
which will be denoted by ψ_B and ψ_D respectively.


Assume that (C, ψ_C, X, Y) satisfies all the above conditions. Let f be an automorphic form on G(). Then for any g ∈G(), 

    ∫_[B] f(vg) ψ_B^-1(v) dv = ∫_(Y ∩ C) ()  Y()∫_[D] f(vyg) ψ_D^-1(v)  dv dy .



For simplicity, we will use ψ_C to denote its extensions ψ_B and ψ_D when using the lemma.






§ PROOF OF THEOREM <REF>


Before we prove Theorem <ref> in this section, let us first introduce a few definitions and useful lemmas. 

Let V_i be the unipotent radical of the parabolic subgroup of type (1^i,n-i), that is, the parabolic subgroup with Levi subgroup (_1)^i × GL_n-i together with a determinant one condition. Then, N= V_n = V_n-1 is the unipotent radical of the Borel subgroup and V_i can be seen as the first i rows of N.
For 1 ≤ i ≤ n-1, let α_i = e_i-e_i+1 be the i-th simple root of _n, and let ψ_α_i be the character of N defined by 

    ψ_α_i(n)= ψ(n_i,i+1), ∀ n ∈ N()   .

For a list of simple roots, we let ψ_α_i_1, …, α_i_m = ψ_α_i_1⋯ ψ_α_i_m and we also regard ψ_α_j for j ≤ i as a character of V_i via restriction. 

Also, let R_i+1 be the subgroup of V_i+1, consisting of the elements v with conditions that v_p,q=0, for all 1 ≤ p ≤ i and p < q ≤ n, that is R_i+1 consists of the row i+1 in V_i+1. It is clear that R_i+1 V_i  V_i+1 is an abelian subgroup of V_i+1.

For a character ψ_N on N, we say that ψ_N is trivial along a simple root α_i if the restriction of ψ_N to R_i is identically zero.


    For _5 we have that 
    
    V_3 = {[ 1 * * * *;   1 * * *;     1 * *;       1  ;         1;   ]}    
            R_3 = {[ 1        ;   1      ;     1 * *;       1  ;         1 ]}  .



Thus, we have that [R_i] ≅ (F )^n-i and the dual of [R_i] is F^n-i, which can be identified with the nilpotent subalgebra ^t  r_i(F) = log(^tR_i(F)), where ^tR_i(F) is the transpose of R_i(F).
Given y ∈^t r_i(F), the corresponding character ψ_y on [R_i] is given by (<ref>) as

    ψ_y(x) = ψ( (y log x)),   ∀ x ∈ [R_i]   .




     For SL_5 with R_3 above, let
     
    y = 
             [   0                ;       0            ;           0        ;         y_1   0    ;         y_2       0 ]∈^t r_3(F)     
             x =
             [   1                ;       1            ;           1 x_1 x_2;               1    ;                   1 ]∈ [R_3]   .

     Then, ψ_y(x) = ψ((y log x)) = ψ(y_1 x_1 + y_2 x_2).


Define

    (·) = (·) -1/n((·))

and let s = s_V_i

    s_V_i = (2(i-1), 2(i-2), …, 0, -2, …, -2)

for which g^s_1 = ∅ and n_s =  g^s_≥ 2 with the corresponding N_s = V_i. In particular, we have s_N = s_V_n-1= (2(n-2), …, 0, -2)


    
    Let φ be an automorphic form on _n(). Then, for 1 ≤ i ≤ n-2,
    
    ∑_y ∈^t r_i(F) 
     y ≠ 0 _[R_i]φ(xg) ψ^-1_y(x)   dx =
                ∑_γ∈Γ_i _[R_i]φ(x ι(γ) g) ψ^-1_α_i(x)   dx ,

    where Γ_i is defined in (<ref>) and ι(γ) in (<ref>).


We note that the left-hand side of the equation in this lemma equals φ(g) up to constant terms corresponding to y=0.


    With Y ∈_(n-i)× 1(F), we parametrize y ∈^t r_i(F) as
    
    y(Y) =
            [ 0_i-1     0     0;     0     0     0;     0     Y 0_n-i ]  .

    Let Ŷ = ^t(1, 0, …, 0) ∈_(n-i)× 1(F). Then the surjective map _n-i(F) →_(n-i) × 1(F)^× defined by γ↦γ^-1Ŷ gives that 
    
    _(n-i) × 1(F)^× (_n-i(F))_Ŷ_n-i(F) = Γ_i
 from (<ref>).
    We then have that,
    
    ∑_y ≠ 0 _[R_i]φ(xg) ψ^-1_y(x)   dx = ∑_γ∈Γ_i_[R_i]φ(xg) ψ^-1_y(γ^-1Ŷ)(x)   dx   .

    
    We now rewrite the character using that for any Y ∈_n-i(F)
    
    y(γ^-1Y) = [  0_i-1      0      0;      0      0      0;      0 γ^-1 Y  0_n-i ] =
            [ I_i-1     0     0;     0     1     0;     0     0  γ^-1 ][ 0_i-1     0     0;     0     0     0;     0     Y 0_n-i ][ I_i-1     0     0;     0     1     0;     0     0     γ ] = 
            l^-1 y l ,

    where we have introduced l = ι(γ) and denoted y(Y) simply as y, which according to (<ref>) gives, for any x ∈ [R_i], that
    
    ψ_ y(γ^-1 Y)(x) = ψ_l^-1 y l(x) = ψ_y(l x l^-1)   .


    The element l is in the Levi subgroup of the parabolic subgroup corresponding to V_i, meaning that it preserves V_i under conjugation. In particular, it also normalizes R_i since for x ∈ R_i parametrized by X ∈_1× n-1
    
    l x(X) l^-1 = 
            [ I_i-1     0     0;     0     1     0;     0     0     γ ][ I_i-1     0     0;     0     1     X;     0     0 I_n-i ][ I_i-1     0     0;     0     1     0;     0     0  γ^-1 ] =
            [  I_i-1      0      0;      0      1 X γ^-1;      0      0  I_n-i ] = x(X γ^-1)  .


    We can thus make the variable substitution lxl^-1→ x in (<ref>) to obtain
    
    ∑_γ∈Γ_i_[R_i]φ(x l g) ψ^-1_y(Ŷ)(x)   dx   ,

    where we have used the fact that φ is left-invariant under l^-1. Noting that ψ_y(Ŷ) = ψ_α_i this proves the lemma.


We will now state a similar lemma for the last row R_n-1, that needs to be treated separately. 
The freedom in choosing a character ψ_0 in this lemma will be of importance later.


    
    Let φ be an automorphic form on _n(). Then, for any character ψ_0 on N trivial on 
    R_n-1 and along (at least) two adjacent simple roots not including α_n-1,
    
    ∑_y ∈^t r_n-1(F) 
     y ≠ 0 _[R_n-1]φ(xg) ψ^-1_y(x)   dx =
                ∑_γ∈Γ_n-1(ψ_0) _[R_n-1]φ(x ι(γ) g) ψ^-1_α_n-1(x)   dx ,

    where Γ_n-1(ψ_0) is defined in (<ref>).

 

    With Y ∈ F, we parametrize y ∈^t r_n-1(F) as
    
    y(Y) =
            [ 0_n-2     0     0;     0     0     0;     0     Y     0 ]  .

    
    We recall from page tpsi0 that T_ψ_0 is the subgroup of diagonal elements in _n(F) stabilizing ψ_0 under conjugation of its argument and that y ∈^t  r_n-1(F)  F. 
    The map T_ψ_0→^t  r_n-1(F)^× : h ↦ h^-1 y(1) h is surjective, which can be shown as follows.
    The character ψ_0 is, by assumption, trivial along at least two adjacent simple roots not including α_n-1. 
    Pick such a pair α_j-1 and α_j where 2 ≤ j ≤ n-2 and for an arbitrary m ∈ F^× let h = (1, …, 1, m, 1, … 1, 1/m) where the first non-trivial element is at the jth position. 
    Then h ∈ T_ψ_0 since y_0 ∈^t  n corresponding to ψ_0 is zero at both rows and columns j and n and h ↦ y(m) 
    
    Because of (<ref>) we have that the centralizer of y(1) in T is T_ψ_α_n-1, and thus,
    
    ^t  r_n-1(F)  (T_ψ_0∩ T_ψ_α_n-1)  T_ψ_0 = Γ_n-1(ψ_0)   .


    We then have that 
    
    ∑_y ∈^t r_n-1(F) 
     y ≠ 0 _[R_n-1]φ(xg) ψ^-1_y(x)   dx = ∑_γ∈Γ_n-1(ψ_0) _[R_n-1]φ(xg) ψ^-1_γ^-1 y(1) γ(x)   dx 
         = ∑_γ∈Γ_n-1(ψ_0) _[R_n-1]φ(xg) ψ^-1_y(1)(γ x γ^-1) = ∑_γ∈Γ_n-1(ψ_0) _[R_n-1]φ(x γ g) ψ^-1_α_n-1(x)   ,

    after making the variable change γ x γ^-1→ x, which concludes the proof.



    
    For n ≥ 5 any character ψ_0 on N that is non-trivial along at most a single simple root which is not α_n-1 satisfies the character condition in lemma <ref>.


The following lemma will be used to iteratively expand in rows. The lemma, which is valid for any automorphic representation, will be followed by two corollaries that specialize to the minimal and next-to-minimal representations respectively.


    
    Let φ be an automorphic form on _n(), 1 ≤ i ≤ n-2, and ψ_0 be a character on N trivial on the complement of V_i in N. For i = n-2 we also require that ψ_0 is trivial along (at least) two adjacent simple roots not including α_n-1. Then, 
    
    _[V_i]φ(vg) ψ^-1_0(v) dv = _[V_i+1]φ(vg) ψ^-1_0(v)   dv +
       + ∑_γ∈Γ_i+1(ψ_0) _[V_i+1]φ(v ι(γ) g) ψ^-1_0(v) ψ^-1_α_i+1(v)   dv   .


 
    For x ∈ R_i+1(F) and v ∈ V_i() we have that φ(xvg) = φ(vg) and can thus Fourier expand along the abelian unipotent R_i+1 as

    φ(vg) = ∑_y ∈^t  r_i+1(F)_[R_i+1]φ(xvg) ψ^-1_y(x)   dx   .

Then, using lemma <ref> (for i+1 ≤ n-2) or lemma <ref> (for i+1 = n-1)

    φ(vg) = _[R_i+1]φ(xvg)   dx + ∑_γ∈Γ_i+1(ψ_0) _[R_i+1]φ(xι(γ)vg) ψ^-1_α_i+1(x)   dx 
        .

    
Let v ∈ V_i be parametrized as

    v = 
        [       A       B;       0 I_n-i-1 ] ,

where A ∈_(i+1) × (i+1) is upper unitriangular and B ∈_(i+1) × (n-i-1) with the elements in the last row being zero. 
Since B does not intersect the abelianization [N,N] N (that is, the Lie algebra of B does not contain any generator of a simple root), we have, by assumption, that ψ_0 only depends on A. 
Similarly, we parametrize x ∈ R_i+1 as

    x =
        [   I_i+1      B';       0 I_n-i-1 ] ,

where B' ∈(i+1) × (n-i-1) with non-zero elements only in the last row. 
Then,

    xv =
        [       A  B + B';       0 I_n-i-1 ]  ,

which means that ψ_0(v) = ψ_0(xv), and since ψ_α_i+1 only depends on the first column in B' which is the same as for B + B', we also have that ψ_α_i+1(x) = ψ_α_i+1(xv).



    
  * For 1 ≤ i ≤ n-3 with γ∈Γ_i+1, l = ι(γ) is in the Levi subgroup corresponding to V_i and we will now show that ψ_0(l^-1 v l) = ψ_0(v) for v ∈ [V_i]. 
We have that

    l^-1vl = 
        [ I_i+1     0;     0  γ^-1 ][       A       B;       0 I_n-i-1 ][ I_i+1     0;     0     γ ] = 
        [       A     B γ;       0 I_n-i-1 ]

and ψ_0(v) only depends on A. 

  * For i = n-2 with γ∈Γ_n-1(ψ_0), l = ι(γ) = γ is in the stabilizer T_ψ_0 which normalizes V_i and, by definition, means that ψ_0(v) = ψ_0(lvl^-1).


Thus, for 1 ≤ i ≤ n-2,

    _[V_i]φ(vg) ψ^-1_0(v) dv = _[V_i]_[R_i+1]φ(xvg) ψ^-1_0(v)   dx  dv +
     + ∑_γ∈Γ_i+1(ψ_0) _[V_i]_[R_i+1]φ(x v l g) ψ^-1_α_i+1(x) ψ^-1_0(v)   dx   dv ,

where we have made the variable change l v l^-1→ v.

Using that R_i+1 V_i = V_i+1 the above expressions simplifies to

    _[V_i+1]φ(vg) ψ^-1_0(v)   dv + ∑_γ∈Γ_i+1(ψ_0) _[V_i+1]φ(v ι(γ) g) ψ^-1_0(v) ψ^-1_α_i+1(v)   dv   .





    
    Let π be an irreducible minimal automorphic representation of _n(), φ∈π, and ψ_0 be a character on N trivial on the complement of V_i in N, 1 ≤ i ≤ n-2. 
    Then, ℱ_ψ_0∫_[V_i]φ(vg) ψ^-1_0   dv can be further expanded as follows.

    
            
  *  If ψ_0 = 1, then
            
                    ℱ_ψ_0    = _[V_i+1] φ(vg) dv + ∑_γ∈Γ_i+1 _[V_i+1] φ(v ι(γ) g) ψ^-1_α_i+1(v)  dv,    
            
             where Γ_i+1(ψ_0) with Γ_i+1 = Γ_i+1(1) 
    is defined in (<ref>).
            
  *  If ψ_0 = ψ_α_j (1 ≤ j ≤ i), then
            
                    ℱ_ψ_0    = _[V_i+1] φ(vg) ψ^-1_0(v)  dv.   
            
    



    We will use lemma <ref> where all the considered ψ_0 satisfy the character condition for the last row according to remark <ref>.
    
    For ψ_0 = 1, the expression is already in the form of lemma <ref>. 
    This proves case <ref>.
    
    For ψ_0 = ψ_α_j with 1 ≤ j ≤ i we have that ψ_0(v) ψ_α_i+1(v) = ψ_α_j, α_i+1(v) = ψ_u(v) for some u ∈ g which is in the next-to-minimal orbit. 
    Theorem <ref> with the Whittaker pair (s_V_i+1, u) gives that ℱ_s_V_i+1, u(φ) vanishes for φ in the minimal representation which leaves only the constant (or trivial) mode in lemma <ref>. 
    This proves case <ref>.


[2]


    
    Let π be an irreducible next-to-minimal automorphic representation of _n(), φ∈π, and ψ_0 be a character on N trivial on the complement of V_i in N, 1 ≤ i ≤ n-2.
    Then, ℱ_ψ_0∫_[V_i]φ(vg) ψ^-1_0   dv can be further expanded as follows.
    
    
        
  *  If ψ_0 = 1, then
        
                ℱ_ψ_0    = _[V_i+1] φ(vg) dv + ∑_γ∈Γ_i+1 _[V_i+1] φ(vι(γ) g) ψ^-1_α_i+1(v)  dv .    
        
        
  *  If ψ_0 = ψ_α_j (1 ≤ j < i), then
        
                ℱ_ψ_0    = _[V_i+1] φ(vg) ψ^-1_α_j(v) dv +  ∑_γ∈Γ_i+1(ψ_α_j)   _[V_i+1] φ(vι(γ) g) ψ^-1_α_j, α_i+1(v)  dv .    
        
        
  *  If ψ_0 = ψ_α_i, then
        
                ℱ_ψ_0    = _[V_i+1] φ(vg) ψ^-1_α_i(v)  dv .    
        
        
  *  If ψ_0 = ψ_α_j, α_k (1 < j+1 < k ≤ i), then
        
                ℱ_ψ_0    = _[V_i+1] φ(vg) ψ^-1_α_j, α_k(v)  dv .    
        
    
    Where Γ_i+1(ψ_0) with Γ_i+1 = Γ_i+1(1) is defined in (<ref>).


    We will use lemma <ref> where the considered ψ_0 in cases <ref>–<ref> satisfy the character condition for the last row according to remark <ref>.
    
    
    
  * For ψ_0 = 1, the expression is already in the form of lemma <ref>.
    This proves case <ref>.

    
  * For ψ_0 = ψ_α_j with 1 ≤ j < i we get that ψ_0(v) ψ_α_i+1(v) = ψ_α_j, α_i+1(v).
    This proves case <ref>.

    
  * For ψ_0 = ψ_α_i we get that ψ_0(v) ψ_α_i+1(v) = ψ_α_i, α_i+1(v) = ψ_u(v) for some u ∈ g belonging to an orbit higher than the next-to-minimal. 
    Theorem <ref> with the Whittaker pair (s_V_i+1, u) gives that ℱ_s_V_i+1, u(φ) vanishes both for φ in the minimal and next-to-minimal representations which leaves only the constant mode in lemma <ref>. 
    This proves case <ref>.

    
  * Lastly, for ψ_0 = ψ_α_j,α_k with 2 ≤ j+1 < k ≤ i we first consider i ≤ n-3 with lemma <ref>. 
    We get that ψ_0(v) ψ_α_i+1(v) = ψ_α_j, α_k, α_i+1(v) = ψ_u(v) for some u ∈ g belonging to an orbit higher than the next-to-minimal.
    Theorem <ref> with the Whittaker pair (s_V_i+1, u) gives that ℱ_s_V_i+1, u(φ) vanishes for φ in next-to-minimal representation which leaves only the first term in (<ref>). 
    
    For i = n-2, we expand along the last row and obtain a sum over characters ψ_u = ψ_0 ψ_y on N for all y ∈^t  r_n-1(F) where only y = 0 gives a u ∈ g belonging to an orbrit in the closure of the next-to-minimal orbit.
    Again, using theorem <ref> only the constant mode remains.
    This proves case <ref> and completes the proof.




    Since φ(x_1g) = φ(g) for x_1 ∈ V_1(F) we can make a Fourier expansion on V_1 and then use lemma <ref> to obtain
    
    φ(g) = _[V_1]φ(v g)   dv + ∑_γ_1 ∈Γ_1_[V_1]φ(v ι(γ_1) g) ψ^-1_α_1(v)   dv   .

    
    We will now make an iteration in the rows of the nilpotent, starting with the row i = 1 and continue until we reach the last row i = n - 1.
    
    
    
  * For case <ref>, that is, with φ in the minimal representation, the first step, using corollary <ref>, is
    
    φ(g) = [V_2]φ(vg)   dv + ∑_γ_2 ∈Γ_2 _[V_2]φ(v ι(γ_2) g)ψ^-1_α_2(v)   dv +
    
            + ∑_γ_1 ∈Γ_1 _[V_2]φ(v ι(γ_1) g) ψ^-1_α_1(v)   dv   ,

    where we note that the extra second term comes from the constant term on V_1. We will, after the iteration end up with
    
    φ(g) = _[N]φ(ng)   dn + ∑_i=1^n-1∑_γ∈Γ_i _[N]φ(n ι(γ) g) ψ^-1_α_i(n)   dn   .

    This completes the proof for the minimal representation.

    
  * For case <ref>, where φ is in the next-to-minimal-representation, we start again from (<ref>) and expand using corollary <ref>.
    We get, for the first step, that
    
    φ(g) = ( _[V_2]φ(v g)   dv + ∑_γ_2∈Γ_2_[V_2]φ(v ι(γ_2) g) ψ^-1_α_2(vg)   dv ) +
    
            +∑_γ_1 ∈Γ_1_[V_2]φ(v ι(γ_1) g) ψ^-1_α_1(v)   dv   ,

    where the parenthesis comes from the expansion of the constant term in (<ref>). Expanding in the next row as well, this becomes
    
    (_[V_3]φ(vg)   dv   + ∑_γ_3∈Γ_3 _[V_3]φ(v ι(γ_3) g) ψ^-1_α_3(v)   dv   + ∑_γ_2∈Γ_2 _[V_3]φ(v ι(γ_2) g) ψ^-1_α_2(v)   dv )  +
    
                + ∑_γ_1 ∈Γ_1( _[V_3]φ(v ι(γ_1) g) ψ^-1_α_1(v)   dv + ∑_γ_3 ∈Γ_3(ψ_α_1) _[V_3]φ(v ι(γ_3) ι(γ_1) g) ψ^-1_α_1, α_3(v)   dv ) .


    For each expansion adding a row i, the constant term gives an extra sum over Γ_i of a Fourier integral with character ψ_α_i, and from all terms with characters ψ_α_j with j < i - 1 we get an extra sum over Γ_i(ψ_α_j) together with a character ψ_α_j, α_i. 
    Corollary <ref> <ref> implies that these terms with characters non-trivial along two simple roots do not receive any further contributions. 
    Thus, after repeatedly using corollary <ref> to the last row, we get that
    
    φ(g) = _[N]φ(ng)   dn + ∑_i=1^n-1∑_γ∈Γ_i _[N]φ(n ι(γ) g) ψ^-1_α_i(n)   dn +
    
            + ∑_j=1^n-3∑_i=j+2^n-1∑_γ_i ∈Γ_i(ψ_α_j) 
    γ_j ∈Γ_j _[N]φ(n ι(γ_i) ι(γ_j) g) ψ^-1_α_j, α_i (n)   dn   ,

    which completes the proof of Theorem <ref>.
    








§ PROOF OF THEOREM <REF>


In this section, we prove Theorem <ref> which relates Fourier coefficients on a maximal parabolic subgroup with Whittaker coefficients on the Borel subgroup.
Recalling that the constant terms are known from <cit.>, we only focus on non-trivial characters, but first we need to introduce some notation and lemmas.


For 1 ≤ m ≤ n-1, let U_m be the unipotent radical of the maximal parabolic subgroup P_m with Levi subgroup L_m isomorphic to the subgroup of _m ×_n-m defined by {(g,g')∈_m ×_n-m: (g) (g')=1}. U_m is abelian and is isomorphic to the set of all m × (n-m) matrices. Write U_m as 

    U_m = {[   I_m     X;     0 I_n-m ] : X ∈_m × (n-m)}  .
 
Let U_m = ^t U_m be the unipotent radical of the opposite parabolic P_m. Then the Lie algebra of U_m can be written as

    𝔲_m = ^t  u_m = 
        {y(Y) = [   0_m     0;     Y 0_n-m ] : Y ∈_(n-m) × m}.

It is clear that the character group of U_m can be identified with ^t  u_m. 
L_m acts on ^t  u_m via conjugation and with (<ref>) this becomes a conjugation of the corresponding character's argument. 
Because of (<ref>), the Fourier coefficients for characters in the same L_m(F)-orbit are related by translates of their arguments, which means that we only need to compute one Fourier coefficient for each orbit.
We will therefore now describe the L_m(F)-orbits of elements y(Y) ∈^t  u_m but leave the details to be proven in appendix <ref>.

Starting first with F the number of L_m( F)-orbits is min(m,n-m)+1 and the orbits are classified by the rank of the (n-m) × m matrix Y. 
A representative of an L_m( F)-orbit corresponding to rank r can be chosen as y(Y_r) where Y_r is an (n-m) × m matrix, zero everywhere except for the upper right r × r submatrix which is anti-diagonal with all anti-diagonal elements equal to one.
For each rank r, 0 ≤ r ≤min(m,n-m), the corresponding G(F)-orbit is parametrized by the partition [2^r 1^n-2r]. 

As shown in appendix <ref>, the L_m(F)-orbits are characterized by the same data as the G(F)-orbits with ([2^r 1^n-2r], d), 0 ≤ r ≤min(m,n-m), d ∈ F^×/(F^×)^k and k ∈([2^r 1^n-2r]) with representatives y(Y_r(d)) where Y_r(d) is of the same form as Y_r above, but with the lower left element in the r × r matrix equal to d

    Y_r(d) = 
        [                                      0 [       1;     ⋱  ;   1    ; d       ];                                      0                                      0 ]∈_(n-m) × m(F)  .

We will continue to write Y_r(1) = Y_r. 
Note that for 0 ≤ r ≤ 2 and n ≥ 5, k is equal to 1.
Each such L_m(F)-orbit is also part of the G(F)-orbit of the same data. 

From (<ref>) the corresponding character on U_m is

    ψ_y(Y_r)(u)=ψ( (y(Y_r) log (u))),    u ∈ U_m().

Let s_m be the semisimple element (1,1,…, 1, -1,-1, …, -1) with m copies of 1's and (n-m) copies of -1's. Then, for any automorphic form φ on _n(), the following Fourier coefficient

    ∫_[U_m]φ(ug) ψ_y(Y_r(d))^-1(u)  du

is exactly the degenerate Fourier coefficient _s_m,y(Y_r(d))(φ). 

Note that in this paper, we focus on minimal and next-to-minimal representations, hence we only need to consider the cases of 0 ≤ r ≤ 2. Indeed, for 3 ≤ r ≤min(m,n-m), by definition, the generalized Fourier coefficient attached to the partition [2^r1^n-2r] is identically zero for minimal and next-to-minimal representations. By Theorem <ref> and since y(Y_r(d)) is in the G(F)-orbit [2^r1^n-2r], all the Fourier coefficients _s_m,y(Y_r)(φ) are also identically zero.

This leaves r ∈{1, 2} and with our assumption that n ≥ 5, we thus only need to consider the representatives y(Y_1) and y(Y_2) with d = 1 since ([2^r1^n-2r])=1. 

The above arguments proves the first part of theorem <ref>, that there exists an element l ∈ L_m(F) such that ℱ_U(φ, ψ_U; g) = ℱ_U(φ, ψ_y(Y_r); lg) (note the slight difference in notation ψ_y(Y_r) instead of ψ_Y_r), and that all ℱ_U(φ, ψ_y(Y_r); lg) for r > r_π vanish identically where r_π_min = 1 and r_π_ntm = 2.

We will now determine the remaining Fourier coefficients ℱ_U(φ, ψ_y(Y_r); g) in terms of Whittaker coefficients.
For 1 ≤ m ≤ n-1, 0 ≤ i ≤ m-1, let U_m^i be the unipotent radical of the parabolic of type (m-i,1^i,n-m). Note that U_m^0=U_m. 
Note that the character ψ_y(Y_1) can be extended to a character of any
subgroup of N containing U_m, still denoted by ψ_y(Y_1). 
Let C_m-i be the subgroup of U_m^i+1 consisting of elements with u_p,q=0 except when q = m-i and the diagonal elements. Note that C_m-i is an abelian subgroup and its character group can be identified with ^t 𝔠_m-i, the Lie algebra of ^t C_m-i. 
Write C_m-i as 

    C_m-i = { c(X) = 
        [ I_m-i-1       X       0;       0       1       0;       0       0 I_n-m+i ]}

and ^t 𝔠_m-i as

    ^t 𝔠_m-i = { y(Y)=
        [ 0_m-i-1       0       0;       Y       0       0;       0       0 0_n-m+i ]}  .
 
For each y ∈^t 𝔠_m-i, the corresponding character ψ_y of C_m-i is defined by ψ_y(c)=ψ((y log(c)). 
For any g ∈_m-i-1, let 

    ι(g)=
        [       g       0       0;       0 I_n-m+i       0;       0       0  (g)^-1 ]∈_n   .



    For _5 we have that
    
    U_3 =
            {[ 1     * *;   1   * *;     1 * *;       1  ;         1 ]}    
            U_3^1 =
            {[ 1   * * *;   1 * * *;     1 * *;       1  ;         1 ]}    
            C_3 =
            {[ 1   *    ;   1 *    ;     1    ;       1  ;         1 ]}  .



Note that U_m^m-1 = V_m and U_m^i+1 = C_m-i U_m^i. We will sometimes use j = m-i instead to denote column as follows U_m^m-j+1 = C_j U_m^m-j.

We will now construct a semi-simple element s = s_U_m^i for which g^s_1 = ∅ and such that n_s =  g^s_≥ 2 corresponds to N_s = U_m^i. These conditions are satisfied by

    (2i, …, 2i, 2(i-1) …, 2, 0, -2, …, -2)

with m-i copies of 2i and n-m copies of -2.

Note that any character ψ on N trivial on the complement of U_m^i in N is also a character on U_m^i by restriction and can be expressed as ψ_y with y ∈ g^s_-2 where s = s_U_m^i such that (s,y) forms a Whittaker pair. 
Indeed, we have that y ∈ g^s_N_-2 where s_N = (2(n-1), 2(n-2), …, 0,-2) from (<ref>) and the complement of U_m^i is described by s-s_N meaning that [y, s-s_N] = 0 for ψ to be trivial on the complement and thus [y, s] = [y, s_N] = -2y. 


    
    Let φ be an automorphic form on _n() and 2 ≤ j ≤ n.
    Let also ψ_0 be a character on N which, if j = 2, should be trivial along α_1 and (at least) two adjacent other simple roots.
    Then, 
    
    ∑_y ∈^t  c_j(F) 
     y ≠ 0_[C_j]φ(xg) ψ_y^-1(x)   dx = ∑_γ∈Λ_j-1(ψ_0)_[C_j]φ(x ι̂(γ) g) ψ^-1_α_j-1(x)   dx   .

    where Λ_j(ψ_0) is defined in (<ref>) and only depends on ψ_0 for j = 2.



    The proof is similar to those of lemmas <ref> and <ref>.
    
    
  * For 2 < j ≤ n, we parametrize y ∈^t  c_j(F) by row vectors Y ∈_1×(j-1)(F) with representative X̂ = (0, …, 0, 1) such that ψ_y(X̂) = ψ_α_j-1. 
        
    The surjective map _j-1(F) →^t  c_j(F)^× : γ↦X̂γ gives that ^t  c_j(F)^× (_j-1(F))_X̂_j-1(F) = Λ_j-1. 

    As in lemma <ref>, we can write the action as a conjugation y(X̂γ) = ι̂(γ)^-1 y(X̂) ι̂(γ) and, using (<ref>), ψ_y(X̂γ)(x) = ψ_y(X̂)(ι̂(γ) x ι̂(γ)^-1). Since ι̂(γ) normalizes C_j a variable change gives the wanted expression.

    
  * For j = 2, with y ∈^t  c_2(F)  F we instead consider the map T_ψ_0→^t  c_2(F)^× : h ↦ h^-1 y(1) h which is surjective by similar arguments as in lemma <ref> and thus gives ^t  c_2(F)^× (T_ψ_0∩ T_ψ_α_1)  T_ψ_0 = Λ_1(ψ_0). Writing the conjugation of y as a conjugation of the character's argument and then substituing variables in the Fourier integral thus proves the lemma.
    



    
Let φ be an automorphic form on _n(), 1 ≤ m ≤ n-1, 2 ≤ j ≤ m and ψ_0 a character on N trivial on the complement of U_m^m-j in N. 
For j = 2, ψ_0 should also be trivial along (at least) two adjacent simple roots other than α_1.

    _[U_m^m-j]φ(ug) ψ_0^-1(u)   du = 
    
        = _[U_m^m-j+1]φ(ug) ψ_0^-1(u)   du + ∑_γ∈Λ_j-1(ψ_0) _[U_m^m-j+1]φ(u ι̂(γ) g) ψ_0^-1(u) ψ_α_j-1^-1(u)   du .




    For 2 ≤ j ≤ m we have that φ(xug) = φ(ug) for x ∈ C_i(F) and u ∈ U_m^i() and since C_j is abelian
    
    φ(ug) = ∑_y ∈^t  c_j(F)_[C_j]φ(xug) ψ_y^-1(x)   dx   .

    Using lemma <ref>, we get that
    
    φ(ug) = _[C_j]φ(xug)   dx + ∑_γ∈Λ_j-1(ψ_0)_[C_j]φ(x ι̂(γ) ug) ψ_α_j-1^-1(x)   dx   .

   
    Let u ∈ U_m^m-j be parametrized as
    
    u =
            [ I_j-1     B;     0     A ]

    where A ∈_(n-j+1)×(n-j+1) is upper unitriangular (with several upper triangular elements being zero) and B ∈_(j-1)×(n-j+1) with elements in the first column being zero. 
    Since B does not intersect the abelianization [N,N] N (that is, the Lie algebra of B does not contain any generator of a simple root), we have, by assumption, that ψ_0 only depends on A.
    We also have that x ∈ C_j can be parametrized as
    
    x =
            [   I_j-1      B';       0 I_n-j+1 ]

    where B' ∈_(j-1)×(n-j+1) with only the first column non-zero. Thus,
    
    xu =
            [    I_j-1 B + B' A;        0        A ]

    which means that ψ_0(u) = ψ_0(xu).
    The first column of B is zero and A is upper unitriangular which means that the first column of B+B'A is the same as the first column of B' and since ψ_α_j-1 only depends on the first column of B' this implies that ψ_α_j-1(x) = ψ_α_j-1(xu).
    
    
    
  * For  3 ≤ j ≤ m with γ∈Λ_j-1 and l = ι̂(γ),
    
    l u l^-1 = 
            [       γ       0;       0 I_n-j+1 ][ I_j-1     B;     0     A ][    γ^-1       0;       0 I_n-j+1 ] = 
            [ I_j-1   γ B;     0     A ]

    and since ψ_0, by assumption, only depends on A we have that ψ_0(u) = ψ_0(l u l^-1).
    
  * For j = 2 with γ∈Λ_1 and l = ι̂(γ) = γ is in the stabilizer T_ψ_0 which, by definition, means that ψ_0(u) = ψ_0(lul^-1).
    
    
    Hence, for 2 ≤ j ≤ m, and after making a variable change lul^-1→ u, we get that
    
    _[U_m^m-j]_[C_j]φ(x l ug) ψ_0^-1(u) ψ_α_j-1^-1(x)  dx   du =  
       = _[U_m^m-j]_[C_j]φ(xulg) ψ_0^-1(u) ψ_α_j-1^-1(x)  dx   du 
       = _[U_m^m-j]_[C_j]φ(xulg) ψ_0^-1(xu) ψ_α_j-1^-1(xu)  dx   du 
       = _[U_m^m-j+1]φ(ulg) ψ_0^-1(u) ψ_α_j-1^-1(u)  du   .
 
    After similar manipulations for the constant term we obtain
    
    _[U_m^m-j]φ(ug) ψ_0^-1(u)   du = 
    
            = _[U_m^m-j+1]φ(ug) ψ_0^-1(u)   du + ∑_γ∈Λ_j-1(ψ_0) _[U_m^m-j+1]φ(ulg) ψ_0^-1(u) ψ_α_j-1^-1(u)  du   .




    
    We note that if ψ_0 is trivial along α_1 but not along at least two adjacent other simple roots we cannot use lemma <ref>, but we could still make an expansion over C_2 and keep the sum over y ∈^t  c_2(F)  F in the proof above.
    Since the character ψ_y has the same support as ψ_α_1 on N we still have that ψ_y(x) = ψ_y(xu) for x ∈ C_j() and u ∈ U_m^m-j() and since ψ_0 is still a character on N trivial on the complement of U_m^m-j it is still true that ψ_0(u) = ψ_0(xu). Thus, using (<ref>)
    
    _[U_m^m-2]φ(ug) ψ_0^-1(u)   du 
                   = ∑_y ∈^t  c_2(F)_[U_m^m-2]_[C_2]φ(xug) ψ_0^-1(u) ψ_y^-1(x)  dx   du 
       = ∑_y ∈^t  c_2(F)_[U_m^m-2]_[C_2]φ(xug) ψ_0^-1(xu) ψ_y^-1(xu)  dx   du 
       = ∑_y ∈^t  c_2(F)_[V_m]_[C_2]φ(vg) ψ_0^-1(v) ψ_y^-1(v)   dv   .




Assume that π is an irreducible minimal automorphic representation of _n(), φ∈π. 
For 1 ≤ m ≤ n-1, 0 ≤ i ≤ m-2, and g ∈_n(), 

    ∫_[U_m^i]φ(ug) ψ^-1_y(Y_1)(u)  du = ∫_[U_m^i+1]φ(ug) ψ^-1_y(Y_1)(u)  du .



    Using lemma <ref> with ψ_0 = ψ_y(Y_1) = ψ_α_m we get that
    
    _[U_m^i]φ(ug) ψ_0^-1(u)   du = _[U_m^i+1]φ(ug) ψ_0^-1(u)   du + ∑_γ∈Λ_m-i-1(ψ_0)ℱ(φ; m,i,γ,g)   ,

    where we have introduced
    
    ℱ(φ; m,i,γ,g) = _[U_m^i+1]φ(u ι̂(γ) g) ψ_0^-1(u) ψ_α_m-i-1(u) du   .


    Let s = s_U_m^i+1 from (<ref>), and let u ∈sl_n with two non-zero entries, both being 1, at positions (m-i, m-i-1) and (m+1, m).
    Then, ℱ(φ; m,i,γ,g) = ℱ_s,u(φ)(ι̂(γ)g) and since u is not in the closure of the minimal orbit, theorem <ref> gives that ℱ_s,u(φ) is identically zero leaving only the constant mode in (<ref>).


[3]

Proof of Theorem <ref>.



  * Minimal representation.
Assume that π be an irreducible minimal automorphic representation of _n(), and φ∈π. Applying Lemma <ref> repeatedly, 
we get that for each 1 ≤ m ≤ n-1, 

    ∫_[U_m]φ(ug)ψ_y(Y_1)^-1(u) du
    = ∫_[U_m^m-1]φ(ug)ψ_y(Y_1)^-1(u) du .

Note that U_m^m-1=V_m and ψ_y(Y_1) = ψ_α_m. 
Applying corollary <ref>
repeatedly, 
we get that for each 1 ≤ m ≤ n-1, 

    ∫_[U_m^m-1]φ(ug)ψ_y(Y_1)^-1(u) du=
    ∫_[N]φ(ng)ψ_y(Y_1)^-1(n) dn ,

which is exactly

    ∫_[N]φ(ng)ψ_α_m^-1(n) dn .



  * Next-to-minimal representation - rank 1. 
    Let π be an irreducible next-to-minimal automorphic representation of _n() and let φ∈π.
    Recalling that U_m = U_m^0 and applying lemma <ref> with ψ_0 = ψ_y(Y_1) = ψ_α_m we get
    
    _[U_m]φ(ug) ψ_y(Y_1)^-1(u)   du =
            _[U_m^1]φ(ug) ψ_0^-1(u)   du

    since ψ_0 ψ_α_m-1 = ψ_α_m,α_m-1 = ψ_u for some u that is not in the closure of the next-to-minimal orbit and thus the non-constant modes in lemma <ref> can be expressed as Fourier coefficients ℱ_s,u with s = s_U_m^1 from (<ref>) which vanish according to theorem <ref>.
    
    Let us make an iteration in 1 ≤ i ≤ m-2.
    Using lemma <ref> we have that
    
    _[U_m^i]φ(ug) ψ_0^-1(u)   du = 
    
            = _[U_m^i+1]φ(ug) ψ_0^-1(u)   du + ∑_γ∈Λ_m-i-1(ψ_α_m) _[U_m^i+1]φ(uι̂(γ) g) ψ_α_m-i-1, α_m^-1(u)   du   .

    
    Since ψ_α_m, α_m-i-1 is a character on N trivial on the complement of U_m^i+1 we can expand the second term further with lemma <ref> (or remark <ref> if m-i-1 = 2 and ψ_α_m, α_m-i-1 is not trivial along at least two adjacent roots other than α_1). This would lead to characters ψ_u = ψ_α_m, α_m-i-1, α_m-i-2 [3](or ψ_u = ψ_α_m, α_m-i-1ψ_y with y ∈^t  c_2(F) respectively) where u is not in the closure of the next-to-minimal orbit. 
    Then, ℱ_s,u with s = s_U_m^i+2 from (<ref>) vanishes according to theorem <ref> and the second term only receives the constant mode contribution.
    Repeating these arguments for the second term in (<ref>), it becomes
    
    ∑_γ∈Λ_m-i-1(ψ_α_m) _[V_m]φ(uι̂(γ) g) ψ_α_m-i-1, α_m^-1(u)   du .


    Iterating over i, starting from i = 1 above, we get that
    
    _[U_m]φ(ug) ψ_y(Y_1)^-1(u)   du = 
    _[V_m]φ(ug) ψ_α_m^-1(u)   du + ∑_j=1^m-2∑_γ∈Λ_j(ψ_α_m) _[V_m]φ(uι̂(γ) g) ψ_α_j, α_m^-1(u)   du   .

    
    For m = 1, U_1 = V_1 and for m = 2 we only get the first term in (<ref>).

    We will now use the methods of section <ref> to expand along rows.
    Using corollary <ref> case <ref>, we see that the second term in (<ref>) does not get any further contributions when expanding to N.
    Starting with the first term in (<ref>) and using corollary <ref> first with case <ref> to V_m+1 and then repeatedly with cases <ref> and <ref> it becomes
    
    _[V_m+1]φ(ug) ψ_α_m^-1(u)   du = 
    
            = _[N]φ(ng) ψ_α_m^-1(n)   dn + ∑_i=m+2^n-1∑_γ∈Γ_i(ψ_α_m) _[N]φ(n ι(γ) g) ψ_α_m, α_i^-1(n)   dn .


    Lastly,
    
    _[U_m]φ(ug) ψ_y(Y_1)^-1(u)   du = _[N]φ(ng) ψ_α_m^-1(n)   dn +
     
            + ∑_j=1^m-2∑_γ∈Λ_j(ψ_α_m) _[N]φ(nι̂(γ) g) ψ_α_j, α_m^-1(n)   dn +
     
            + ∑_i=m+2^n-1∑_γ∈Γ_i(ψ_α_m) _[N]φ(n ι(γ) g) ψ_α_m, α_i^-1(n)   dn   .



  * Next-to-minimal representation - rank 2.
Let π be an irreducible next-to-minimal automorphic representation of _n() and let φ∈π. 
We start from the integral

    ∫_[U_m]φ (ug) ψ_y(Y_2)^-1(u)  du .


For each root α, let X_α be the corresponding one-dimensional root subgroup in _n. Let 

    C_1 = X_e_m-e_m+2∏_i=1^m-2 X_e_i - e_m+2 ,

and 

    R_1 =  X_e_m-1-e_m∏_i=1^m-2 X_e_m-1 - e_i .

Then C_1 is a subgroup of U_m. Let U_m' be the subgroup of U_m 
with C_1-part identically zero. Then one can see that the quadruple 

    (U_m', C_1, R_1, ψ_y(Y_2))

satisfies all the conditions of Lemma <ref>. By this lemma, 

    ∫_[U_m]φ (ug) ψ_y(Y_2)^-1(u) du
    
    =     ∫_C_1()∫_[R_1U_m']φ (ucg) ψ_y(Y_2)^-1(u)  du dc .


Let 

    C_2 = ∏_i=1^m-2 X_e_i - e_m+1 ,

and 

    R_2 =  ∏_i=1^m-2 X_e_m - e_i .

Then C_2 is a subgroup of R_1U_m'. Let U_m” be the subgroup of R_1U_m' 
with C_2-part identically zero. Then one can see that the quadruple 

    (U_m”, C_2, R_2, ψ_y(Y_2))

satisfies all the conditions of Lemma <ref>. Applying this lemma and by changing of variables, 

    ∫_C_1()∫_[R_1U_m']φ (ucg) ψ_y(Y_2)^-1(u)  du dc
    
    =     ∫_C_1()∫_C_2()∫_[R_2U_m”]φ (uc_2c_1g) ψ_y(Y_2)^-1(u)  du dc_2 dc_1
    
    =     ∫_(C_1C_2)()∫_[R_2U_m”]φ (ucg) ψ_y(Y_2)^-1(u)  du dc .


Let ω be the Weyl element sending torus elements 

    (t_1, t_2, …, t_n)

to torus elements 

    (t_m-1, t_m+2, t_m, t_m+1, t_1, t_2, …, t_m-2, t_m+3, t_m+4, …, t_n) .

Conjugating ω cross from left, the integral
in (<ref>) becomes

    ∫_C()∫_[U_m^ω]φ (uω cg) ψ_y(Y_2)^ω, -1(u)  du dc ,

where U_m^ω = ω R_2U_m”ω^-1, C=C_1C_2,
for u ∈ U_m^ω, 
ψ_y(Y_2)^ω(u) = ψ_y(Y_2)(ω^-1 u ω).  

    U_m^ω = U_m^ω,1V_1 ,

where elements u ∈ U_m^ω,1 have the following form

    [ I_2   0;   0  u' ] ,

and U_m^ω,1 normalizes V_1. 
Recall that  V_i be unipotent radical of parabolic subgroup of type (1^i,n-i).
Note that ψ_y(Y_2)^ω|_V_1=ψ_α_1, 
ψ_y(Y_2)^ω|_U_m^ω,1=ψ_α_3. 
Recall that α_1=e_1-e_2, α_3=e_3-e_4. 
Hence, the integral
in (<ref>) becomes

    ∫_C()∫_[U_m^ω,1]∫_[V_1]φ (vuω cg)ψ_α_1^-1(v) ψ_α_3^-1(u)  dv du dc .



Since π is an irreducible next-to-minimal automorphic representation of _n(), 
by corollary <ref>, case <ref>, 
the integral
in (<ref>) becomes

    ∫_C()∫_[U_m^ω,1]∫_[V_2]φ (vuω cg)ψ_α_1^-1(v) ψ_α_3^-1(u)  dv du dc .

U_m^ω,1 still normalizes V_2, and 

    U_m^ω,1V_2 = U_m^ω,2V_3 ,

where elements u ∈ U_m^ω,2 have the following form

    [ I_4   0;   0  u” ] ,

u” is in the radical of the parabolic subgroup of type (m-2,n-m-2) in _n-4, 
and U_m^ω,2 normalizes V_3. 
Note that ψ_y(Y_2)^ω|_V_3=ψ_α_1,α_3 and 
ψ_y(Y_2)^ω|_U_m^ω,2 is the trivial character. 
By corollary <ref>, case <ref>, 
the integral
in (<ref>) becomes

    ∫_C()∫_[U_m^ω,2]∫_[V_4]φ (vuω cg)ψ_α_1,α_3^-1(v)  dv du dc .


Applying corollary <ref>, case <ref>, repeatedly,
the integral
in (<ref>) becomes

    ∫_C()∫_[U_m^ω,2]∫_[N]φ (nuω cg)ψ_α_1,α_3^-1(n)  dn du dc ,

which becomes 

    ∫_C()∫_[U_m^ω,2]∫_[N]φ (nω cg)ψ_α_1,α_3^-1(n)  dn du dc ,

by changing of variables. 
Since ∫_[U_m^ω,2]du=1, we have obtained that 

    ∫_[U_m]φ (ug) ψ_y(Y_2)^-1(u)  du
    = ∫_C()∫_[N]φ (nω cg)ψ_α_1,α_3^-1(n)  dn dc .


This completes the proof of Theorem <ref>. 




§ PROOF OF THEOREMS <REF> AND <REF>


    
Proof of Theorem <ref>.
Let π be any irreducible automorphic representation of _n() and let φ∈π. The generalized Fourier coefficient of φ attached to the partition [21^n-2] has been defined in Section <ref>. We recall it as follows. 

Let s=(1, -1, 0, …, 0), and let u = J_[21^n-2] which is a matrix zero everywhere except the (2,1) entry being 1.  Then the generalized Fourier coefficient of φ attached to the partition [21^n-2] is as follows:

    ℱ^[211…] (φ;g)  = _s,u (φ;g)= ∫_[N_s]φ(ng)ψ_u^-1(n)  dn ,

where elements in the one-dimensional unipotent N_s have the form 

    [     1     *     0;     0     1     0;     0     0 I_n-2 ] .


Let X=∏_i=3^n X_e_1-e_i and Y=∏_i=3^n X_e_i-e_2. 
Then one can see that Y(F) can be identified with the character space of [X] as follows:
given y ∈ Y(F), ψ_y(x)=ψ_u([x,y]), for any x ∈ [X]. 
Note that both X and Y normalize N_s. Taking the Fourier expansion of _s,u (φ)(g) along [X], we obtain that 

    _s,u (φ;g)=∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xng)ψ_u^-1(n) ψ_y^-1(x) dn dx .

Since y^-1∈ Y(F) and φ is automorphic, 
the above integral becomes

    ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xng)ψ_u^-1(n) ψ_y^-1(x) dn dx
    
    =     ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(y^-1xng)ψ_u^-1(n) ψ_y^-1(x) dn dx
    
    =     ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(y^-1xnyy^-1g)ψ_u^-1(n) ψ_y^-1(x) dn dx
    
    =     ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xn' y^-1g)ψ_u^-1(n) ψ_y^-1(x) dn dx ,

where n'=n+[x,y]. 
By changing variables, we obtain that 

    ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xn' y^-1g)ψ_u^-1(n) ψ_y^-1(x)dndx
    
    =     ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xn y^-1g)ψ_u^-1(n) 
    ψ_u^-1(-[x,y]) 
    ψ_y^-1(x) dn dx .


Note that 

    ψ_u^-1(-[x,y]) 
    ψ_y^-1(x)
    
    =     ψ_u([x,y]) 
    ψ_u(-[x,y]) 
    
    =     1 .

Hence, we have that 

    _s,u (φ;g) = ∑_y∈ Y(F)∫_[X]∫_[N_s]φ(xn y^-1g)ψ_u^-1(n) 
     dn dx .

Note that XN_s=U_1 and ψ_u=ψ_α_1. 
Therefore, we have that 

    _s,u (φ;g) = ∑_y∈ Y(F)∫_[U_1]φ(u y^-1g)ψ_α_1^-1(u) 
     du .


This completes the proof of Theorem <ref>. 




Proof of Theorem <ref>.
Let π be any irreducible automorphic representation of _n() and let φ∈π. The generalized Fourier coefficient of φ attached to the partition [2^21^n-4] has also been defined in Section <ref>. We recall it as follows. 

Let s=(1, -1, 1, -1, 0, …, 0), and let u = J_[2^21^n-4] which is a matrix zero everywhere except the (2,1) and (4,3) entries being 1.  Then the generalized Fourier coefficient of φ attached to the partition [2^21^n-4] is as follows:

    ℱ^[221…](φ;g) = 
    _s,u (φ;g)= ∫_[N_s]φ(ng)ψ_u^-1(n)  dn ,

where elements in N_s have the form 

    [     1     *     0     *     0;     0     1     0     0     0;     0     *     1     *     0;     0     0     0     1     0;     0     0     0     0 I_n-4 ] .

Let ω be the Weyl element sending the torus element

    (t_1, t_2, …, t_n)

to the torus element

    (t_1, t_3, t_4, t_2, t_5, t_6, …, t_n) .

Conjugating ω across from left, we obtain that 

    _s,u (φ;g)= ∫_[N_s^ω]φ(nω g)ψ_u^ω,-1(n)  dn ,

where N_s^ω = ω N_s ω^-1, and for n ∈ N_s^ω,
ψ_u^ω(n) =ψ_u(ω^-1 n ω). 
Elements in n ∈ N_s^ω have the following form 

    n=n(z)=[   I_2     z     0;     0   I_2     0;     0     0 I_n-4 ] ,

and ψ_u^ω(n)=ψ(z_1,2+z_2,1). 


Let 

    X'=∏_i=5^n X_e_1-e_i∏_i=5^n X_e_2-e_i

 and 

    Y'=∏_i=5^n X_e_i-e_4∏_i=5^n X_e_i-e_3 .

Then one can see that Y'(F) can be identified with the character space of [X'] as follows:
given y ∈ Y'(F), ψ_y(x)=ψ_u^ω([x,y]), for any x ∈ [X']. 
Note that both X' and Y' normalize N_s. Taking the Fourier expansion of _s,u (φ)(g) along [X'], we obtain that 

    _s,u (φ;g)=∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xnω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx .

Since y^-1∈ Y'(F) and φ is automorphic, 
the above integral becomes

    ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xnω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx
    
    =     ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(y^-1xnω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx
    
    =     ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(y^-1xnyy^-1ω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx
    
    =     ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xn' y^-1ω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx ,

where n'=n+[x,y]. 
By changing variables, we obtain that 

    ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xn' y^-1ω g)ψ_u^ω,-1(n) ψ_y^-1(x) dn dx
    
    =     ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xn y^-1ω g)ψ_u^ω,-1(n) 
    ψ_u^ω,-1(-[x,y]) 
    ψ_y^-1(x) dn dx .


Note that 

    ψ_u^ω,-1(-[x,y]) 
    ψ_y^-1(x)
    
    =     ψ_u^ω([x,y]) 
    ψ_u^ω(-[x,y]) 
    
    =     1 .

Hence, we have that 

    _s,u (φ;g) = ∑_y∈ Y'(F)∫_[X']∫_[N_s^ω]φ(xn y^-1ω g)ψ_u^-1(n) 
     dn dx .

Note that X'N_s^ω=U_2 and ψ_u=ψ_y(Y_2), using the notation from section <ref>.
Therefore, we have that 

    _s,u (φ;g) = ∑_y∈ Y'(F)∫_[U_2]φ(u y^-1ω g)ψ_y(Y_2)^-1(u) 
     du dx .


This completes the proof of Theorem <ref>. 




§ APPLICATIONS



As is evident from table <ref>, the case _5 appears in the list of symmetry and duality groups in string theory. It is related to compactification of type II string theory on a three-torus T^3 from ten to seven spacetime dimensions. Fourier coefficients of automorphic forms on _5 are related to non-perturbative effects as discussed in the introduction. Therefore we analyse here in some detail the structure of Fourier coefficients for automorphic forms attached to a minimal or next-to-minimal automorphic representation of _5 that are relevant to the first two higher-derivative corrections in four-graviton scattering amplitudes.

We will give a detailed description of how the formalism developed above can be used to calculate explicit expressions for Fourier coefficients on maximal parabolic subgroups for automorphic forms attached to a minimal or next-to-minimal automorphic representation. Following a general discussion, we will treat two explicit examples for n = 5.



 §.§ Generalities

With applications to string theory in mind, throughout this section we are restricting to F = and let ≡_. The types of expressions that are of interest are of the form:

    ℱ^(φ, ψ; g) = _U()  U()φ(ug) ψ^-1(u)   du ,

where U() is a parabolic subgroup of G(), ψ is some rank-1 or rank-2 character on U() and φ is an automorphic form in the minimal- or next-to-minimal automorphic representations of G(). Any such coefficient can be brought to a standard form using the action of the arithmetic Levi subgroup L(). For rank-1 this form is ψ = ψ_y(kY_1) for some integer k≠ 0 and for rank-2 one has ψ(k_1 u_m,m+1+k_2 u_m-1,m+2) for integers k_1 and k_2, cf. (<ref>). For simplicity, we will restrict ourselves to the case Y_y(Y_2) corresponding to k_1=k_2=1 and demonstrate how to apply theorem <ref>. The techniques demonstrated here allow for the calculation of all such Fourier coefficients for automorphic forms in the minimal and next-to-minimal representations on _n.

In order to apply theorem <ref>, we first perform an adelic lift <cit.>

    ℱ^(φ, ψ; g_∞) = ℱ^(φ, ψ; (g_∞, I_n, I_n, ⋯)) = _[U]φ(u(g_∞, I_n, I_n, …)) ψ^-1(u)   du 
        .

The theorem now gives ℱ^ in terms of adelic Whittaker functions. These Whittaker functions will then be evaluated using the adelic reduction formula

    W_ψ(λ, a) = ∑_w_c w_0' ∈𝒞_ψa^(w_c w_0')^-1λ + ρ M(w_c^-1, λ) W_ψ^a'(w_c^-1λ, 1)

of <cit.>. The power of this formula lies in that it expresses a degenerate Whittaker function evaluated on the Cartan torus of a group G() as a sum of generic Whittaker functions on a subgroup G'(). This subgroup G'() is determined by deleting all nodes in the Dynkin diagram of G() on which ψ is not supported. λ denotes the weight of the Eisenstein series, w_0' denotes the longest Weyl word on G', 𝒞_ψ denotes the set

    𝒞_ψ = { w ∈𝒲 |  w Π' < 0 }

where Π' is the set of simple roots of G' and w_c is hence the summation variable and corresponds to a specific representative of the quotient Weyl group 𝒲/𝒲' described in <cit.>. ρ denotes the Weyl vector, M denotes the intertwiner

    M(w, λ) = ∏_α > 0 
     wα < 0ξ(⟨λ | α⟩)/ξ(⟨λ | α⟩ + 1)

as featured in the Langlands constant term formula, where ξ is the completed Riemann zeta function and ψ^a denotes the “twisted character” both defined in appendix <ref>.

The evaluation of a real Fourier coefficient ℱ^ over a unipotent schematically looks like

    ℱ^(φ, ψ; g_∞) = ℱ^(φ, ψ; (g_∞, I_n, I_n, ⋯))       Adelic lift
    
            =   ∑_ψ∑_l∈Λ or  l∈Γ W_ψ(l(g_∞, I_n, I_n, ⋯))       Theorem <ref>
    
            =   ∑_ψ∑_l∈Λ or  l∈Γ W_ψ((n_∞ a_∞ k_∞, n_2 a_2 k_2, n_3 a_3 k_3, ⋯))       Iwasawa-decomposition
    
            =   ∑_ψ( ∏_p≤∞ψ_p(n_p) ) ∑_l∈Λ or  l∈Γ W_ψ((a_∞, a_2,a_3, ⋯))        W_ψ(nak) = ψ(n) W_ψ(a) 
    
            =   ∑_ψψ_∞(n_∞) ∑_l∈Λ or  l∈Γ∑_w a^… M(⋯) W_ψ^a'(⋯, 1)       Reduction formula (<ref>)
        .

The fourth line extracts the unipotent n_p-dependence at each of the local places p ≤∞. In the fifth line we have used that only the archimedean unipotent n_∞ contributes. The reason that the p-adic unipotent matrices n_p of the p-adic Iwasawa-decomposition of l ∈G(F) ⊂G(_p) above drop out is as follows. In using theorem <ref>, we will be faced with evaluating Whittaker functions such as

    W_α_j, α_m(ι̂(λ_j) g)      for   j ≤ m-2   where  λ_j∈Λ_j  and  
    
            W_α_m, α_i(ι(γ_i) g)      for   i ≥ m+2   where  γ_i ∈Γ_i 
            .

We have that γ_i and λ_j are embedded in _n as (cf. (<ref>))

    ι̂(λ_j) = 
        (
        [   λ_j;       I_n-j ])
          and  ι(γ_i) = 
        (
        [ I_i;     γ_i ]) 
        .

It is clear from their block-diagonal form that the unipotent n_p in the p-adic Iwasawa-decomposition of ι̂(λ_j) (and ι(γ_i)) will feature the same block-diagonal form. Since W_α_j, α_m (and W_α_m, α_i) is only sensitive to the unipotent on rows j and m ≥ j+2 > j (on rows i and m ≤ i-2 ≤ i), the block diagonal structure of n_p implies ψ_α_j, α_m; p(n_p) = 1 (and ψ_α_m, α_i; p(n_p) = 1).

For a real matrix g ∈_n(), we will denote its Iwasawa-decomposition

    g = n_∞ a_∞ k_∞ = 
        ( 
        [        1     x_12        ⋯        ⋯     x_1n;                 1        ⋱        ⋱        ⋮;                          ⋱        ⋱        ⋮;                                   1 x_n-1, n;                                            1 ])
        ( 
        [         y_1                                                ;                 y_2/y_1                                    ;                                   ⋱                        ;                                     y_n-1/y_n-2            ;                                                     1/y_n-1 ])
        k_∞ 
        .

Similarly, for a p-adic matrix g ∈_n(_p) we denote it as

    g = n_p a_p k_p = 
        n_p
        ( 
        [            η_1, p                                                                        ;                       η_2, p/η_1, p                                                      ;                                                     ⋱                                    ;                                                       η_n-1, p/η_n-2, p                  ;                                                                                1/η_n-1, p ])
        k_p 
        .

Appendix <ref> contains closed formulae for the x's and the y's, as well as a closed formula for the p-adic norm |η_i, p|_p of the η's.

In what follows, we will make use of all formulae that are derived or stated in appendices <ref>, <ref> and <ref> along with the following notation

    
  * A prime on a variable, eg. x', generally denotes x' ≠ 0.
    
  * For sums we write ∑_x ≡∑_x ∈.
    
  * We write ∑_x' f(x) ≡∑_x ∈{0} f(x) and ∑_x' ∈ f(x) ≡∑_x ∈{0} f(x). Note that the prime is used to indicate whether or not zero is included in the sum but the prime is omitted in the summand.
    
  * For products we write ∏_p ≡∏_p  prime. Writing ∏_p ≤∞ denotes the product over all primes p (the non-archimedean places) as well as the element p = ∞ (the archimedean place).
    
  * For x ∈ we denote x≡ e^2π i x.





 §.§ Example: Rank-1 coefficient of pimin on Palpha4 in SL5


Here, we will calculate the real rank-1 Fourier coefficient (<ref>) for the minimal Eisenstein series E(λ; g) with λ = 2sΛ_1 - ρ in the maximal parabolic

    P_α_4 = (4) ×(1) × U_α_4⊂(5)   subject to   ((4) ×(1)) = 1 ,

associated with removing the “last” node in the Dynkin diagram of (5). The unipotent radical is

    U() = U_α_4() = 
        {(
        [ 1       *;   1     *;     1   *;       1 *;         1 ])
        } 
        .


Theorem <ref> gives for the unramified character ψ_y(Y_1) that

    ℱ^(E(2sΛ_1 - ρ), ψ_y(Y_1); g) =    W_α_4(g)  
        .



The Whittaker function is found by the reduction formula with data given in table <ref>. In this case, there is no diagonally embedded rational matrix l, or equivalently l = I_5, in the general procedure and hence we have |η_1, p|_p = |η_2, p|_p = |η_3, 4|_p = |η_4, p|_p = 1. We get

    W_α_4(λ; (g_∞, I_5, I_5, ⋯)) = 
    
        =   x_45( y_4^5-2sξ( 2s-3 )/ξ( 2s )∏_p < ∞ |η_4, p|_p^5-2s) B_s-3/2( y_4^2/y_3, 1 ) 
       ×∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-3/2)) 1-p^-2(s-3/2)+1|η_4, p^2/η_3, p|_p^2(s-3/2)-1/1-p^-2(s-3/2)+1
    
        =   x_45 y_4^5-2s1/ξ( 2s ) 2 | y_4^2/y_3|_∞^s-2 K_s-2( 2π| y_4^2/y_3|_∞) 
    
        =    2 x_45 y_3^2-s y_4 1/ξ( 2s )  K_s-2( 2π| y_4^2/y_3|_∞) = ℱ^(E(2sΛ_1 - ρ), ψ_y(Y_1); g_∞) .

The x's and y's are the Iwasawa coordinates for the matrix g_∞ as in (<ref>). The function B_s that appears is a more compact way of writing the _2 Whittaker vector defined explicitly in (<ref>).

Parameterizing g_∞ as

    g_∞ = ue = 
        [ I_4   Q;   0   1 ][ r^-1/4e_4         0;         0         r ]  where   e_4 ∈_4() ,

we get in particular that

    y_3 = r^-3/4||N e_4||   and   y_4 = r^-1 ,

where N = 
[ 0 0 0 1 ] so that N e_4 is equal to the last row in e_4. This is obtained using the formula (<ref>). We get in particular that

    y_3^2-s y_4 = r^2s-5( r^-5/4 ||N e_4 || )^s-2  and  y_4^2/y_3 = r^-5/4 ||N e_4 ||
         .

The more general (real) ramified Fourier coefficient has the expression

    ∫ E( 2sΛ_1 - ρ; 
            [   1             u_1;       1         u_2;           1     u_3;               1 u_4;                   1 ]
            g_∞)
            m_1 u_1 + m_2 u_2 + m_3 u_3 + m'_4 u_4 d^4u
            = 
    
            =   x_45 r^2s-52/ξ(2s)σ_4-2s(k) ( r^-5/4 ||N e_4 || )^s-2 K_s-2( 2π r^-5/4||N e_4|| )
    
            =   x_45 r^3s/4-5/22/ξ(2s)σ_2s-4(k)/|k|^s-2 ||Ñ e_4 ||^s-2 K_s-2( 2π|k|  r^-5/4||Ñ e_4|| )

for integer m's while for non-integer rational m's it vanishes. Here g_∞ has been parametrized as above, N = 
[  m_1  m_2  m_3 m'_4 ]=k Ñ, k = (N) and m'_4 ≠ 0. This expression can also be found by starting from ψ_y(kY_1) for the standard Fourier coefficient instead. This corresponds to N = 
[ 0 0 0 k ]  and its L() orbit gives the general expression (<ref>). 
Formula (<ref>) agrees with <cit.> where the Fourier coefficients were computed by Poisson resummation technique after a translation of conventions.





 §.§ Example: Rank-1 coefficient of pintm on Palpha4 in SL5


Here, we will calculate the real rank-1[There is no rank-2 character for this parabolic.] Fourier coefficient (<ref>) for the next-to-minimal Eisenstein series E(λ; g) with λ = 2sΛ_2 - ρ in the maximal parabolic

    P_α_4 = (4) ×(1) × U_α_4⊂(5)   subject to   ((4) ×(1)) = 1

associated with removing the “last” node in the Dynkin diagram of (5). The unipotent radical is

    U() = U_β_4() = 
        {(
        [ 1       *;   1     *;     1   *;       1 *;         1 ])
        } 
        .


Theorem <ref> gives

    ℱ^(E(2sΛ_2 - ρ), ψ_y(Y_1); g) = 
    
          =    W_α_4(g) + ∑_λ_1 ∈Λ_1 W_α_1, α_4(λ_1 g) + ∑_λ_2 ∈Λ_2 W_α_2, α_4(λ_2 g) 
    
          =    W_α_4(g) 
       +∑_z' W_α_1, α_4(
          (
              [   z;       1;           1;               1;                 1/z ])
          _l_z
          g
          )
          
       +∑_x', y W_α_2, α_4( 
          (
          [ x^-1;    y    x;            I_3;      ])
          _l_xy
          g
          )
          
       +∑_x' W_α_2, α_4(
          (
          [     0 -x^-1;     x     0;               I_3;       ])
          _l_x
          g
          ) ,

using the representatives derived in appendix <ref>.



The first Whittaker function is found by the reduction formula with the data of table <ref>. In this case, there is no diagonally embedded rational matrix l, or equivalently l = I_5, and hence we have |η_1, p|_p = |η_2, p|_p = |η_3, 4|_p = |η_4, p|_p = 1. We get
    W_α_4(λ; (g_∞, I_n, I_n, ⋯)) = 
    
        =   x_45 B_s-1( y_4^2/y_3, 1 ) ( y_1^2s-1 y_4^4-2sξ( 2s-2 )/ξ( 2s )∏_p < ∞ |η_1, p|_p^2s-1|η_4, p|_p^4-2s. 
       + . y_1^3-2s y_2^2s-2 y_4^4-2sξ(2s-2)^2/ξ(2s) ξ(2s-1)∏_p < ∞ |η_1, p|_p^3-2s |η_2, p|_p^2s-2 |η_4, p|_p^4-2s. + 
       + . y_2^4-2s y_3^2s-3 y_4^4-2sξ(2s-3)ξ(2s-2)/ξ(2s)ξ(2s-1)∏_p < ∞ |η_2, p|_p^4-2s |η_3, p|_p^2s-3 |η_4, p|_p^4-2s)  
       ∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1|η_4, p^2/η_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1
    
        =   x_45( y_1^2s-1 y_4^4-2s1/ξ( 2s ) + y_1^3-2s y_2^2s-2 y_4^4-2sξ(2s-2)/ξ(2s) ξ(2s-1). 
       + . y_2^4-2s y_3^2s-3 y_4^4-2sξ(2s-3)/ξ(2s)ξ(2s-1)) 2 | y_4^2/y_3|_∞^s-3/2 K_s-3/2( 2π| y_4^2/y_3|_∞) 
    
        =    2 x_45( y_1^2s-1 y_3^3/2-s y_4 1/ξ( 2s ) + y_1^3-2s y_2^2s-2 y_3^3/2-s y_4 ξ(2s-2)/ξ(2s) ξ(2s-1). 
       + . y_2^4-2s y_3^s-3/2 y_4 ξ(2s-3)/ξ(2s)ξ(2s-1)) K_s-3/2( 2π| y_4^2/y_3|_∞) .

The x's and y's are the Iwasawa coordinates for the matrix g_∞ as in (<ref>).




The second Whittaker function is found by the reduction formula with the data given in table <ref>. The p-adic Iwasawa-decomposition of l_z has

    |η_1, p|_p = |η_2, p|_p = |η_3, p|_p = |η_4, p|_p = |z|_p 
        .

We get

    ∑_z'    W_α_1, α_4( λ; l_z (g_∞, I_n, I_n, ⋯) ) = 
    
        = ∑_z'   x_12 + x_45 y_1^3-2s y_2^2s-2 y_4^4-2sξ(2s-2)/ξ(2s)
        B_s-1/2( y_1^2/y_2, 1 ) B_s-1( y_4^2/y_3, 1 ) ∏_p < ∞ |η_1, p|_p^3-2s |η_2, p|_p^2s-2 |η_4, p|_p^4-2s
       ∏_p < ∞γ_p( η_1, p^2/η_2, p) ( 1 - p^-2(s-1/2)) 1-p^-2(s-1/2)+1| η_1, p^2/η_2, p|_p^2(s-1/2)-1/1-p^-2(s-1/2)+1×
       ×∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_4, p^2/η_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1
    
        =∑_z' ∈   x_12 + x_45 y_1^3-2s y_2^2s-2y_4^4-2sξ(2s-1)/ξ(2s)∏_p < ∞ |z|_p^5-2s
        4| y_1^2/y_2|_∞^s-3/2| y_4^2/y_3|_∞^s-2 K_s-3/2( 2π| y_1^2/y_2|_∞) K_s-2( 2π| y_4^2/y_3|_∞) 
       σ_-2(s-1/2)+1(|z|_∞) σ_-2(s-1)+1(|z|_∞) 
    
        =∑_z' ∈    4 x_12 + x_45 y_2^s-1/2y_3^2-sξ(2s-1)/ξ(2s) |z|_∞^2s-5
        K_s-3/2( 2π| y_1^2/y_2|_∞) K_s-2( 2π| y_4^2/y_3|_∞) σ_2-2s(|z|_∞) σ_3-2s(|z|_∞) .

The x's and y's are the Iwasawa coordinates for the matrix l_z g_∞.



The third and fourth Whittaker functions are found by the reduction formula with the data from table <ref>. The p-adic Iwasawa-decomposition of l_xy has

    |η_1, p|_p^-1 = max{|y|_p, |x|_p}and
        |η_2, p|_p = |η_3, p|_p = |η_4, p|_p = 1 
        .


We get

    ∑_x', y    W_α_2, α_4( λ; l_xy (g_∞, I_n, I_n, ⋯) ) = 
    
          = ∑_x', y   x_23 + x_45 y_2^4-2s y_3^2s-3 y_4^4-2sξ(2s-2)^2/ξ(2s)ξ(2s-1)
        B_s-1( y_2^2/y_1 y_3, 1 ) B_s-1( y_4^2/y_3, 1 )∏_p < ∞ |η_2, p|_p^4-2s |η_3, p|_p^2s-3 |η_4, p|_p^4-2s
       ∏_p < ∞γ_p( η_2, p^2/η_1, pη_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_2, p^2/η_1, pη_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1×
       ×∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_4, p^2/η_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1
    
        = ∑_x', y ∈   x_23 + x_45 y_2^4-2s y_3^2s-3 y_4^4-2s1/ξ(2s)ξ(2s-1) 4 | y_2^2/y_1 y_3|_∞^s-3/2| y_4^2/y_3|_∞^s-3/2
        K_s-3/2( 2π|y_2^2/y_1 y_3|_∞) K_s-3/2( 2π| y_4^2/y_3|_∞) σ_-2(s-1)+1(k) 
    
        = ∑_x', y ∈    4 x_23 + x_45 y_1^3/2-s y_2^1 y_4^11/ξ(2s)ξ(2s-1)
        K_s-3/2( 2π|y_2^2/y_1 y_3|_∞) K_s-3/2( 2π| y_4^2/y_3|_∞) σ_3-2s(k) ,

where k = (|y|, |x|). Here, the x's and y's are the Iwasawa coordinates for the matrix l_xy g_∞.

The p-adic Iwasawa-decomposition of l_x has

    |η_1, p|_p^-1 = max{|0|_p, |x|_p} = |x|_p
        and
        |η_2, p|_p = |η_3, p|_p = |η_4, p|_p = 1 
        .

We get

    ∑_x'    W_α_2, α_4( Λ; l_x (g_∞, I_n, I_n, ⋯) ) = 
    
          = ∑_x'   x_23 + x_45 y_2^4-2s y_3^2s-3 y_4^4-2sξ(2s-2)^2/ξ(2s)ξ(2s-1)
        B_s-1( y_2^2/y_1 y_3, 1 ) B_s-1( y_4^2/y_3, 1 )∏_p < ∞ |η_2, p|_p^4-2s |η_3, p|_p^2s-3 |η_4, p|_p^4-2s
       ∏_p < ∞γ_p( η_2, p^2/η_1, pη_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_2, p^2/η_1, pη_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1
       ∏_p < ∞γ_p( η_4, p^2/η_3, p) ( 1 - p^-2(s-1)) 1-p^-2(s-1)+1| η_4, p^2/η_3, p|_p^2(s-1)-1/1-p^-2(s-1)+1
    
        = ∑_x' ∈   x_23 + x_45 y_2^4-2s y_3^2s-3 y_4^4-2s1/ξ(2s)ξ(2s-1) 4 | y_2^2/y_1 y_3|_∞^s-3/2| y_4^2/y_3|_∞^s-3/2
        K_s-3/2( 2π|y_2^2/y_1 y_3|_∞) K_s-3/2( 2π| y_4^2/y_3|_∞) σ_-2(s-1)+1(|x|_∞) 
    
        = ∑_x' ∈    4 x_23 + x_45 y_1^3/2-s y_2^1 y_4^11/ξ(2s)ξ(2s-1)
        K_s-3/2( 2π|y_2^2/y_1 y_3|_∞) K_s-3/2( 2π| y_4^2/y_3|_∞) σ_3-2s(|x|_∞)   .

The x's and y's are the Iwasawa coordinates for the matrix l_x g_∞.

The complete Fourier coefficient ℱ^(E(2sΛ_2 - ρ), ψ_y(Y_1); g_∞) is then given by the combination of (<ref>), (<ref>), (<ref>) and (<ref>). We note that the our final result differs formally from the one given in <cit.> where the result is given as a convoluted integral over two Bessel functions whereas we do not have any remaining integral. The two results need not be in actual disagreement as there are many non-trivial relations involving infinite sums or integrals of Bessel functions.

The automorphic form

    lim_s→ 1/22ζ(3)ξ(2s-3)/ξ(2s) E(2sΛ_2-ρ;g) = 2ζ(3) E(3Λ_4-ρ)

lies in a minimal automorphic representation and controls the first non-trivial corrections that string theory predicts to the four-graviton scattering amplitude beyond standard general relativity <cit.>. The Fourier coefficients that we computed above can then be used to to extract so-called 1/2 BPS instanton contributions in the string perturbation limit of the amplitude. More precisely, they represent non-perturbative corrections to the scattering amplitude that, albeit smooth, are not analytic in the string coupling constant around vanishing coupling. They are therefore not visible in standard perturbation theory for small coupling but represent important correction nonetheless. Their interpretation is in terms of specific Dp-branes (p≤ 2) that are extended (p+1)-dimensional objects that can extend on non-trivial cycles of the torus T^3 that is present when _5 is the duality group. The detailed structure of the Fourier coefficient, in particular the arithmetic divisor sums appearing, can shed some light on the combinatorics of these D-branes similar to what is happening in the _2 case <cit.>.

For the next non-trivial correction to the four-graviton scattering amplitude one requires an automorphic form in the next-to-minimal automorphic representation <cit.>. This function is not a single Eisenstein series of the type we have analysed above but a very special combination of two formally divergent Eisenstein series with some Fourier coefficients computed using the Mellin transform of a theta lift in <cit.>.




§ EULER PRODUCTS AND TWISTED CHARACTERS



This appendix contains details and explanations for section <ref>, which is why we restrict to the field F = with the corresponding ring of adeles = _.

An Euler product is a product over the primes. The p-adic norm is denoted |·|_p and is defined for the p-adic numbers _p. The absolute value norm or “infinity norm” is denoted |·|_∞ and is defined for real numbers = _∞. The p-adic numbers as well as the real numbers (being completions of the rational numbers) all contain the rational numbers: ⊂_p for all p prime. The norm of an adele x = (x_∞, x_2, x_3, x_5, …) ∈ is denoted |·| (without ornaments) and is the product of norms at the local places

    |x| = ∏_p ≤∞ |x_p|_p 
        .

The rational numbers  are diagonally embedded into the adeles 

    ⊂  in the sense that   (q, q, q, q, …) ∈  for   q ∈ 
        .






  
Product of norms
For a rational number x ∈ with a decomposition into primes as 

    x = ±∏_p p^m^(p) ,

we get a particularly simple result for the adelic norm of x, namely

    ∏_p ≤∞ |x|_p = |x|_∞∏_p p^-m^(p) = |x|_∞ |x|_∞^-1 = 1 
      .

This is most often used as

    x ∈  ⇒  ∏_p |x|_p = |x|_∞^-1 
        .






  
Greatest common divisor
For a set of natural numbers { x_i } where each x_i has a decomposition into primes as

    x_i = ∏_p p^m^(p)_i ,

one can express the greatest common divisor k as

    k ≡({x_i }) = ∏_p p^min_i { m_i^(p)}
      .

Together with

    | x_i |_p = p^-m_i^(p) ,

we are led to the expression

    k = ∏_p p^min_i { m_i^(p)} = ∏_p min_i{ p^ m_i^(p)} = ∏_p min_i{ |x_i|_p^-1} = ∏_p ( max_i {| x_i |_p})^-1
      .

We also have the formula

    | k |_p = max_i {| x_i |_p }
      .


Note that

    (x_1, ⋯, x_n, 0) = (x_1, ⋯, x_n) ,

since every (nonzero) integer divides 0. Additionally, we define

    (x) = x   ∀ x ∈ ,

including x = 0.





  
Divisor sum
We have the identity 

    ∏_p 1-p^-s| m |_p^s/1 - p^-s = ∑_d|md^-s≡σ_-s(m) ,

for s ∈ and m ∈.





  
The completed Riemann zeta function
The Riemann zeta function

    ζ(s) = ∑_n=1^∞ n^-s ,   (s) >1

can be written as an Euler product as

    ζ(s) = ∏_p 1/1 - p^-s,   (s)>1

and can be analytically continued to the whole complex plane except at s = 0 and s = 1 where it has simple poles. This is done by defining the completed Riemann zeta function

    ξ(s) ≡Γ( s/2)π^-s/2ζ(s)

which obeys the functional relation

    ξ(s) = ξ(1-s)

as shown by Riemann.





  
p-adic gaussian
The p-adic gaussian γ_p: _p →{0, 1} is defined as

    γ_p(x) = 
        
            1,     |x|_p ≤ 1 
    
            0,     |x|_p > 1 
    
        =
        
            1,     x ∈_p 
    
            0,     x _p .

For a rational number x we then get

    ∏_p γ_p(x) = 
        
            1,     x ∈_p ∀ p 
    
            0,    else
        =
        
            1,     x ∈
    
            0,    else .


Notice also that for rational numbers x_1,…, x_n∈ and picking an x ∈ such that for all primes p

    |x|_p = max{|x_1|_p, ⋯, |x_n|_p } ,

we have

    γ_p(x) = ∏_i = 1^n γ_p(x_i)
        .

A consequence of this is that for an eulerian function depending only on the p-adic norms of its argument

    f(x) = ∏_p f_p(|x|_p) ,

then with x as in (<ref>), we have

    f(x) ∏_p γ_p(x) = ∏_p f_p(|x|_p) γ_p(x) = ∏_p f_p(|k|_p) γ_p(x) = f(k) ∏_p γ_p(x) ,

where

    k = (|x_1|_∞, ⋯, |x_n|_∞) .

This equation makes sense as ∏_p γ_p(x) ensures that the left- and right hand sides are nonzero only when each x_i is integer for which k is well defined. We now see how a sum over rationals with x as in (<ref>) can collapse to a sum over integers due to the p-adic gaussian

    ∑_x_1, …, x_n f(x) ∏_p γ_p(x) = ∑_x_1, …, x_n f(k) ∏_p γ_p(x) = ∑_x_1, …, x_n ∈ f(k)  .







  
(2) Whittaker function
The ramified (meaning m not necessarily unity) (2, ) Whittaker function evaluated at

    g = ( g_∞, , , …) = 
      ( 
      (
      [ 1 x; 0 1 ])
      (
      [   y   0;   0 1/y ])
      k, , , …)

written as an Euler product reads

    W_α( 2sΛ - ρ, m; g ) = W_α( 2sΛ - ρ, m;
        ( 
        (
        [ 1 x;   1;   ])
        (
        [   y    ;     1/y;     ])
        k, , , …)
        ) = 
    
        =   mx
        B_s(m, y)
        ∏_p γ_p(m) ( 1-p^-2s)1-p^-2s+1|m|_p^2s-1/1 - p^-2s+1 ,

where α is the simple root and Λ is the fundamental weight. Here

    B_s (m, y) ≡2π^s/Γ(s) y^1/2 |m|^s-1/2 K_s-1/2( 2π |m| y )

should be seen as the archimedean (2)-Whittaker function and each factor in the Euler product as the non-archimedean Whittaker functions. The product

    ∏_p γ_p(m)

restricts to m ∈ as explained above. The expression can then be written as

    W_α( 2sΛ - ρ, m; g ) = mx2/ξ(2s)y^1/2 |m|^s-1/2σ_1-2s(|m|) K_s-1/2( 2π |m| y )  
      .

Notice how the factors of the Eulerian expression for the Riemann zeta function in the non-archimedean part combines with π^s/Γ(s) in the archimedean part to form a completed Riemann zeta function.





  
Twisted character
Let m ∈ and ψ_p,m be an additive character on _p defined as

  ψ_∞, m(x)    = e^2πi m x ;        m, x    ∈        for real numbers  

  ψ_p, m(x)    = e^-2πi [ m x ]_p ;       m, x    ∈_p         for p-adic numbers . 

An unitary multiplicative character on the unipotent radical N() of the Borel subgroup of (n, ) can then be parametrized by m_1, …, m_n-1∈ as 

    ψ( n ) = ψ( e^∑_α∈Δ_+ u_α E_α ) = ψ( e^∑_α∈Π u_α E_α ) = ∏_p ≤∞∏_i=1^n ψ_p, m_i( ( u_α_i)_p ) ,

where Δ_+ is the set of positive roots and Π = {α_1, …, α_n-1}⊂Δ_+ is the set of simple roots. The second equality is due to the fact that the additive character is only sensitive to the abelianization of N(). In the final equality, ( x_α_i)_p denotes the p-adic (or real) component of the adelic coordinate x_α_i. 

For an element a ∈ A(), we would like to evaluate the twisted character

    ψ^a(n) ≡ψ( a n a^-1)   .


Let x_α(t) = exp(t E_α) where t ∈ and E_α is the positive Chevalley generator for the root α. 
For t ∈^×, define w_α(t) = x_α(t) x_-α(-t^-1) x_α(t) and h_α(t) = w_α(t) w_α(1)^-1. 
An element a ∈ A() is then parametrized by <cit.>

    a = ∏_i=1^n-1 h_α_i(y_i)      y_i ∈^×  ,

where the different generators h_α commute for all simple roots α∈Π and are multiplicative in t_α.
For a simple root α and a root β, we have that <cit.>

    h_α(y) x_β(u) h_α(y)^-1 = x_β(y^β(H_α) u) .


Since the character ψ on N is sensitive only to the x_β with β a simple root, it is enough to consider 

    h_α_i(y) x_α_j(u) h_α_i(y)^-1 = x_α_j(y^A_ij u)

for the simple roots α_i and α_j, where A_ij is the Cartan matrix.

We then have that

    ψ(a n a^-1) = ψ(exp(∑_j = 1^n-1(∏_i = 1^n-1 y_i^A_ij) x_α_j E_α_j) ) = 
    
            = ψ(exp(y_1^2/y_2 x_α_1 E_α_1 +  ∑_j=2^n-2y_j^2/y_j-1 y_j+1 x_α_j E_α_j + y_n-1^2/y_n-2 x_α_n-1 E_α_n-1 + …) )   .


We can interpret the transformation ψ→ψ^a as that the parameters m_i  transform according to

    m_i    → m_i' = ( y_i^2/y_i-1y_i+1) m_i,    i = 1, …, n-1 ,

where we have defined y_0 = y_n = 1. Note that starting with rational parameters m_i, the transformed parameters m_i' are no longer necessarily rational.




§ IWASAWA-DECOMPOSITION


Proof of the following results can be found in <cit.>. For a real matrix g ∈_n() written in Iwasawa form

    g = n_∞ a_∞ k_∞ = 
        ( 
        [        1     x_12        ⋯        ⋯     x_1n;                 1        ⋱        ⋱        ⋮;                          ⋱        ⋱        ⋮;                                   1 x_n-1, n;                                            1 ])
        ( 
        [         y_1                                                ;                 y_2/y_1                                    ;                                   ⋱                        ;                                     y_n-1/y_n-2            ;                                                     1/y_n-1 ])
        k_∞ ,

we have

    x_μν =    y_ν-1^2 ϵ( V_μ, V_ν+1, …, V_n; V_ν, V_ν+1, …, V_n ),   μ < ν, and
    
    y_μ^-2 =   ϵ( V_μ+1, …, V_n; V_μ+1, …, V_n ) ,

where V_μ is the μth row of g (regarded as an n-vector). Furthermore, ϵ denotes the totally antisymmetric product

    ϵ( A_1, …, A_m; B_1, …, B_m ) = δ_a_1 a_m^i_1 i_m( A_1)^a_1…( A_m )^a_m( B_1)_i_1…( B_m )_i_m ,

where the A's and B's are n-vectors and 

    δ_a_1 a_m^i_1 i_m = m! δ_[ a_1.^i_1…δ_. a_m]^i_m = 1/(n-m)!ϵ_a_1  a_m α_m+1α_nϵ^i_1  i_m α_m+1α_n

denotes the generalized Kronecker delta. Put in words, ϵ takes two sets of vectors and returns the sum of every possible product of scalar products between the two sets weighted by the signs of the given permutations. For example

    ϵ(A_1, A_2; B_1, B_2) = (A_1· B_1)(A_2· B_2) - (A_1· B_2)(A_2· B_1) 
        .


For a p-adic matrix g ∈_n(_p), the Iwasawa-decomposition

    g = n_p a_p k_p = 
        n_p
        ( 
        [            η_1, p                                                                        ;                       η_2, p/η_1, p                                                      ;                                                     ⋱                                    ;                                                       η_n-1, p/η_n-2, p                  ;                                                                                1/η_n-1, p ])
        k_p

is no longer unique. The p-adic norms of the η's however are constant across the family of decompositions and are given by

    | η_n-k|_p =
    (
    max_σ∈Θ_k^n{|
      g(
      [ n-k+1     …     n;  σ(1)     …  σ(k);       ])
    |_p })^-1 ,
    where
    k ∈{1, …, n-1} ,

and Θ_k^n detones the set of all ordered subsets of {1, …, n} of order k. Here, 
g(
[ r_1   … r_k; c_1   … c_k;     ])
denotes a minor of order k, given as the determinant of the submatrix of g obtained by only picking the k rows {r_i} and k columns {c_i}. For example, the matrix

    [  1/1  1/2  1/3  1/4;  1/5  1/6  1/7  1/8;  1/9 1/10 1/11 1/12; 1/13 1/14 1/15 1/16;      ] 
        .

has

    |η_2|_3 =   max{| |
                [  1/9 1/10; 1/13 1/14;      ]| |_3, 
                | |
                [  1/9 1/11; 1/13 1/15;      ]| |_3, 
                | |
                [  1/9 1/12; 1/13 1/16;      ]| |_3, 
                | |
                [ 1/10 1/11; 1/14 1/15;      ]| |_3, 
                | |
                [ 1/10 1/12; 1/14 1/16;      ]| |_3, 
                | |
                [ 1/11 1/12; 1/15 1/16;      ]| |_3
            } = 
    
            =   max{| 1/4095|_3, 
                | 8/19305|_3, 
                | 1/1872|_3, 
                | 1/5775|_3, 
                | 1/3360|_3, 
                | 1/7920|_3, 
            }
    
            =   max{ 3^2, 3^3, 3^2, 3^1, 3^1, 3^2 } = 3^3 = 9 
            .






§ PARAMETRIZING GAMMA_I AND LAMBDA_J


Recall the definitions

    Γ_i(ψ_0) 
            (_n-i(F))_Ŷ_n-i(F)     1 ≤ i ≤ n-2 
    
            (T_ψ_0∩ T_ψ_α_n-1)  T_ψ_0    i = n-1 ,

where

    (_n-i(F))_Ŷ = {[   1 ξ^T;   0   h ]
            : h ∈_n-i-1(F), ξ∈ F^n-i-1} ;

and

    Λ_j(ψ_0) 
            (_j(F))_X̂_j(F)     2 < j ≤ n 
    
            (T_ψ_0∩ T_ψ_α_1)  T_ψ_0    j = 2 ,

where

    (_j(F))_X̂ = {[ h ξ; 0 1 ]
            : h ∈_j-1(F), ξ∈ F^j-1} 
        .

In this appendix, we will find convenient representatives for these coset spaces. We begin with a lemma.


    
    Let S_k(F) denote the set of all k× k matrices m over the field F satisfying m = 1. The coset space _k(F)  S_k(F) can then be parametrized as
    
    _k(F)  S_k =
            ⋃_a = 0^k-1{( 
                [       0       0       0;     I_a       0       0;       0       v I_k-a-1 ])
                : v ∈ F^k-a-1} .



    We will use induction to prove the lemma. Assume that the result holds up to and including matrices of size k× k and consider the coset space _k+1(F)  S_k+1(F). Start with a matrix m_k+1∈ S_k+1(F). Left action of the group _k+1(F) ∋ h_k+1 taking m_k+1→ h_k+1m_k+1 is equivalent to performing Gauss elimination among the rows of m_k+1. Since we have m_k+1 = 1 we can bring m_k+1 to the form
    
    m_k+1→[ 0 0; v m ] ,

    where v ∈ F^k and m satisfies m_k ≤ 1. We cannot have m ≥ 2 as we could then perform additional row manipulations to produce two zero rows in m and hence another zero row in m_k+1 which violates m_k+1∈ S_k+1(F).
    

  §.§.§ Case 1: m = 0
 
    Here m is invertible and we can bring m_k+1 to the form
    
    [   0   0;   v m_k ]→[   0   0;   v I_k ]

    having relabelled v. We get the contribution
    
    .
            {[         0         0         0;       I_a         0         0;         0         v I_k+1-a-1 ]
                : v ∈ F^k+1-a-1}|_a=0 
            .

    

  §.§.§ Case 2: m = 1
 
    We now have m = m_k for some m_k ∈ S_k and we can apply the induction assumption which leads us to consider matrices of the form
    
    [       0       0       0       0;   v^(1)       0       0       0;   v^(2)     I_a       0       0;   v^(3)       0       u I_k-a-1 ] ,
              where   a ∈ [0, k-2] ∩ 
            .

    We see that we must have v^(1)≠ 0 and with further row manipulations we can thus bring this to the form
    
    [       0       0       0       0;   v^(1)       0       0       0;   v^(2)     I_a       0       0;   v^(3)       0       u I_k-a-1 ]→[       0       0       0       0;       1       0       0       0;       0     I_a       0       0;       0       0       u I_k-a-1 ] 
            .

    We get the contributions
    
    ⋃_a = 0^k-2{[       0       0       0       0;       1       0       0       0;       0     I_a       0       0;       0       0       v I_k-a-1 ]
                : v ∈ F^k-a-1}
            =
            ⋃_a = 1^k-1{( 
                [         0         0         0;       I_a         0         0;         0         v I_k+1-a-1 ])
                : v ∈ F^k+1-a-1} 
            .

    This combines with the contribution from case 1 to give the form stated in the lemma.
    
    That the base case k = 1 has the correct form is trivial. Peano's axiom of induction now establishes the lemma.



    
    The coset space
    
    (_n-i(F))_Ŷ_n-i(F)    1 ≤ i ≤ n-2

    can be parametrized as

    (_n-i(F))_Ŷ_n-i(F) = 
    
            =   {[   x'^-1       0       0;       y      x'       0;       v       0 I_n-i-2 ]
                : x' ∈ F^×, y ∈ F, v ∈ F^n-i-2}
    ∪   ⋃_a = 0^n-i-2{[              0              0 (-1)^a+1 x'^-1              0;             x'              0              0              0;              0            I_a              0              0;              0              0              v      I_n-i-a-2 ]
                : x' ∈ F^×, v ∈ F^n-i-a-2} .



    Denote k ≡ n-i. Consider a matrix
    
    G =  
            [   s T^T;   B   m ]∈_k(F) ,

    where s is a scalar, m is a (k-1)× (k-1)-matrix and T and B (for “top” and “bottom”) are (k-1)-column vectors.
    The action of an element
    
    M =  
            [   1 ξ^T;   0   h ]∈ (_k(F))_Ŷ

    on G is
    
    G → MG = 
            [   s + ξ^T B T^T + ξ^T m;         h B         h m ] 
            .

    Parametrizing the coset space (_k(F))_Ŷ_k(F) amounts to choosing ξ∈ F^k-1 and h ∈_k-1(F) such that the product MG takes a particularly nice form, manifestly with at most k degrees of freedom which is the dimension of this coset space.

    Even though h ∈_k-1(F) we will proceed with h ∈_k-1(F) and restore the unit determinant of h at the end by left multiplication of the matrix
    [     1     0     0;     0    x'     0;     0     0 I_k-2 ]
    where 0 ≠ x' = ( h)^-1. By having h ∈_k-1(F) we are free to perform Gauss elimination among the bottom k-1 rows in G.

    We consider the two cases m = 0 and m = 1. Note that the cases m ≥ 2 do not arise as with row elimination it would then be possible to produce two zero-rows in m and hence a zero-row in G which violates G ∈_k(F).
    
    

  §.§.§ Case 1: m = 0

    We choose h = m^-1 and ξ^T = -T^T m^-1. Since h has full rank, we can redefine hB → B without loss of generality and redefine s + ξ^T B → s. This leads to the representative
    
    G →[     s     0;     B I_k-1 ] 
            .

    We now restore the unit determinant to h
    
    G →[     1     0     0;     0    x'     0;     0     0 I_k-2 ][     s     0;     B I_k-1 ]
            =
            [     s     0     0;     y    x'     0;     v     0 I_k-2 ] ,

    where we have split the (k-1)-vector into a scalar y and a (k-2)-vector v. The condition G = 1 now sets s = x'^-1 leading to
    
    G 
            =
            [ x'^-1     0     0;     y    x'     0;     v     0 I_k-2 ] 
            .

    This is a nice form of the representative G which manifestly has k degrees of freedom.

    

  §.§.§ Case 2: m = 1

    We can no longer choose h = m^-1. Having promoted h to be an element of _k-1(F), we can make use of lemma <ref> which leads us to consider representatives of the form
    
    G →[       s  T^(1)T   T^(2)  T^(3)T;   B^(1)       0       0       0;   B^(2)     I_a       0       0;   B^(3)       0       v I_k-a-2 ]  for   a ∈ [0, k-2] ∩  and   v ∈ F^k-a-2 
            .

    We see that B^(1)≠ 0 in order for G to remain non-singular. With further row elimination we can therefore bring this to the form
    
    [       s T^(1) T   T^(2) T^(3) T;   B^(1)       0       0       0;   B^(2)     I_a       0       0;   B^(3)       0       v I_k-a-2 ]→[       s T^(1) T   T^(2) T^(3) T;       1       0       0       0;       0     I_a       0       0;       0       0       v I_k-a-2 ] 
            .

    Next, using the ξ-freedom we can bring this to the form
    
    [       s T^(1) T   T^(2) T^(3) T;       1       0       0       0;       0     I_a       0       0;       0       0       v I_k-a-2 ]→[       1   ξ^(1) ξ^(2) T ξ^(3) T;       0       1       0       0;       0       0     I_a       0;       0       0       0 I_k-a-2 ][       s T^(1) T   T^(2) T^(3) T;       1       0       0       0;       0     I_a       0       0;       0       0       v I_k-a-2 ]
    
                =   [         s + ξ^(1) T^(1) T + ξ^(2) T T^(2) + ξ^(3) T v T^(3) T + ξ^(3) T;                 1                 0                 0                 0;                 0               I_a                 0                 0;                 0                 0                 v           I_k-a-2 ]→[       0       0   T^(2)       0;       1       0       0       0;       0     I_a       0       0;       0       0       v I_k-a-2 ] ,

    with a suitable choice of ξ and having redefined T^(2). We now restore the unit determinant to h
    
    [       0       0   T^(2)       0;       1       0       0       0;       0     I_a       0       0;       0       0       v I_k-a-2 ]→[     1     0     0;     0    x'     0;     0     0 I_k-2 ][       0       0   T^(2)       0;       1       0       0       0;       0     I_a       0       0;       0       0       v I_k-a-2 ]→ 
            =
            [       0       0   T^(2)       0;      x'       0       0       0;       0     I_a       0       0;       0       0       v I_k-a-2 ] 
            .

    The condition G = 1 now sets T^(2) = (-1)^a+1 x'^-1, leading to the representative
    
    [              0              0 (-1)^a+1 x'^-1              0;             x'              0              0              0;              0            I_a              0              0;              0              0              v        I_k-a-2 ] 
            .




    
    The coset space
    
    (_j(F))_X̂_j(F)    2 < j ≤ n

    can be parametrized as

    (_j(F))_X̂_j(F) = 
    
            =   {[ I_j-2     0     0;     0    x'     0;   v^T     y x'^-1 ]
                : x' ∈ F^×, y ∈ F, v ∈ F^j-2}
    ∪   ⋃_a = 0^j-2{[              I_a                0                0                0;                0                0          I_j-a-2                0;                0                0                0               x';                0 (-1)^j+a+1 x'^-1              v^T                0 ]
                : x' ∈ F^×, v ∈ F^j-a-2} .



    Denote k ≡ j. Consider a matrix
    
    G =  
            [   m   T; B^T   s ]∈_k(F) ,

    where s is a scalar, m is a (k-1)× (k-1)-matrix and T and B (for “top” and “bottom”) are (k-1)-column vectors.
    The action of an element
    
    M =  
            [   h h ξ;   0   1 ]∈ (_k(F))_X̂

    on G is
    
    G → MG = 
            [ h(m + ξ B^T)   h(T + s ξ);          B^T            s ] 
            .

    Parametrizing the coset space (_k(F))_X̂_k(F) amounts to choosing ξ∈ F^k-1 and h ∈_k-1(F) such that the product MG takes a particularly nice form, manifestly with at most k degrees of freedom which is the dimension of this coset space.

    Even though h ∈_k-1(F) we will proceed with h ∈_k-1(F) and restore the unit determinant of h at the end by left multiplication of the matrix
    [ I_k-2     0     0;     0    x'     0;     0     0     1 ] ,
    where 0 ≠ x' = ( h)^-1. By having h ∈_k-1(F) we are free to perform Gauss elimination among the top k-1 rows in G.

    We consider the two cases s ≠ 0 and s = 0.

    

  §.§.§ Case 1: s = s' ≠ 0

    We choose ξ = -1/s' T. This leads to the representative
    
    G →[ m - 1/s'T B^T             0;           B^T            s' ] 
            .

    From the condition
    
    1 =  G = ( m - 1/s'T B^T) s'

            we get that
        
    ( m - 1/s'T B^T) ≠ 0 ,

    and hence the matrix m - 1/s'T B^T can be inverted using our h-freedom which leads to the representative
    
    [ m - 1/s'T B^T             0;           B^T            s' ]→[ I_k-1     0;   B^T    s' ] 
            .

    We now restore the unit determinant to h
    
    G →[ I_k-2     0     0;     0    x'     0;     0     0     1 ][ I_k-1     0;   B^T    s' ]
            =
            [ I_k-2     0     0;     0    x'     0;   v^T     y    s' ] ,

    where we have split B into a scalar y and a (k-2)-vector v. The condition G = 1 now sets s' = x'^-1 leading to
    
    G 
            =
            [ I_k-2     0     0;     0    x'     0;   v^T     y x'^-1 ] 
            .

    This is a nice form of the representative G which manifestly has k degrees of freedom.

    

  §.§.§ Case 2: s = 0

    We can no longer eliminate T with our ξ-freedom. The group element G takes the form
    
    G 
            =
            [    m   T'; B'^T    0;      ] ,

    where the vectors T' and B' must be non-zero (as indicated by the primes) in order for G to be non-singular. A ξ-transformation takes the form
    
    [    m   T'; B'^T    0;      ]→[ m + ξ B'^T         T';       B'^T          0;            ] 
            .

    We now consider the k-1 distinct cases labelled by a ∈ [0, k-2] ∩ defined by that B'^T takes the form B'^T = 
    [ 0_1× a     b'      v ], where v is a k-a-2-vector and 0 ≠ b' ∈ F. The ξ-transformation then lets us eliminate the (a+1)th column of m. This works since the (a+1)th column of the matrix ξ B'^T is b' ξ where b' ≠ 0 by assumption. We are led to the representative
    
    [    m   T'; B'^T    0;      ]→[ m_1   0 m_2  T';   0  b' v^T   0;     ] ,

    where m_1 is a (k-1)× a-matrix and m_2 is a (k-1)× (k-a-2)-matrix.

    The (k-1)× (k-1) matrix [ m_1   0 m_2 ] clearly has column-rank at most k-2. Since column-rank and row-rank for matrices are equal, we know that the row-rank is also at most k-2 and with row manipulations we can thus produce a zero row
    
    [ m_1   0 m_2  T';   0  b' v^T   0;     ]→[    0    0    0   t'; m_21    0 m_22  T^-;    0   b'  v^T    0;      ] ,

    where 0 ≠ t' ∈ F in order for G to be non-singular. With further row manipulations we can then eliminate the vector T^- and bring this to the form
    
    [    0    0    0   t'; m_21    0 m_22  T^-;    0   b'  v^T    0;      ]→[    0    0    0    1; m_21    0 m_22    0;    0   b'  v^T    0;      ] 
            .

    The (k-2)× (k-2)-matrix [ m_21 m_22 ]
    must have full rank in order for G to be non-singular and can thus be inverted, leading to the representative
    
    [    0    0    0    1; m_21    0 m_22    0;    0   b'  v^T    0;      ]→[       0       0       0       1;     I_a       0       0       0;       0       0 I_k-a-2       0;       0      b'     v^T       0;         ] 
            .

    We permute the first k-1 rows to get
    
    [       0       0       0       1;     I_a       0       0       0;       0       0 I_k-a-2       0;       0      b'     v^T       0;         ]→[     I_a       0       0       0;       0       0 I_k-a-2       0;       0       0       0       1;       0      b'     v^T       0;         ] 
            .

    Lastly, we restore the unit determinant to h
    
    [     I_a       0       0       0;       0       0 I_k-a-2       0;       0       0       0       1;       0      b'     v^T       0;         ]→[ I_k-2     0     0;     0    x'     0;     0     0     1 ][     I_a       0       0       0;       0       0 I_k-a-2       0;       0       0       0       1;       0      b'     v^T       0;         ]→ 
            =
            [     I_a       0       0       0;       0       0 I_k-a-2       0;       0       0       0      x';       0      b'     v^T       0;         ] 
            .

    The condition G = 1 now sets b' = (-1)^k+a+1 x'^-1, leading to the representative
    
    [              I_a                0                0                0;                0                0          I_k-a-2                0;                0                0                0               x';                0 (-1)^k+a+1 x'^-1              v^T                0;                  ] 
            .




    Another way of parametrizing the coset (_k(F))_X̂_k(F) is to parametrize the coset _k(F) / (_k(F))_X̂ which works completely analogously to the how the coset (_k(F))_Ŷ_k(F) was parametrized in lemma <ref> and then invert the resulting matrices.




§ LEVI ORBITS


Let P_m for 1 ≤ m ≤ n-1 be the maximal parabolic subgroup of _n associated to the simple root α_m with Levi decomposition L U_m (where we drop the subscript for L for convenience) and let y ∈^t u_m(F) be parametrised by a matrix Y ∈_(n-m) × m(F) as in (<ref>).
We will now study the L(F)-orbits of elements y.

We parametrise an element l ∈ L(F) by the two matrices A ∈_m× m(F) and B ∈_(n-m)×(n-m)(F) with (A) (B) = 1 as l = (A, B^-1).
This element acts by conjugation on y as ly(Y)l^-1 = y(AYB). 
Using unit determinant matrices A and B we may perform standard row and column additions to put Y on a form which has zero elements everywhere except for an anti-diagonal r × r matrix with non-vanishing determinant in the upper right corner where 0 ≤ r ≤min(m, n-m) is the rank of Y. 
This can be seen as follows.

If the upper right element is zero, pick any non-zero element whose row-column position we denote (i, j) and add multiples of row i to the first row, and column j to the last column to make the upper right element non-zero.
Then, use the non-zero upper right element to cancel all remaining non-zero elements on the first row and last column by further row and column additions.
Repeat the procedure for the matrix obtained by removing the first row and last column.
The induction terminates when we run out of rows or columns, or when the remaining elements are all zero.

We will now rescale the anti-diagonal elements by conjugating y with diagonal matrices l, meaning that the ith diagonal element in l rescales both row i and column i (inversely).
Each rescaling of an element in the anti-diagonal of the r × r matrix then leaves two less diagonal elements in l for further rescalings.
Since the non-zero elements of Y at this stage do not share any rows or columns (because of the anti-diagonal r × r submatrix), we may then perform any and all rescalings until we run out of free diagonal elements in l. 
The number of free diagonal elements in l is n-1 because of the determinant condition which means we can make n-1/2 rescalings. We have that 1 ≤ m ≤ n-1 and r ≤min(m, n-m) ≤n/2.  Thus, it is possible to rescale all anti-diagonal elements unless n = 2r = 2m, for which there will be one remaining anti-diagonal element d.
Using conjugations with l = (a, 1, …, 1, 1/a), a ∈ F^× this d can be shown to be in F^× / (F^×)^2. 

We have now shown that the L_m(F)-orbits of elements y(Y) ∈^t u_m(F) are characterized by matrices Y = Y_r(d) ∈_(n-m) × m(F), where Y_r(d) which is non-zero only for an anti-diagonal r× r matrix in its upper right corner whose elements are all one except the lower left which is d

    Y_r(d) = 
        [                                      0 [       1;     ⋱  ;   1    ; d       ];                                      0                                      0 ]  .

For n = 2r = 2m, d ∈ F^× / (F^×)^2 and otherwise d = 1.
For convenience we will denote Y_r(1) as Y_r. 
Thus, the L_m(F)-orbits on ^t u_m(F) are characterized by the same data as the _n(F)-orbits ([2^r1^n-2r], d) with d ∈ F^×/(F^×)^k, k = ([2^r1^n-2r]) and 0 ≤ r ≤min(m, n-m).

By conjugations with the Weyl element w ∈_n(F) mapping torus elements (t_1, t_2, …, t_n) ↦ (t_1, … t_m-r, t_m, t_m+1, t_m-1, t_m+2, …, t_m-r+1, t_m+r, t_m+r+1, … t_n), where we have underlined the changed elements, we see that y(Y_r(d)) is put on the form of the standard representative for the _n(F)-orbit ([2^r1^n-2r], d) shown in proposition <ref>. 

In this paper we will always be able to find a representative on the form Y_r(1), that is, to rescale all elements, since we consider n ≥ 5 and r ≤ 2 where the latter restriction comes from the fact that higher rank elements have vanishing associated Fourier coefficients in a next-to-minimal or minimal automorphic representation according to theorem <ref>.

Lastly, we note that if we instead consider L(F)-orbits the last remaining rescaling in the maximal rank n=2r case would be possible by conjugation with l = (√(d), 1, …, 1, 1/√(d)).




utphys-alpha


