Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

where [U]∶=(U ∩ )/U denotes the compact quotient of U.W e l l -s t u d i e ds p e c i a l cases of this definition arise when U is the unipotent N of a Borel subgroup and in that case the Fourier coefficients are called Whittaker coefficients,see(1.5)belo w . Another common case is when U is the unipotent of a (nonminimal) parabolic subgroup P = LU ⊂ G, and we shall refer to (1.1)i nt h a tc a s ea saparabolic Fourier coefficient.

Generally, when U is nonabelian, the coefficient F χU only captures a part of the Fourier expansion of η.T oreconstructη from its coefficients, one needs to consider a series of subgroups U i0 ={1}⊂U i0-1 ⊂⋅⋅⋅⊂U 1 = U with successive abelian quotients U i /U i+1 . Two examples are the derived series of U, and the lower central series of U.De n o t eb yX i the set of all nontrivial unitary characters of U i that are trivial on U i+1 and on U i ∩ . The complete Fourier expansion of η with respect to U takes the form

where F 1U [η] is the constant term with respect to U = U 1 .Th es i m p l e s tc a s eo f an o n a b e l i a nU is one that admits a Heisenberg structure, i.e., [U , U] is a onedimensional group, and this will be an important tool for us when we analyze groups of type E 8 that do not admit any abelian unipotents U as radicals of parabolic subgroups. In this case, the lower central series coincide with the derived series. Namely, we take i 0 = 3andU 2 to be [U , U] and call the Fourier coefficients F χU1 [η] the abelian Fourier coefficients and those for U 2 the nonabelian Fourier coefficients.

A natural approach to studying Fourier coefficients is to try to express them in terms of simpler coefficients, as in the celebrated results of Piatetski-Shapiro and Shalika [PS79,Sha74]. Unfortunately, this kind of reduction procedure does not seem to work in the full generality of (1.1) and no explicit formulas are known in general. However, the problem becomes more tractable when restricting to the subclass of coefficients given by Whittaker pairs as in Section 1.2. In this case, the techniques of [GGS17,GGS] allow one to develop a useful reduction theory, which is studied in the companion paper [GGK + ].

In this paper, we will analyze Fourier coefficients and expansions in the case of special classes of automorphic forms on split, simply-laced Lie groups. Specifically, we consider automorphic forms η attached to so-called minimal or next-to-minimal automorphic representations π min and π ntm of the adelic group G.Th i sm ea n sth a t all Fourier coefficients attached to nilpotents outside of a union of Zariski closures of minimal or next-to-minimal nilpotent orbits vanish. We refer to Section 2.1 below for the precise definitions. We note that in type D n , there are two next-to-minimal complex orbits for n > 4 and three next-to-minimal orbits for D 4 , while in types A and E, the next-to-minimal orbit is unique. Minimal orbits are unique in all simple Lie algebras. A sufficient condition for π to be minimal or next-to-minimal is that one of its local components is minimal or next-to-minimal, see Lemma 2.0.7 below. For minimal representations, this condition is also shown to be necessary under some additional assumptions on G,see [GS05,KS15].

Even though we shall not rely on explicit automorphic realizations of minimal and next-to-minimal representations, it might be instructive to indicate how they can be obtained. Minimal representations of π min have been studied extensively in the literature due to their role in establishing functoriality in the form of theta correspondences. In [GRS97] they were obtained as residues of degenerate principal series and used to construct global Eulerian integrals; see also [GRS11,Gin06,Gin14]. Next-to-minimal representations have not been analyzed as extensively though in recent years, this has started to change, partly due to their importance in understanding scattering amplitudes in string theory [GMV15, Pio10, FKP14, GKP16, FGKP18]; see Section 1.9 below for more details on this connection. Next-to-minimal representations exist for all next-to-minimal orbits, see, e.g., Section 5 below and [FGKP18]. They can be obtained as residues of degenerate principal series, see [GMV15,Pio10]f o rt y p eE.I nt y p e sA, E 6 ,a n d for one of the orbits in type D, there are one-parameter families of next-to-minimal representations.

In [GGS17,GGS] it was shown that there exist G-equivariant epimorphisms between different spaces of Fourier coefficients, thus determining their vanishing properties in terms of nilpotent orbits. In [GGK + ] we determined exact relations (instead of only showing the existence of such) between different types of Fourier coefficients. In this paper, we apply the techniques of [GGK + ]t o relate maximal parabolic Fourier coefficients, which are hard to compute, to a more manageable class of coefficients such as the known Whittaker coefficients with respect to the unipotent radical of a Borel subgroup. Furthermore, we express minimal and next-to-minimal automorphic forms through their Whittaker coefficients.

In the next subsection, we discuss the class of Fourier coefficients studied in [GGS17, GGS, GGK + ]. This class includes parabolic coefficients, coefficients of lower central series (but not the derived series) for unipotent radicals of parabolics, and the coefficients considered in [GRS11,Gin06,Gin14,JLS16].

Fourier coefficients associated to Whittaker pairs

Assume throughout this paper that G is a split simply-laced reductive group defined over K. In order to explain our main results in more detail, we briefly introduce some terminology. Denote by g the Lie algebra of G(K).AWhittaker pair is an ordered pair (S, φ)∈g × g * ,whereS is a semisimple element with eigenvalues of ad(S) in Q and ad * (S)(φ)=-2φ. This implies that φ is necessarily nilpotent and corresponds to a unique nilpotent element f = f φ ∈ g by the Killing form pairing. Each Whittaker pair (S, φ) defines a unipotent subgroup N S ,φ ⊂ G given by (2.2) below and a unitary character χ φ on N S ,φ by χ φ (n)=χ(φ(log n)) for n ∈ N S ,φ .

Our results are applicable to a wide space of functions on G, that we denote by C ∞ ( /G) and call the space of automorphic functions. This space consists of functions f that are left -i n v a r i a n t ,s m o o t hw h e nr e s t r i c t e dt ot h ep r e i m a g ei n G of ∏ infinite ν G(K ν ) a n dfi n i t eu n d e rt h er i g h ta c t i o no ft h ep r e i m a g ei nG of ∏ finite ν G(o ν ) where o is the ring of integers of K.N o t et h a tw ed on o ti n c l u d e the usual requirements of moderate growth and finiteness under the center z of the universal enveloping algebra. Such automorphic functions arise, for example, in applications in string theory [GV06,DGV15,FGKP18].

Following [MW87, GRS97, GRS11, GGS17], we attach to each Whittaker pair (S, φ) and automorphic function η on G the following Fourier coefficient

We note that the integrals we consider in this paper are well-defined for automorphic functions as they are either compact integrals or represent Fourier expansions of periodic functions.

Remark 1.2.1 Note that the unipotent group N S ,φ is not necessarily the unipotent radical of a parabolic subgroup of G. Consider, for example, the case of G = E 8 and let P = LU ⊂ E 8 be the Heisenberg parabolic such that the semisimple part of the Levi is E 7 and the unipotent radical U is the 57-dimensional Heisenberg group with one-dimensional center C =[U , U]. Then the Fourier coefficient F S ,φ can include the "nonabelian" coefficient corresponding to N S ,φ = C and χ φ anontrivialcharacter on C. This case is relevant for applications to physics; see Section 1.9 below.

If a Whittaker pair (h, φ) corresponds to a Jacobson-Morozov sl 2 -triple (e, h, f φ ), we say that it is a neutral Whittaker pair and call the corresponding coefficient a neutral Fourier coefficient. This is the class studied in [GRS11,Gin06,Gin14,JLS16] and referred to there simply as a Fourier coefficient.

We denote by WO(η) the set of nilpotent orbits O such that there exists a neutral pair (h, φ) such that F h,φ [η]/ ≡ 0andφ ∈ O,seeDefinition2.0.6 below. It was shown in [GGS17,Th e o r e mC]t h a tt h ev a n i s h i n go fF h,φ [η] implies the vanishing of any F S ,φ [η] where (S, φ) is a Whittaker pair that is not necessarily neutral. Let WS(η) be the set of maximal elements in WO(η) called the Whittaker support of η.I fa n automorphic function η min has a Whittaker support which contains a minimal orbit but no larger orbit, we say that it is a minimal automorphic function and similarly for a next-to-minimal automorphic function η ntm as detailed further in Section 2.1.

Statement of Theorem A

Choose a K-split maximal torus T ⊂ G and a set of positive roots. Let h be the Lie algebra of T ∩ .F o rasim p ler oo tα, we denote by P α the corresponding maximal parabolic subgroup, by L α is standard Levi subgroup, and by U α its unipotent radical. In other words, u α ∶= Lie U α is spanned by the root spaces whose expression in terms of simple roots contains α with positive coefficient. Define S α ∈ h by α(S α )=2andβ(S α )=0forallothersimplerootsβ.

(1.4) It will follow from the definition of N S ,φ that for any φ ∈ g * such that ad * (S α )φ =-2φ, we have that N Sα ,φ = U α . This means that the Fourier coefficient F Sα ,φ is the parabolic Fourier coefficient with respect to the unipotent subgroup U α and the character χ φ . Let S Π ∶= ∑ α∈Π S α , where Π is the set of all simple roots. Then the associated unipotent subgroup is the radical N of the Borel subgroup defined by the choice of simple roots. For any φ ∈ g * with ad * (S Π )φ =-2φ, and any automorphic function η defines the associated Whittaker coefficient by

Theorem A Let η min be a minimal automorphic function on a simply-laced split group G and (S α , φ) a Whittaker pair with S α determined by a simple root α as above. Depending on the orbit of φ, we have the following statements for the corresponding Fourier coefficient.

(i) The restriction of F Sα ,0 [η min ] to the Levi subgroup L α is a minimal or a trivial automorphic function. (ii) If φ is minimal, then there exists γ 0 ∈ ∩ L α that conjugates φ to an element φ ′ of weight -αbyAd * (γ 0 )φ = φ ′ and for any such γ 0 we have

For part (i), we remark that F Sα ,0 [η min ] is the usual constant term in a maximal parabolic. For Eisenstein series, it can be computed using the results of [MW95]. It can also be expressed through Whittaker coefficients using Theorem B below.

Remark 1.3.1 We note that the formula (1.6) is compatible with the expected equivariance of the Fourier coefficient F Sα ,φ [η min ](g), i.e., it satisfies

for all u ∈ U α .Forthistohold,onerequiresthatγ -1 0 uγ 0 ∈ U α for all u ∈ U α and χ φ (u)=χ φ ′ (γ -1 0 uγ 0 ), (1.8) which indeed holds due to the fact that γ 0 ∈ ∩ L α .

Remark 1.3.2 The notation W S ,φ and W φ a r eu s e di n [ GGS17,GGS] to denote something quite different. The present notation is however consistent with [FGKP18].

is necessarily a minimal automorphic function on the Levi subgroup L α . This does not necessarily hold for general G. For example, if G = GL 2 (A)×GL 2 (A), η min depends only on t h ev a r i a b l eo ft h es e c o n df a c t o r ,a n dα is a root of the second copy of GL 2 ,t h e n F Sα ,0 [η min ] is constant. Furthermore, if the restriction of η min to the second copy of GL 2 (A) is cuspidal, then F Sα ,0 [η min ] vanishes.

One can also obtain an expression for the minimal automorphic function itself. Thisisthesubjectofthenextsubsection.

Statement of Theorem B

For any root ε, denote by

)ω for all h ∈ h} the corresponding subspace of g * and by g × ε the set of nonzero elements of this subspace. Note that g * ε is a one-dimensional linear space over K.W esaythatasimple Let I =(β 1 ,...,β n ) be an enumeration of the simple roots of g in some order, and let l i be the Levi subalgebra with simple roots {β 1 ,...,β i }.WewillsaythatI is abelian if each β i is abelian in l i ,andtha tI is quasi-abelian if each β i is quasi-abelian in l i . From the table, we see that the Bourbaki enumeration is quasi-abelian if g = E 8 and abelian if g is simple (simply-laced) and different from E 8 .Wealsonotethatl i ⊂ l j for i < j.

Let I =(β 1 ,...,β n ) be any quasi-abelian enumeration of the simple roots of g. Given an automorphic function η on /G, we define functions

Let L i-1 be the Levi subgroup of G with Lie algebra l i-1 , and let Q i-1 be the parabolic subgroup of L i-1 whose Lie algebra is given by the nonpositive eigenspaces of ad(β ∨ i ) in l i-1 .InLemma3.2.1 below, we show that Q i-1 is the stabilizer in L i-1 of the root space g * -β i , as an element of the projective space of l * i .W elet i-1 =(L i-1 ∩ )/(Q i-1 ∩ ) , and put for i ∈{1,...,n}

is independent of the choice of a representative for γ,sinceQ i-1 ∩ stabilizes g × -β i . Thus, A i [η] is well-defined. We will use similar summations over cosets in the future without further comment.

If β i is a Heisenberg root of l i , then we define

Note that Ω i is a commutative subgroup of .Denotebyα i max the highest root for thesimplecomponentofl i containing β i ,andlets β i and s α i max denote the reflections with respect to the roots β i and α i max .Th e ns β i s α i max s β i is an involutive Weyl group element that switches β i and α i max .W efixarepresentativeγ i ∈ for s β i s α i max s β i and define

(1.11) Finally, we define

Theorem B Let η min be a minimal automorphic function on G. Then, for any choice of a quasi-abelian enumeration, we have

where α i is given by the Bourbaki labeling. Note that β 4 = α 2 is a Heisenberg root in G, while β i for 1 ≤ i ≤ 3 is an abelian root for the Levi subgroup L i with simple roots β 1 ,...,β i .UsingTheoremB we get that

where Ω 4 is defined in (1.10), γ 4 is defined above (1.11), and i-1 above (1.9). For this example, we get that the Lie algebra of Ω 4 is g -α2-α1 ⊕ g -α2-α3 ⊕ g -α2-α4 ⊕ g -2α2-α1-α3-α4 , γ 4 is a representative of the Weyl word s 1 s 3 s 2 s 4 s 2 s 1 s 3 in with simple reflections s i ,and 0 = 1 = 2 ={1} while 3 ≅(P 1 (K)) 3 .

W ep i c k e dt h ea b o v ee x a m p l et oi l l u s t r a t et h ea p p e a r a n c eo faH e i s e n b e r gt e r m A i + B i in (1.13) and because the right-hand side of (1.14)i sm a n i f e s t l yt r i a l i t y invariant.

Remark 1.4.3 As one can see from Table 1, every simple group of type different from A has a unique Heisenberg root. This can also be shown conceptually. This fact gives a way to choose an almost canonical quasi-abelian enumeration in the following inductive way. Let β 1 be the Heisenberg root of any simple component g ′ of g which is notoftypeA. If there are no such components, let β 1 be α 1 in any simple component of g. Then, let g 2 be the Levi subalgebra of g obtained by excluding the root β 1 ,choose the root β 2 of g 2 in the same way and continue by induction. This enumeration is closely related to the notion of Kostant's cascade,a sw e l la st oH-tower subgroups [Sal07, Section 3.2].

Let us now formulate analogs of Theorems A and B for next-to-minimal automorphic functions.

Statement of Theorem C

As before, let α be a simple root of g,a n dl e t(S α , ψ) be a Whittaker pair such that ψ ∈ g × -α and S α is given by (1.4) associated to the maximal parabolic subgroup corresponding to α.LetI (⊥α) =(β 1 ,...,β m ) be a quasi-abelian enumeration of the simple roots orthogonal to α which is always possible to find, see Table 1.F o ra n y 1 ≤ i ≤ m, we also define i-1 and γ i as above, but with the enumeration I (⊥α) ,a nd given an automorphic function η on /G we set

For any 1 ≤ i ≤ m with β i a Heisenberg root in the Levi subalgebra given by β 1 ,...,β i , we furthermore set

(1.16) Finally, we define

(1.17) Furthermore, let b be the Lie algebra of the negative Borel spanned by h and the root spaces of negative roots. For an element γ ∈ , we define

where g S >1 for a semisimple element S denotes the sum of all eigenspaces of ad S with eigenvalue > 1, see (2.1).

Remark 1.5.1 Since i is a partial flag variety for L i , it coincides with the group of K-points of the corresponding projective algebraic variety. By the valuation criterion for properness [Har77, Chapter II, Theorem 4.7], it then coincides with the (integral) O K -points of the same variety.

Theorem C Let η ntm beanext-to-minimalautomorphicfunctiononG,let(S α , φ) be a WhittakerpairwithS α as above and I (⊥α) =(β 1 ,...,β m ) a quasi-abelian enumeration a sa b o v e .D e p e n d i n go nt h eo r b i to fφ ,w eh a v et h ef o l l o w i n gs t a t e m e n t sf o rt h e corresponding Fourier coefficient.

(i) For trivial φ = 0, the restriction of F Sα ,0 [η ntm ] to the Levi subgroup L α is a trivial, or minimal, or next-to-minimal automorphic function. (ii) For φ in the minimal orbit, there exists γ 0 ∈ L α ∩ such that ψ ∶= Ad * (γ 0 )φ ∈ g × -α . For any such γ 0 ∈ L α ∩ ,wehave

If φ is next-to-minimal, then there exist orthogonal simple roots α ′ and α ′′ ,a n d an element γ 0 ∈ that is a product of an element of L α ∩ and a Weyl group representative, such that ψ ∶= Ad * (γ 0 )φ ∈ g × -α ′ + g × -α ′′ .F o ra n ys u c hγ 0 ,α ′ , and α ′′ ,wehave

Colloquially, we will refer to the condition in (iv)asφ beinginanorbitlargerthan next-to-minimal.

Remark 1.5.2 (i) For Theorem C(i), we remark that the coefficient F Sα ,0 [η ntm ] is the usual constant term that can be determined for Eisenstein series using the results of [MW95]. We note also that the restriction of F Sα ,0 [η ntm ] to the Levi subgroup L α can be expressed through Whittaker coefficients using Theorem B above and Theorem D below. (ii) Different choices of γ 0 c a nl e a dt os p a c e sV γ0 of different dimensions, some of which may be simpler for explicitly evaluating the integral, see for example (5.10). (iii) We stress that, similarly to (1.6), the right-hand side of the formula (1.20)i s compatible with the equivariance of the Fourier coefficient, i.e., satisfies

for all u ∈ U α . Equivariance of the Fourier coefficient is ensured by the integration over V γ0 .

Example 1.5.3

Let G = SO 4,4 (A) with = SO 4,4 (K) and η ntm a next-tominimal automorphic function on G.L e tα = α 1 and take the abelian enumeration I (⊥α1) =(β 1 , β 2 )=(α 3 , α 4 ). Fix a minimal element φ min ∈ g × -α1-α2 and let γ min 0 be a representative of the simple reflection s 2 in which means that ψ min ∶= Ad * (γ min 0 )φ min ∈ g × -α1 .

From Theorem C(ii), we get that

In order to obtain the last line, we note that i-1 is defined above (1.9)replacingI with I (⊥α1) and evaluates to 0 = 1 ={1} in this case. Now, fix a next-to-minimal element φ ntm ∈ g × -α1-α2-α3 + g × -α1-α2-α4 and let γ ntm 0 be a representative of the Weyl word s 2 s 1 such that ψ ntm ∶= Ad * (γ ntm 0 )φ ntm ∈ g × -α3 + g × -α4 . Using Theorem C(iii), we get that

where V γ ntm 0 is defined in (1.18)a n di t sL i ea l g e b r ah e r ee v a l u a t e st og -α2 (A)⊕ g -α1-α2 (A).

There are in fact three next-to-minimal (complex) orbits which are all related by triality. If the Whittaker support of η ntm does not include the orbit of φ ntm, the corresponding Fourier coefficient F Sα 1 ,φntm [η ntm ] is trivial and so is also the Whittaker coefficient W ψntm [η ntm ].Theresult(1.23) is therefore only nontrivial when the Whittaker support includes this orbit.

Example 1.5.4 LetusalsoconsiderG and η ntm as above but now with α = α 2 .W e have that I (⊥α2) is empty. Thus, for any minimal φ min ,w i t ha na s s o c i a t e de l e m e n t γ min 0 ∈ and canonical form ψ min ∶= Ad * (γ min 0 )φ min ∈ g × -α2 , we get from Theorem C(ii)that

It is interesting to ask which Fourier coefficients are Eulerian [Gin06,Gin14]. The expectation, based on the reduction formula of [FKP14] For example, in the case of Eisenstein series attached to the minimal representation of E 6 , E 7 , E 8 ,itwasshownin [FKP14]thatW φ [η] is given by just a single Whittaker coefficient on SL 2 ,whichiswellknowntobeEulerian.Seealso [FGKP18,Chapter10] for more details on these and other examples. By Theorem A,t h i si m p l i e st h a t the parabolic Fourier coefficient F Sα ,φ [η min ] of an Eisenstein series in the minimal representation calculated in the unipotent of a maximal parabolic determined by α should be Eulerian for simply-laced split groups.

Conversely, [KS15]s h o w st h a ti fG is linear, simply connected and absolutely simple, and the form η min generates an irreducible representation π = ⊗ π ν with all local components π ν minimal then F Sα ,φ [η min ] is Eulerian for any abelian root α and nonzero φ with ad * (S α )φ =-2φ.ByTheoremA, this implies that the corresponding Whittaker coefficient is Eulerian.

We expect that Theorem C will be useful to prove similar Eulerianity results for next-to-minimal representations. By contrast, if φ ∉ WS(η),t h eW h i t t a k e r coefficients and Fourier coefficients corresponding to φ are not expected to be Eulerian. In a follow-up paper [GGK + 20], we prove the Eulerianity of various types of Fourier coefficients, along the lines suggested above. In particular, we deduce from [FKP14, FGKP18, KS15] that maximal rank Whittaker coefficients of minimal and next-to-minimal Eisenstein series on simply-laced groups are Eulerian.

We can also express any next-to-minimal automorphic function in terms of its Whittaker coefficients, similar to Theorem B that treats the case of minimal automorphic functions. This is the subject of the next subsection.

Statement of Theorem D

Notation 1.6.1 Let α be a simple root.

(i) Let Q α denote the parabolic subgroup of L α with Lie algebra (l α ) α ∨ ≤0 .B yLemma 3.2.1 below, Q α is the stabilizer in L α of the line g * -α as an element of the projective space of g * .Let α denote the quotient of

We also fix a representative γ α ∈ for the Weyl group element s α s αmax s α ,wheres α and s αmax denote the corresponding reflections.

Theorem D Let η ntm be a next-to-minimal automorphic function on G, and let α be anicesimplerootofg.

(i) If α is an abelian root and ⟨α, α max ⟩>0 then η ntm = A, where

If α is an abelian root and ⟨α, α max ⟩=0 then η ntm = A + B, where

(1.28)

Part (i) of the above theorem only arises in type A when α is an extreme root of the diagram, part (ii)appliestoallotherrootsintypeA andtoallabelianrootsintypesD and E.Part(iii)onlyappliestotypeE and more specifically to root α 2 for E 6 ,rootα 1 for E 7 and root α 8 for E 8 using Bourbaki numbering. Note that δ α appearing in parts (ii)and(iii)areasdefinedinNotation1.6.1(iv)anddiffersinthetwoparts.

The right-hand sides of (1.26), (1.27), and (1.28) can be expressed in terms of Whittaker coefficients. Indeed, F Sα ,φ+ψ [η ntm ] and F Sα ,φ [η ntm ] can be expressed using Theorem C, while F Sα ,0 [η ntm ] defines a next-to-minimal function on L α ,thatcanthen be further decomposed using Theorem D by induction on the rank of G.T opresent this decomposition, we will need some further notation.

Statements of Theorems E, F,andG

Notation 1.7.1 Let β 1 ,...,β n be a quasi-abelian enumeration such that β 1 ,...,β n-1 is an abelian enumeration for L n-1 .Inthisnotation,wedefinethetermsA ij , B nj to be used in the next two theorems.

(i) For any i ≤ n, we define A ii in the following way. Let α i max denote the highest root for the simple component of l i containing β i .Ifβ i is abelian in l i and α i max is not orthogonal to β i ,we set A ii = 0. Otherwise, we define δ i to be the root δ β i of l i , and fix a g i ∈ that normalizes the torus and conjugates β i and δ i to orthogonal simple roots. Such a g i exists by Corollary 3.0.4.DefineV g i as in (1.18)andset

where Λ β i is the quotient of L i-1 ∩ defined as in Notation 1.6.1(v)a bove.Asin Remark 1.5.2(ii), the definition is independent of the choice of g i .

Heisenberg, fix a representative γ n ∈ for the Weyl group element s βn s α n max s βn ,wheres βn and s α n max denote the corresponding reflections. (iv) We will write j iif⟨β j , β i ⟩=0. (v) For any index j with j n, we define B nj in the following way. If β n is abelian, we set B nj ∶= 0.ForHeisenbergβ n , we define L ′ j to be the Levi subgroup of G given by the roots β k with k < jandk n, Q ′ j-1 to be the subgroup of L ′ j-1 that stabilizes the root space g * -β j ,and

is abelian, we define B nn to be zero. If β n is Heisenberg and nice, we define

which is again independent of the choice of g n .

Recall also the notation A i , B i from (1.9)a n d( 1.11). Applying Theorem D by induction and using also Theorems B and C, we obtain the following theorem.

Theorem E Fix a quasi-abelian enumeration β 1 ,...,β n such that β 1 ,...,β n-1 is an abelian enumeration for L n-1 ,andβ n is a nice quasi-abelian root. Let η ntm be a nextto-minimal automorphic function on G. Then

We note t hat if g hasatmostonecomponentoftypeE 8 , then an enumeration as in Theorem E is always possible. For example, one can take the Bourbaki enumeration on each component. Note that the right-hand side of (1.33) is entirely expressed in terms of Whittaker coefficients.

One can simplify the expression in (1.33)b ya l l o w i n go n e s e l ft ou s ei nt h efi n a l expression not only Whittaker coefficients but also constant terms with respect to parabolic nilradicals, that in turn can be determined for Eisenstein series using [MW95]. In this way, one obtains the following statement.

Theorem F Assume that [g, g] is simple of rank n, and fix the Bourbaki enumeration of its simple roots. Let η ntm be a next-to-minimal automorphic function on G. Then (i) In type A, we have

(ii) In types D, E 6 , and E 7 , we have

Using Lemma 2.0.7 below on the connection of Fourier coefficients to wave-front sets of local components, we derive from Theorem E the following one.

Theorem G Let the rank of G be greater than 2 and let π be an irreducible representation of G with decomposition π = ⊗ π ν into local components. Suppose that there exists νs u c ht h a tπ ν is minimal or next-to-minimal. Then π cannot be realized in cuspidal automorphic forms on G.

Remark 1.7.2 Forclassicalgroups,strongerstatementsareknownforcuspidalrepresentations. Namely, in type A, all cuspidal representations were shown to be generic by Shalika and Piatetski-Shapiro [Sha74,PS79]. For other classical groups, cuspidal representations are nonsingular, by [Li92]. This means that they possess nonvanishing Fourier coefficients with respect to nondegenerate characters of the Siegel parabolic. Thus, they cannot have minimal or next-to-minimal local components if the rank of G is greater than 2 (and G is classical).

For G of type E 6 or E 7 , the case of minimal representations of Theorem G is proven in [MS12].

It is possible that Theorem G holds for all quasi-split groups of rank greater than 2, not only simply-laced ones. In light of the result [Li92] mentioned above, it is left to prove it for F 4 .Thiscasemightfollowfrom[GGS, Theorem 8.2.1(ii)], which states that Whittaker supports of cuspidal representations consist of K-distinguished orbits.

For statements on the possibility of decomposing π = ⊗ π ν into local factors for covering groups, see [Wei18,Section8].

Motivation from string theory

The results of this paper have applications in string theory. In short, string theory predicts certain quantum corrections to Einstein's theory of general relativity. These quantum corrections come in the form of an expansion in curvature tensors and their derivatives. The first nontrivial correction is of fourth order in the Riemann tensor,

where E n /K n is a particular symmetric space, the classical moduli space of the theory. The parameter n = d + 1 contains the number of spacetime dimensions d that have been compactified on a torus T d .Th eg r o u p sE n are all split real forms of rank n complex Lie groups (see Table 2).

In the full quantum theory, the classical symmetry E n (R) is broken to an arithmetic subgroup E n (Z),calledtheU-duality group, which is the Chevalley group of integer points of E n [HT95]. Thus, the coefficient functions η n are really functions on the double coset E n (Z)/E n (R)/K n and, in certain cases, they can be uniquely determined. For the two leading order quantum corrections, corresponding to R 4 and ∂ 4 R 4 , the coefficient functions η n are, respectively, attached to the minimal and next-tominimal automorphic representations of E n [Pio10,GMV15]. Fourier expanding η n with respect to various unipotent subgroups U ⊂ E n reveals interesting information about perturbative and nonperturbative quantum effects. Of particular interest are the cases when U is the unipotent radical of a maximal parabolic P α ⊂ G corresponding to asimplerootα at an "extreme" node (or end node) in the Dynkin diagram. Consider the sequence of groups E n displayed in Table 2 and the associated Dynkin diagram in "Bourbaki labeling. " The extreme simple roots are then α 1 , α 2 ,andα n (this is slightly modified for the low rank cases where the Dynkin diagram becomes disconnected). The Fourier expansions of the automorphic form η with respect to the corresponding maximal parabolics then have the following interpretations (see Figure 2 for the associated labeled Dynkin diagrams):

• P = P α1 : String perturbation limit. In this case, the constant term of the Fourier expansion corresponds to perturbative terms (tree level, one-loop, etc.) with respect to an expansion around small string coupling, g s → 0. The nonconstant Fourier coefficients encode nonperturbative effects of the order e -1/gs and e -1/g 2 s arising from so-called D-instantons and NS5-instantons.

• P = P α2 : M-theory limit. This is an expansion in the limit of large volume of the M-theory torus T d+1 . The nonperturbative effects arise from M2-and M5-brane instantons. • P = P αn : Decompactification limit. Thisisanexpansioninthelimitoflargevolume of a single circle S 1 in the torus T d (or T d+1 in the M-theory picture). The nonperturbative effects encoded in the nonconstant Fourier coefficients correspond to so called BPS-instantons and Kaluza-Klein instantons.

For the reasons presented above, it is of interest in string theory to have general techniques for explicitly calculating Fourier coefficients of automorphic forms with respect to arbitrary unipotent subgroups.

In string theory, the abelian and nonabelian Fourier coefficients of the type defined in (1.1) typically reveal different types of nonperturbative effects (see for instance [PP09, BKN + 10, Per12]). The archimedean and nonarchimedean parts of the adelic integrals have different interpretations in terms of combinatorial properties of instantons and the instanton action, respectively. For example, in the simplest case of an Eisenstein series on SL 2 , the nonarchimedean part is a divisor sum σ k (n)=∑ d|n d k and corresponds to properties of D-instantons [GG97, GG98, KV98, MNS00]( s e e also [FGKP18] for a detailed discussion in the present context). Theorem F provides explicit expressions for the Fourier coefficients of the automorphic coupling of the next-to-minimal ∂ 4 R 4 higher derivative correction in various limits; see Section 5.2 foramoredetaileddiscussioninthecaseofE 8 . amplitudes. The theorem ensures that this can never happen as there are no cusp forms in the minimal or next-to-minimal spectrum.

Structure of the paper

In Section 2, we give the definitions of the notions mentioned above.

In Section 2.2,weintroducetheresultsof[GGK + ] that relate Fourier coefficients corresponding to different Whittaker pairs, in particular Theorem 2.2.6,whichisthe main tool of the current paper. Two more results from [GGK + ]t h a tw er e c a l li n Section 2.2 andheavilyuseintherestofthepaperareProposition2.2.7 that expresses any automorphic function through Heisenberg parabolic Fourier coefficients and a geometric Lemma 2.2.8.

In Section 3,wededuceTheoremsA-C from Section 2.2. For Theorem A(i)weshow that any Fourier coefficient F S ,ψ of the constant term equals a Fourier coefficient F H,ψ of η for some H,andthusvanishesunlessψ is minimal or zero. We first deduce from Lemma 2.2.8 that any minimal φ ∈(g * ) Sα -2 ca nbeco n j uga t edin t og × -α using L α ∩ (Corollary 3.1.3). This, together with Theorem 2.2.6, implies Theorem A(ii). Part (iii) of Theorem A follows from the definition of minimality and Corollary 2.2.5,whic h says that any Fourier coefficient is linearly determined by a neutral Fourier coefficient corresponding to the same orbit.

To prove The orem B, assume first that α ∶= β n is an abelian root. In this case, we decompose the form η min into Fourier series with respect to U α .E a c hF o u r i e r coefficient is of the form F Sα ,φ .F o rφ = 0, the restriction of this coefficient to L α is minimal and we use the theorem for L α (by induction on rank). For nonzero and nonminimal φ, F Sα ,φ vanishes by Theorem A(iii). For minimal φ, the expressions for F Sα ,φ are given by Theorem A(ii). We group them together using Corollary 3.1.3.Ifα is a Heisenberg root, we express η min through parabolic Fourier coefficients F Sα ,φ using Proposition 2.2.7.F o rφ ≠ 0, F Sα ,φ is given by Theorem A,a n df o rφ = 0b y induction.

Theorem C(i)i sp r o v e ns i m i l a r l yt oTh e o r e mA(i). To prove Theorem C(ii), we restrict F Sα ,φ [η ntm ] to L α , show that it is a minimal automorphic function, and apply Theorem B.TheoremC(iii)andC(iv) follow from Theorem 2.2.6 and Corollary 2.2.5, respectively. For Theorem C(iii), we also use a geometric lemma saying that any nextto-minimal φ ∈(g * ) Sα -2 c a nb ec o n j u g a t e di n t og × -α + g × -β for some positive root β orthogonal to α using L α ∩ (Lemma 3.3.6).

In Section 4,wefirstproveTheoremD using the same strategy as in the proof of Theorem B. However, we need two additional geometric propositions (Propositions 4.0.1 and 4.1.2) that describe the action of L α on next-to-minimal elements of (g * ) Sα -2 . We prove these in Section 4.3.InSection4.2, we derive Theorems E, F,andG from Theorems B, C,andD.

In Section 5 we provide examples of Theorems A-D f o rg r o u p so ft y p eD 5 and E 8 computing the expansions of automorphic function and Fourier coefficients with respect to different parabolic subgroups of interest in string theory and compare our E 8 resultstotheavailableliterature[BP17, GKP16, KP04].

Definitions and preliminaries

W euseasimilarsetupasinthecompanionpaper[GGK + ]butfromSection2.1 and onwards, we will restrict to split simply-laced groups. Let K be a number field, A = A K its ring of adeles and o its ring of integers. Fix a nontrivial unitary character χ of A, which is trivial on K. This additive character χ defines an isomorphism between A and Â via the map a ↦ χ a , where the hat denotes the character group and χ a (b)=χ(ab)

Let G be a reductive group defined over K, G(A) the group of adelic points of G and G a finite central extension of G(A). We assume that there exists a section G(K)→G of the covering p ∶ G → G(A), and we fix such a section and denote its image by . Recall that a semisimple element S is called rational if ad(S) has eigenvalues in Q. For any rational semisimple S ∈ g and i ∈ Q,weset

We wi l l a ls o us e simi lar not at ion for (g * ) S i . We wi l l s ay t hat an element of g * is nilpotent if it is given by the Killing form pairing with a nilpotent element of g.Equivalently,φ ∈ g * is nilpotent if and only if the Zariski closure of its coadjoint orbit includes zero. Because of the eigenvalue equation for φ in Definition 2.0.1,if(S, φ) is a Whittaker pair, then φ is nilpotent.

For any φ ∈ g * , we define an antisymmetric form ω φ of g by ω φ (X, Y)=φ([X, Y]) and given a Whittaker pair (S, φ) on g,wesetu ∶= g S >1 ⊕ g S 1 and define

where g φ is the centralizer of φ in g under the coadjoint action. Note that n S ,φ is nilpotent, an ideal in u with abelian quotient, and that φ defines a character of n S ,φ . Define an automorphic character on N S ,φ by χ φ (exp X)∶=χ(φ(X)).

We ca l l a f unc t ion on G an automorphic function if it is left -invariant, finite under therightactionofthepreimageinG of ∏ finite ν G(o ν ), and smooth when restricted t ot h ep r e i m a g ei nG of ∏ infinite ν G(K ν ). We denote the space of all automorphic functions by C ∞ ( /G). Definition 2.0.2 For an automorphic function η,wedefinetheFourier coefficient of η with respect to a Whittaker pair (S, φ) to be

where, for a unipotent subgroup U ⊂ G, we denote by [U] the quotient (U ∩ )/U. Definition 2.0.3 AWhittakerpair(H, φ) is called a neutral Whittaker pair if either (H, φ)=(0, 0),o rH can be completed to an sl 2 -triple (e, H, f ) such that φ is the Killing form pairing with f.Equivalently ,H can be completed to some sl 2 -triple and the map X ↦ ad * (X)φ defines an epimorphism g H 0 ↠(g * ) H -2 . See for example [Bou75, Section 11] for details on sl 2 -triples over arbitrary fields of characteristic zero. Definition 2.0.4 We call a Whittaker pair (S, φ) standard if N S ,φ is the unipotent radical of a Borel subgroup of G.By[GGK + ,Corollary2.1.5],anilpotentφ ∈ g * can be completed to a standard Whittaker pair if and only if it is a principal nilpotent element of some K-Levi subgroup of G. Here, principal means that the dimension of its centralizer equals the rank of the group. We call such φPL-nilpotentand their orbits PL-orbits . For a standard pair (S, φ), we call the Fourier coefficient F S ,φ a Whittaker coefficient and denote it W S ,φ or W φ if S defines the fixed Borel subgroup, see (1.5).

Remark 2.0.5 (i) In [GGS17, Section 6] the integral (2.4) above is called a Whittaker-Fourier coefficient, but in this paper, we call it Fourier coefficient for short. The Whittaker coefficients are called in [GGS17, Section 6] principal degenerate Whittaker-Fourier coefficients. The notations W S ,φ and W φ are used in [GGS17,GGS]to denote something quite different. (ii) Note that for G = GL n ,allorbitsO are PL-orbits. In general, this is, however, not the case, see [GGK + ,AppendixA]. (iii) We refer the readers interested in the definitions of principal nilpotents, PLnilpotents, and standard pairs for nonquasi-split groups to [GGK + ,Section2.1].

In [GGK + , Section 2.3] we defined a partial order for -orbits which will be used in the following definition. It is a refinement of the partial order for complex orbits defined by the Zariski closure. Definition 2.0.6 Let η be an automorphic function. We denote by WO(η) the set of nilpotent -orbits O in g * such that F h,φ [η]/ ≡ 0 for some neutral Whittaker pair (h, φ) with φ ∈ O. Furthermore, we define the Whittaker support of η, denoted by WS(η), to be the set of maximal elements in WO(η).

The following well-known lemma relates these notions to the local notion of wavefront set. For a survey on this notion, and its relation to degenerate Whittaker models, we refer the reader to [GS19,Section4].

Lemma 2.0.7 Suppose that η is an automorphic form in the classical sense, and that it generates an irreducible representation π of G. Let π = ⊗ ν π ν be the decomposition of π to local factors. Let O ∈ WO(η). Then, for any ν, there exists an orbit O ′ ν in the wave-front set of π ν such that O lies in the Zariski closure of O ′ ν .M o r e o v e r ,i fνi s nonarchimedean, then O lies in the closure of O ′ ν in the topology of g * (K ν ). Proof Acting by G on the argument of η, we can assume that there exists a neutral pair (h, φ) with φ ∈ O such that F h,φ [η](1)≠0. Moreover, decomposing η to a sum of pure tensors, and replacing η by one of the summands, we can assume that η is a pure tensor and F h,φ [η](1)≠0stillholds.Letη = ⊗ ′ μ v μ be the decomposition of η to local factors. Consider the functional ξ on π ν given by ξ(v)∶=F h,φ (v ν ⊗(⊗ ′ μ≠ν v μ ))(1). Substituting the vector v ν , we see that this functional is nonzero. It is easy to see that this ξ is (exp(n h,φ (K ν )), χ φ )-equivariant. The theorem follows now from [MW87, Proposition I.11] and [Var14] for nonarchimedean ν,a ndfr o m[Ros95,Theo r emD] and [Mat87]forarchimedeanν. ∎

It is useful to fix a complex embedding σ ∶ K↪C which will allow us to speak about the complex nilpotent orbit corresponding to an orbit O of in g.Thestruct ur eo f complex orbits is well understood; see for example [CM93]. Using [Ðok98], one can show that the complex orbit corresponding to O does not depend on σ,a l t h o u g h we shall not use this fact. None of our statements depends on the choice of complex embedding σ.

Minimal and next-to-minimal representations

From now on, we assume that G is simply-laced. We call a nonzero complex orbit in g * (C) minimal if its Zariski closure O is a disjoint union of O and the zero orbit. We call a complex orbit O next-to-minimal if O does not intersect any component of g of type A 2 ,andO is a disjoint union of O, minimal orbits, and the zero orbit.

, where O and O ′ are minimal orbits in g j and g l , respectively.

We call a (rational) element of g * or a rational orbit in g * minimal/next-to-minimal if its complex orbit is minimal/next-to-minimal.

Let O {0} be the set containing only the zero G(K)-orbit in g * ,letO {1} be the union of O {0} and the set of minimal G(K)-orbits, and similarly let O {2} be the union of O {0} ,O {1} and the set of next-to-minimal orbits.

We say that an automorphic function η is minimal if WO(η) is a subset of O {1} but not of O {0} .By[GGS17,TheoremC](orbyProposition2.2.4 below), this implies that F H,φ [η]=0foranyWhittakerpair(H, φ) with φ nonzero and nonminimal. We call an automorphic function ηtrivialif WO(η)=O {0} .By[GGS17, Corollary 8.2.2], t h es e m i s i m p l ep a r to fG acts on any trivial automorphic function by ± Id. We call ar e p r e s e n t a t i o no fG in automorphic functions minimal if all the functions in this representation are minimal or trivial.

We say that an automorphic function η is next-to-minimal if WO(η) is a subset of O {2} but not of O {1} .Again,by[GGS17,TheoremC](orbyProposition2.2.4 below), this implies that F H,φ [η]=0foranyWhittakerpair(H, φ) with φ higher than nextto-minimal. We call a representation π of G in automorphic functions next-to-minimal if it includes a next-to-minimal function, and all the functions in this representation are next-to-minimal, minimal, or trivial. By Lemma 2.0.7,ifπ consists of automorphic forms in the classical sense, is nontrivial, irreducible, and has a minimal local factor, then it is minimal. Similarly, if it has a next-to-minimal local factor, then it is minimal or next-to-minimal.

Relating different Whittaker pairs

Lemma 2.2.1 ([GGK + , Lemma 3.3.1]) Let (S, φ) be a Whittaker pair, η an automorphic function and γ ∈ .Then,

(2.5) Definition 2.2.2 Let (H, φ) and (S, φ) be Whittaker pairs with the same φ.W ewill say that (H, φ) dominates (S, φ) if H and S commute and

The following lemma provides two fundamental special cases of domination. Corollary 2.2.5 Let η be an automorphic function and let (S, φ) be Whittaker pair, with φ ∉ WO(η).ThenF H,φ [η]=0. Theorem 2.2.6 ([GGK + ,TheoremC(i)]) Let η be an automorphic function on G, and let φ ∈ WS(η).Let(H, φ) and (S, φ) be Whittaker pairs such that (H, φ) dominates (S, φ).Denote v ∶= g H >1 ∩ g S <1 , and V ∶= Exp(v(A)). (2.7)

We emphasize that the integral over V is an adelic integral. For the next proposition, recall from Section 1.4 that we say that a simple root α is a Heisenberg root if the nilradical of the maximal parabolic subalgebra defined by α is a Heisenberg Lie algebra. All such roots for simple (simply-laced) Lie algebras are listed in the second row of Table 1 in Section 1.4. Proposition 2.2.7 ([GGK + , Proposition 5.1.5]) Let α be a Heisenberg root, and let α max denote the highest root of the component of g corresponding to α. Let Ω α denote the abelian group obtained by exponentiation of the abelian Lie algebra given by the direct sum of the root spaces of negative roots β satisfying ⟨α, β⟩=1.Letγ α be a representative of a Weyl group element that conjugates α to α max .Let

(2.9) Lemma 2.2.8 ([GGK + , Lemma B.0.3]) Let S, Z ∈ g be rational semisimple commuting elements, let φ ∈ g Z 0 ∩ g S -2 and φ ′ ∈ g Z >0 ∩ g S -2 . Assume that φ is conjugate to φ + φ ′ by G(C). Then there exist X ∈ g Z >0 ∩ g S 0 and v ∈ Exp(g Z >0 ∩ g S 0 ) such that ad * (X)(φ)=φ ′ and v(φ)=φ + φ ′ .

Proof of Theorems A, B and C

For the whole section, we assume that G is split and the Dynkin diagram of g is simplylaced, i.e., all the connected components have types A, D,orE.AsinSection1.4,let,for any root δ, g * δ denote the corresponding root-subspace of g * and g × δ the set of nonzero elements of this subspace. (i) If g is of type A or E, thenanytwopairsoforthogonalrootsareW eyl-conjugate. (ii) If g is of type D n with n ≥ 5, then any pair of orthogonal roots is Weyl-conjugate to exactly one of the pairs (α 1 , α 3 ) and (α n-1 , α n ). (iii) If g is of type D 4 , then any pair of orthogonal roots is Weyl-conjugate to exactly one of the pairs (α 1 , α 3 ), (α 1 , α 4 ), and (α 3 , α 4 ).

Proof In types A and E, we apply Lemma 3.0.1 that implies that each of the two pairs can be Weyl-related to a pair where one of the roots is the highest root and the other is orthogonal to it. Since the Dynkin diagram of the root system consisting of roots orthogonal to the highest one is still connected, the stabilizer of the highest root acts transitively on it and one can relate the other elements of the pairs as well, showing that it is possible to relate any two pairs of orthogonal roots.

In type D n , we use the standard realization of roots as

where ε i denotes the unit vector in R n .TheW eylgr o u pactsb yperm u ta tio no fthe indices, and even number of sign changes. The usual choice of simple roots is

Usingreflections,wecanconjugateanypairoforthogonalrootstoapairoforthogonal positive roots. The pairs of orthogonal positive roots have one of the two forms (1)

(2) (ε i ± ε j , ε k ± ε l ) with i < j and k < l all distinct. We can conjugate any pair of type (1) to

It is easy to see that one cannot conjugate a pair of type (1) into a pair of type (2). ∎

We remark t hat in typ e D n ,thepairs(α 1 , α 3 ) and (α n-1 , α n ) correspond to two distinct next-to-minimal orbits, given by the partitions 2 4 1 2n-8 and 31 2n-3 ,respectively. Corollary 3.0.4 Any pair of orthogonal roots in g is Weyl-conjugate to a pair of orthogonal simple roots.

Proof If [g, g] is not simple and the roots lie in different simple components, this follows from Lemma 3.0.1 by conjugating each of them to a simple root. If the roots lie in the same component, this follows from Corollary 3.0.3. ∎

Lemma 3.1.1 If η is a minimal automorphic function and φ

Proof We have g S 1 ={0}=g Sα ≥1 ∩ g S <1 , which implies the lemma by Theorem 2.2.6 ∎ Let L α denote the Levi subgroup of the parabolic subgroup P α of G. Lemma 3.1.2 Any root δ with δ(S α )=-2 can be conjugated to -αu s i n gt h eW e y l group of L α .

Proof We can assume that g is simple. This statement can be proved using the language of minuscule representations, i.e., representations such that the Weyl group has a single orbit on the weights of the representation. By [Bou75, Section VIII.3] these are the fundamental representations corresponding to the abelian roots (see Table 1).

It suffices to show that the representation of the Levi L α on the first internal Chevalley V α ∶= u α /[u α , u α ] is minuscule. These modules are explicitly computed in [MS12, Section 5] and this can be checked case-by-case. For completeness, we give a conceptual argument.

We claim first t hat V α is irreducible with lowest weight α. Evidently, α is a weight of V α with multiplicity one. Also any positive root β of L α involves only simple roots different from α,andthusαβ is not a root. Hence α is a lowest weight of V α .Onthe other hand, any weight of V α is of the form α + γ,whereγ is a sum of positive roots from L α .Thus,α is the unique lowest weight of V α .

The Dynkin diagram of L α is obtained from that of G by removing α,a n de a c h component has exactly one simple root adjacent to α, which is easily checked to be an abelian root for the component. Thus, the corresponding fundamental representations areminuscule,andthussoistheirtensorproductW α .However,W α has highest weight -α,since⟨-α, β⟩ is 1 if β is adjacent to α and zero otherwise. It follows that V α ≃ W * α , and hence V α is minuscule.

,where

Proof (i)L e tz be a generic element of h that is 0 on α a n dn e g a t i v eo no t h e r positive roots. Decompose (g * ) Sα -2 =⊕ k i=0 V k by eigenvectors of z, with eigenvalues 0 = t 0 < t 1 <⋅⋅⋅< t k .N o t et h a tV 0 = g * -α .L e tX ∈(g * ) Sα -2 be a minimal element and X = ∑ i X i its decomposition by eigenvalues of z.ByLemma3.1.2,wecanassume,by replacing X by its L α ∩ -conjugate, that X 0 ≠ 0. By Lemma 2.2.8, X is conjugate to

. Identify Y ′ with some f ∈ g -α using the Killing form, and complete f to an sl 2 -triple e, h, f with e ∈ g α .Then For part (i), suppose that there exists a Whittaker pair (H, ψ) for

Th u s ,t h eo r b i to fψ is minimal in g * and thus by Lemma 3.1.4 also in l * α . ∎

Proof of Theorem C

Suppose that rk(g)>2. Let η be a next-to-minimal automorphic function. Let α be a simple root and let ψ ∈ g × -α . Lemma 3.3.1 Let γ ≠ αb eap o s i t i v er o o t ,a n dl e tφ ′ ∈ g × -γ .L e tO denote the orbit of ψ + φ ′ .ThenO is minimal if ⟨α, γ⟩>0, O is next-to-minimal if ⟨α, γ⟩=0 and O is neither minimal nor next-to-minimal if ⟨α, γ⟩<0.

Proof By Lemma 2.1.3, we can assume that [g, g] is simple. Let h ′ ⊂ h be the simultaneous kernel of α and γ,andletl be its centralizer in g.Thenh ′ has codimension at most 2 in h,hencel isaLevisubalgebraofsemisimplerank≤ 2whoserootsinclude α and γ.NotethatO ∩ l is a principal nilpotent orbit in l. By a straightforward rank 2 calculation, we see that l has type For the proof, we will need the following geometric lemma.

Lemma 3.3.5 Let φ ′ ∈ g ′ * be nilpotent such that φ ′ + ψ belongs to a next-to-minimal orbit in g * .Thenφ ′ belongs to the minimal orbit of g ′ * .

Proof Clearly φ ′ ≠ 0. If the orbit of φ ′ is not minimal, then it belongs to the Slodowy slice of some element ψ ′ of the minimal orbit of (g ′ ) * .Thenφ ′ + ψ ′ belongs to a nextto-minimal orbit of g * ,andφ ′ + ψ belongs to the Slodowy slice of φ ′ + ψ ′ and thus lies in an orbit that is higher than next-to-minimal. ∎ ProofofProposition3.3.4 Let Z ∶= S -S α .NotethatZ vanishes on simple roots in Δ α and on α and is 2 on other simple roots. Suppose that there exists a Whittaker pair

Then, for T big enough, we have

By Proposition 2.2.4 and Lemma 3.3.5, ψ lies in the minimal orbit of g ′ * . ∎ Lemma 3.3.6 (See Section 3.4 below) For any next-to-minimal element φ ∈(g * ) Sα -2 , there exist γ 0 ∈ L α ∩ and a positive root β orthogonal to α s.t. Ad * (γ 0 )φ ∈ g × -α + g × -β ⊂ g * -α ⊕ g * -β . Remark 3.3.7 The above lemma only establishes that any next-to-minimal φ can be mapped to two orthogonal root spaces by L α ∩ .H o w ev e r ,th ea ct i o no fL α ∩ is often even transitive on (g * ) Sα -2 , giving a single orbit. One can show that this happens in all cases except for:

• A 3 and node α 2 . • D 4 and nodes α 1 , α 3 , α 4 (all related by triality).

• D n and when the two orthogonal roots (α, β) are Weyl conjugate under D n to (α n-1 , α n ),seeCorollary3.0.3, corresponding to the orbit 31 2n-3 .Thishappensfor n ≥ 4alwaysfornodeα 1 as well as for nodes α i with 2 ≤ i ≤ n -2ifφ belongs to that orbit. For instance, for A 3 and node α 2 , one has that next-to-minimal are φ ∈ g × -α1 + g × -α3 . The torus element for node i scales elements in g × -α i by rational squares ( i = 1, 3) while keeping the other space unchanged. The torus element for node 2 scales both spaces by rational elements in the same way, and so one cannot use the torus action in L α2 ∩ toarriveatauniquerepresentative.Theothercasescanbeseentoreducetothesame phenomenon.

For A n with n ≥ 4 and all exceptional cases, there is a unique rational representative for next-to-minimal nilpotents in (g * ) Sα -2 .

Proof of Theorem C Part (iv)followsfromProposition2.2.4,sinceη is a next-tominimal function.

For part (iii), by Lemma 3.3.6, we may assume φ ∈ g × -α + g × -β forsomepositiveroots β orthogonal to α.B yCo r o lla ry3.0.4,o neca nco n j uga tethepa iro fr oo ts(α, β) to apa iro fo rthog o nalsim p ler oo ts(α ′ , α ′′ ),usin gtheW eylgr o u p .Leta be the joint kernel of α ′ and α ′′ in h,andletz ∈ a be a generic rational semisimple element. Let S T ∶= α ′ ∨ + α ′′ ∨ + Tz for T ≫ 0 ∈ Q,w h e r eα ′ ∨ and α ′′ ∨ are the dual coroots. Since no linear combination of α ′ ∨ and α ′′ ∨ lies in a, S T is a generic element of a and thus for T big enough, g S T ≥2 is a Borel subalgebra of g that contains h.Thus,itisconjugate under the Weyl group to our fixed Borel subalgebra. The statement follows now from Theorem 2.2.6. We note that different choices for z may give V of different dimensions.

For part (ii), Proposition 3.3.

a minimal or a trivial automorphic function on G ′ .Thestatement follows now from Theorem B applied to η ′ together with the fact that its Whittaker coefficients, obtained by integration over the maximal unipotent subgroup

Part (i)isprovenverysimilarlytoTheoremA(i). ∎

Proof of Lemma 3.3.6

Let α be a simple root. We assume that g is simple.

Note that in simply-laced root systems, orthogonal roots are strongly orthogonal, and thus the sum of two roots is a root if and only if they have scalar product -1. Also, for any two nonproportional roots, the scalar product is in {-1, 0, 1}.Foranyrootε, we denote by ε ∨ the coroot given by the scalar product with ε.

Case 2. General: We can assume ψ ≠ 0. Let H ∈ h be a generic element that has distinct negative integer values on all positive roots. Note that z ⊂ g H >0 and ψ ∈(g * ) H

>0 .Decomposeψ = ∑ i>0 ψ i ,whereψ i ∈(g * ) H i .W eprovethelemma by descending induction on the minimal j for which ψ j ≠ 0. The base of the induction is j that equals the maximal eigenvalue of ad * (H).Inthiscase,ψ = ψ j and we are in Case 1. For the induction step, let j be minimal with ψ j ≠ 0. By Case 1, there exists

By the induction hypothesis, there exists

B yL e m m a3.1.2, we can assume α ∈ F.U s i n g Lemma 3.4.2, we can assume that for any other ε ∈ F,w eh a v e⟨α, ε⟩≤0, i.e., F ⊂ {α}∪Ψ α ,whereΨ α is as in (3.3), namely

Assume first that there exists β ∈ F with (α, β)=0, and let

By (3.11), we see that φ α + φ β lies in the closure of the complex orbit O of φ.Now ,by Lemma 3.3.1, φ α + φ β is next-to-minimal and thus lies in O.Thus,Lemma2.2.8 and (3.11) imply that φ is conjugate to φ α + φ β under L α ∩ .

Let us now show that β ∈ F with ⟨α, β⟩=0 indeed exists. Assume the contrary, i.e., (α, ε)=-1f o ra l lε ∈ F.N o t et h a tF is not empty, since φ is not minimal. Pick any

By (3.11), we see that φ α + φ ω lies in the closure of the complex orbit O of φ and thus is minimal or next-to-minimal. This contradicts Lemma 3.3.1 since ⟨α, ω⟩<0.

Thus, there exists β ∈ F with ⟨α, β⟩=0, and as we showed above, φ is conjugate to

Proof of Theorems D, E, F,andG

Let α be a nice root. Denote by R the set of minimal elements in (g * ) Sα -2 and by X the set of next-to-minimal elements in (g * ) Sα -2 .L e tα max be the highest root of the component of g that includes α.Recallthatδ α denotes α max if α is an abelian root and denotes α maxαβ,wh er eβ is the only simple root nonorthogonal to α,ifα is a nice Heisenberg root. Denote δ ∶= δ α .SeeSection4.3 below for more details on this δ in the Heisenberg case.

We will use the following geometric propositions, that we will prove in Section 4.3 below.

Proposition 4.0.1 (i) If α is abelian and ⟨α, α max ⟩>0, then X is empty.

). Note that this implies that at most one next-to-minimal orbit can intersect X. For the next proposition, we assume that either ⟨α, α max ⟩=0o rα is a nice Heisenberg root. Recall that in these cases , R α denotes the parabolic subgroup of L α with Lie algebra (l α ) δ ≤0 ,andletRQ α = R α ∩ Q α . Denote further by St α the stabilizer in L α ∩ of the plane g * -α ⊕ g * -δ , as an element of the Grassmanian of planes in g * . Proposition 4.0.2 RQ α ∩ is a subgroup of St α of index 2.

Proof of Theorem D

Let η be a next-to-minimal automorphic function on G.

Suppose first that α is an abelian root, i.e., the nilradical U α of the maximal parabolic P α is abelian. Using Fourier transform on U α ,weobtain

(as a point in the projective space of g * ). Thus,

where α denotes the quotient of L α ∩ by Q α ∩ .If⟨α, α max ⟩>0thenbyProposition 4.0.1, X is empty. This implies part (i) of Theorem D. Let us now assume ⟨α, α max ⟩=0 and prove part (ii) of Theorem D.B yP r o p o s i t

). Recall that we denote by Λ α the quotient of L α ∩ by RQ α ∩ .B y Proposition 4.0.2,wehave

From (4.1), (4.2), and (4.3), we obtain

as required where A and B are defined in the statement of Theorem D. Suppose now that α is a nice Heisenberg root. Let γ α be a representative of the Weyl group element s α s αmax s α ,w h e r es α and s αmax denote the corresponding reflections. Since ⟨α, α max ⟩=1, γ α conjugates α to α max .Thus,byProposition2.2.7,

We call the first sum the abelian term,andthesecondsumthenonabelian term. In the same way as above, we obtain

To determine the nonabelian term, we will need a further geometric statement. Recall that M α ⊂ L α denotes the Levi subgroup generated by the roots orthogonal to α.NotethatM α is the standard Levi subgroup of the parabolic Q α of L α . Lemma 4.1.1 The group M α ∩ R α ∩ is the stabilizer in M ∩ of the line g * -δ and of the plane g * -α ⊕ g * -δ . Proof The first assertion follows from Lemma 3.2.1 applied to the root δ.Thesecond one follows from Proposition 4.0.2,sinceM α ∩ R α is a parabolic subgroup of M α . ∎ Denote by X the set of next-to-minimal elements in g

From (4.5), (4.6), and (4.7), we obtain

as required.

P r o o fo fTh e o r e mE

We proceed by induction on the rank of g.Th eb a s ec a s ei s rank 1, that has no next-to-minimal forms, and the statement vacuously holds. For the induction step, let η be a next-to-minimal automorphic function. Let I ={β 1 ,...,β n } be a convenient quasi-abelian enumeration of the roots of g.Denoteα ∶= β n .Theorem C provides the expressions for all the terms in the right-hand side of the expressions in Theorem D,excepttheconstantterm.ByTheoremC(i), the restriction of the constant term F Sα ,0 [η](g) to the Levi subgroup L α is next-to-minimal or minimal or trivial. Thus, we can obtain the expressions for the constant term by Theorem B and the induction hypothesis. Applying Theorem C(ii)toF Sα ,φ for any φ ∈ g × -α ,weget,inthe notation of Theorem C,

Further, for any φ ∈ g × -α and ψ ∈ g × -αmax ,TheoremC(iii) provides an expression for F Sα ,φ+ψ [η]. This expression implies

Assume first that α ∶= β n is an abelian root. Then, using Theorem D,(4.10), (4.11), and the induction hypothesis, we obtain

Suppose now that α is a nice Heisenberg root. Then we need to add the expressions for the nonabelian term in (4.8). These are also provided by Theorem C.Namely ,

The theorem follows now from Theorem D and (4.10)-(4.14). ∎ Theorem F follows in a similar way but without using the induction and omitting some terms that vanish.

Proof of Proposition 4.0.2

Lemma 4.3.1 There exists w in the Weyl group of L α such that w 2 = 1 and w(α)=δ α .

Proof We can assume that g is simple. If α is abelian, we take w to be w 0 ,wherew 0 is the longest element in the Weyl group of L α .Sinceα is the lowest weight of the first internal Chevalley L α -module n α ,andδ α = α max is its highest weight, w 0 (α)=δ α . If α is Heisenberg, we take w to be s β w 0 ,whereβ is the only root attached to α. In this case, the highest weight of n α is α maxα, while the lowest weight is still α. Thus, w 0 (α)=α maxα.Sinceα is a Heisenberg root, β is orthogonal to α max .Thus, s β (α maxα)=α maxα -⟨α maxα, β⟩β = α maxαβ = δ α .T oprovethatw is an involution, we will show that w 0 (β)=-1. To see this, we apply the well-known fact that -w 0 isagraphautomorphismoftheDynkindiagram.Forg of type E 8 ,wehave α = α 8 , L α is of type E 7 , and the Dynkin diagram has no automorphisms. For g of type E 7 ,wehaveα = α 1 , L α is of type D 6 ,andw 0 isknowntobe-1. In the remaining case of g of type E 6 ,wehaveα = α 2 , β = α 4 , L α is of type A 5 ,and-w 0 induces the nontrivial graph automorphism, which however fixes β. Proof By the defining property of minuscule representations, it is enough to show that Φ α corresponds to the set of weights of a minuscule representation of M α .Note that for any root ε, ⟨α, ε⟩=ε(S α )-ε(S β )/2. Thus, ε ∈ Φ α if and only if ε(S β )=4. In other words, Φ α is the set of roots of the L β -module g S β 4 , described in [MS12,Section5] whereitiscalledthesecondinternalChevalleymodule,thereforewehavetoshowthat thesecondinternalChevalleymodulesthatariseareminuscule.

The second Chevalley module for the node β is given by all roots of g with coefficient 2 along β. This can never happen for g of type A, so the second Chevalley module is trivial. For types D and E,andα an extreme node of the Dynkin diagram, not necessarily nice, the second Chevalley module for the adjacent L β is irreducible [MS12]. This irreducible representation can be found uniformly by finding the lowest root θ of g with coefficient 2 along β.Th Proof In the ε notation, we have α = ε kε k+1 ,andΦ α consists of all the roots ε iε j with i < k < k + 1 < j.Thestabilizerofα in the Weyl group of L α permutes all i < k and all j > k + 1 independently. ∎ (ii)W eha vetoshowtha tforan yrootμ, μ(Z)≤2. If μ = α max, then μ(2 -1 S α )= 2a n dμ(β ∨ )=0. If μ = β,t h e nμ(2 -1 S α )=0a n dμ(β ∨ )=2. For any other μ, max(μ(2 -1 S α ), μ(β ∨ )) ≤ 1. ∎

We are now ready to prove Proposition 4.1.2.Letx ∈ X and decompose it to a sum of root covectors x = x α + ∑ ε∈Ψα x ε with x ε ∈ g * -ε .LetF ∶= {ε ∈ Ψ α |x ε ≠ 0}.ByLemma 3.3.1, F intersects Φ α and thus, by Lemma 3.1.2,wecanassumeδ ∈ F.Decomposex = x 0 + x 1 + x 2 with x i ∈(g * ) Z i .W ehavex 0 = x α + x δ . Applying Lemma 2.2.8 to S ∶= S α and Z, we obtain that there exists a nilpotent X ∈(l α ) Z >0 with ad * (X)(x 0 )=x 1 + x 2 . (4.18) Decompose X to a sum of root vectors X = ∑ λ∈Ψ X λ , X λ ≠ 0 ∈ g -λ ,whereΨissome set of roots. Choose some X ∈(l α ) Z >0 satisfying (4.18)suchthatthecardinalityofΨis minimal possible.

Proof Since X ∈(l α ) Z >0 ,w eh a v e⟨β, λ⟩>0f o ra n yλ ∈ Ψ. Suppose by way of contradiction X ∉ m.Th e n⟨α, λ⟩≠0f o rso m eλ ∈ Ψ. Fix such λ.Th e nLe m m a4.3.5(i) implies ⟨α, λ⟩<0andthus⟨α, λ⟩=-1andthusλ + α is a root and [X λ , x α ]≠0. By Lemma 4.3.4(ii), α + λ ∉ Ψ α , and thus this term has to be canceled by [X μ , x δ ] for some μ ∈ Ψ. Thus, μ = α + λδ is a root and thus ⟨α + λ, δ⟩=1. But this contradicts ⟨α + λ, δ⟩=⟨λ, δ⟩=⟨λ, α maxα -β⟩=0 + 1 -⟨λ, β⟩≤0. ∎ Thus, X ∈ m Z >0 .B u tm Z >0 = m Z 1 .Th u sa d * (X)(x 0 )∈(g * ) Z 1 and thus x 2 = 0a n d ad * (X)x 0 = x 1 .Let y ∶= Exp(-X)xx 0 =-ad * (X)(x 1 )+1/2(ad * (X)) 2 (x 0 ). (4.19)

The right-hand side of (4.19)h a so n l yt h e s et w ot e r m sb e c a u s eX ∈ g Z 1 , x ∈ g Z ≥0 and g = g Z ≤2 .Sincead * (X) raises the Z-eigenvalues by 1, we get that y ∈(g * ) Z 2 .Notethatall the roots of y still lie in Ψ α /{δ},sinceX ∈ m.Thus,x 0 + y ∈ X. By the same argument as above, there exists Y ∈ m Z 1 such that ad * (Y)(x 0 )=y.H o w e v e r ,a d * (Y)(x 0 )∈ (g * ) Z

1 and thus y = 0. Thus, Exp(-X)x = x 0 = x α + x δ , i.e., we can conjugate x using Exp(-X)∈M α ∩ into g × -α + g × -δ .ThisprovesProposition4.1.2. ◻ Remark 4.3.8 The assumption that G α is not of type D n is necessary, since in type D n , the Heisenberg root is α 2 and the set Φ α2 ⊂ Ψ α2 intersects both complex nextto-minimal orbits. Indeed, let λ ∶= α

belongs to the orbit given by the partition 2 4 1 n-4 ,a n dg × -α + g × -μ

belongs to the orbit given by the partition 31 n-3 . To see this note that in the ε notation, we have α

Examples for D 5

In the following examples, we will consider G = Spin 5,5 (A) with = Spin 5,5 (K).W e use the conventional Bourbaki labeling of the roots shown in Figure 3.Thecomplex nilpotent orbits for D 5 are labeled by certain integer partitions of 10 with a partial ordering illustrated in the Hasse diagram of Figure 4 where O 1 10 is the trivial orbit and O 2 2 1 6 the minimal orbit. Note that this ordering is based on the closure on complex orbits and not on the partial ordering that we introduced in [GGK + ]. There is no unique next-to-minimal orbit, and both O 2 4 1 2 and O 31 7 can occur as Whittaker supports of automorphic forms arising in string theory. These two complex orbits are usually denoted (2A 1 ) ′ and (2A 1 ) ′′ in Bala-Carter notation [CM93]w i t h2 A 1 indicating two orthogonal simple roots and the primes distinguish the two possible pairs (up to Weyl conjugation, see Lemma 3.0.4). Wewillfocusonexamplesofimportanceinstringtheory.Inparticular,weconsider expansions in the string perturbation limit associated to the maximal parabolic subgroup P α1 and the decompactification limit associated to P α5 discussed in section 1.9. The Fourier coefficients computed in (5.3)and(5.8)belowhavepreviouslybeen computed for particular Eisenstein series in [GMV15] equations (4.84) and (4.88), respectively, although with very different methods using theta lifts. While the Fourier coefficient (5.3) for a minimal automorphic form is readily checked to be of the same form as [GMV15, (4.84)], the comparison between Fourier coefficient (5.8) for a nextto-minimal automorphic form and [GMV15, (4.88)] is a bit more intricate and will be discussed further in Remark 5.1.1 below.

Minimal representation

We will start with considering a minimal automorphic function η min on G = Spin 5,5 (A). Such a minimal automorphic form can for instance be obtained as a residue of a maximal parabolic Eisenstein series [GRS97, GMV15, FGKP18]. We will compute the Fourier coefficients of η min with respect to the unipotent radical of Figure 4: Hasse diagram of nilpotent orbits for D5 with respect to the closure ordering on complex orbits. There are two nonspecial orbits given by 32 2 1 3 and 52 2 1. the maximal parabolic subgroup P α1 associated to the root α 1 ,w h i c hi st h es t r i n g perturbation limit discussed in Section 1.9, and the corresponding Levi subgroup L α1 has semisimple part of type D 4 .

We may describe such Fourier coefficients by Whittaker pairs (S α1 , φ) where S α1 = 2ω ∨ 1 and φ ∈ g * )

Sα 1 -2 . Indeed, the associated Fourier coefficient F Sα 1 ,φ is then the expected period integral over N Sα 1 ,φ = U α1 ,t h eu n i p o t e n tr a d i c a lo fP α1 ,w h e r ew e recall that N Sα 1 ,φ is given by (2.3).

As in previous sections, we will use the shorthand notation [U]=(U ∩ )/U for the compact quotient of a unipotent subgroup U.

Since η min is minimal, Theorem A(iii)g i v e st h a tF Sα 1 ,φ [η min ] is nonvanishing only if φ ∈ O min = O 2 2 1 6 or φ = 0. We will now consider the former. The latter can be computed using Theorem B with G of type D 4 or the results from [MW95]f o r Eisenstein series. By Corollary 3.1.3(i), φ ∈ O min can be conjugated to φ ′ = Ad * (γ 0 )φ ∈ g × -α1 by an element γ 0 ∈ L α1 ∩ . This conjugation leaves the integration domain invariant, or, equivalently, we may use Lemma 2.2.1 to obtain

The unipotent radical U α1 is a subgroup of the unipotent radical N of our fixed Borel subgroup, and we may make further Fourier expansions along the complement of U α1 in N. Of these Fourier coefficients, only the constant term survives since such nontrivial characters, combined with φ ′ , are in a larger orbit than O min and therefore do not contribute according to Corollary 2.2.5. By repeating these arguments, or equivalently use Lemma 3.1.1 based on a special case of Theorem 2.2.6 (where V is trivial), we obtain that

Next-to-minimal representations

Let η ntm be a next-to-minimal automorphic form on G = Spin 5,5 (A).Sincethereare two next-to-minimal orbits for D 5 , there are two cases to consider. We begin with automorphic forms associated with the next-to-minimal orbit WS(η ntm )={O 31 7 } that has dimension 16, also known as (2A 1 ) ′ in Bala-Carter notation. Let also P α1 = L α1 U α1 be the maximal parabolic subgroup of G withrespecttothesimplerootα 1 such that the Levi subgroup L α1 hassemisimplepartoftypeD 4 . Automorphic forms with the above Whittaker support can, for example, be obtained as generic elements of the degenerate principal series of maximal parabolic Eisenstein series associated with P α1 .

We will now compute the Fourier coefficients of η ntm with respect to U α1 using Theorem C. These are described by Whittaker pairs (S α1 , φ) where S α1 = 2ω ∨ 1 and φ ∈ (g * ) Sα 1 -2 .Thecaseφ = 0canbetreatedusingTheoremD with G of type D 4 . According to Theorem C or Corollary 2.2.5,w ea r eth usleftwi thφ being minimal or next-tominimal where the latter in this case only gives nonvanishing Fourier coefficients for φ ∈ O 31 7 and not

where γ min ∈ L α1 ∩ using Corollary 3.1.3(i). From Lemma 2.2.1,wethenhavethat

(5.4) Let I (⊥α1) =(β 1 , β 2 , β 3 )∶=(α 5 , α 4 , α 3 ) and L i be the Levi subgroup of G obtained from a subsequence of simple roots (β 1 ,...,β i ) of I (⊥α1) . Each semisimple part of L i has simple components of type A for which all simple roots are abelian according to Table 1,andthusI (⊥α1) is an abelian enumeration. Using Theorem C(ii), we obtain

As explained in Section 1.4, i-1 is defined as follows. Let Q i-1 denote the parabolic subgroup of L i-1 given by the restriction of β ∨ i to L i-1 .ThenQ i-1 is the stabilizer in L i-1 of the root space g * -β i of L i .Th e n i-1 =(L i-1 ∩ )/(Q i-1 ∩ ) with 0 ={1}. Concretely, we may take the representatives

where w i is a representative in of the simple reflection corresponding to the simple root α i .Thelastequalityin(5.7)istheBruhatdecompositionof 2 and is isomorphic to P 1 (K)×P 1 (K).

Let us now consider next-to-minimal characters φ = φ ntm ∈(g * ) Sα 1 -2 instead. By Proposition 4.0.1, φ ntm can be conjugated using

which are known to be in O 31 7 . Indeed, by Corollary 3.0.2,thereisaW eylwordw that moves the roots α 1 and α max to two orthogonal simple roots, and from the proof of the corollary, we have that these roots have to be α 4 and α 5 .

Lemma 2.2.1 together with Theorem C(iii)foranyofthesechoicesgive F Sα 1 ,φntm [η ntm ](g)=F Sα 1 ,Ad * (γntm)φntm [η ntm ](γ ntm g) = ∫

V W Ad * (wγntm)φntm [η ntm ](vwγ ntm g) dv (5.8) with V = Exp(v)(A) where v = g -α3 ⊕ g -α2-α3 ⊕ g -α1-α2-α3 .

Remark 5.1.1 We may now revisit the comparison between (5.8)a n dt h eF o u r i e r coefficient [GMV15, (4.88)] for a particular Eisenstein series. The latter is expressed in of double divisor sums and a single Bessel function. Specifying to the same Eisenstein series in (5.8), the Whittaker coefficient on the right-hand side resolves to a product of two (single) divisor sums and two Bessel functions (see, for example, [FGKP18]). We expect that the noncompact adelic integral in (5.8) will allow us to relate the two expressions, something that will require further investigation.

Lastly, we will consider the other next-to-minimal orbit O 2 4 1 2 of dimension 20 and Bala-Carter label (2A 1 ) ′′ . That is, consider η ntm such that WS(η ntm )={O 2 4 1 2 }. Such an automorphic form can, for example, be obtained as generic elements of the degenerate principal series of maximal parabolic Eisenstein series associated with P α4 or P α5 . We showed above that all the next-to-minimal elements in (g * ) Sα 1 -2 are in O 31 7 , and thus the corresponding next-to-minimal Fourier coefficients F Sα 1 ,φ [η ntm ] would vanish.

Therefore, we will here consider another parabolic subgroup P α5 = L α5 U α5 associated with the root α 5 such that L α5 has semisimple part of type A 4 .L e tS α5 = 2ω ∨ and φ ntm a next-to-minimal element in (g * ) Sα 5 -2 .B yP r o p o s i t i o n4.0.1, there exists γ ntm ∈ L α5 ∩ such that Ad * (γ ntm )φ ntm ∈ g × -α5 + g × -αmax . Furthermore, by Corollary 3.0.4, there exists a Weyl word w ij , and simple roots α i and α j such that Ad * (w ij γ ntm )φ ntm ∈ g × -α i + g × -α j with the possible choices listed in (5.10) below, up to interchanging the two roots. For any (and therefore all) such choices of simple roots α i and α j ,itisknownthatg × -α i + g × -α j ⊂ O 2 4 1 2 and thus φ ntm ∈ O 2 4 1 2 . For any of the choices, Lemma 2.2.1 together with Theorem C(iii)gives F Sα 5 ,φntm [η ntm ](g)=F Sα 5 ,Ad * (γntm)φntm [η ntm ](γ ntm g) = ∫

V ij W Ad * (w ij γntm)φntm [η ntm ](vw ij γ ntm g) dv, (5.9) where V ij = Exp(v ij )(A) and v ij = v ji canbereadfromthefollowingtableusingthe notation α m1 m2 m3 m4 m5 = ∑ 5 i=1 m i α i . α i α j v ij = v ji α 1 α 3 g -α00010 ⊕ g -α01000 ⊕ g -α01110 ⊕ g -α01111 α 1 α 4 g -α01000 ⊕ g -α01100 ⊕ g -α01101 α 1 α 5 g -α00100 ⊕ g -α00110 ⊕ g -α01100 ⊕ g -α01110 ⊕ g -α01211 α 2 α 4 g -α00100 ⊕ g -α00101 α 2 α 5 g -α00100 ⊕ g -α10000 ⊕ g -α00110 ⊕ g -α11100 ⊕ g -α11110 ⊕ g -α11211

(5.10) Asonecanseefromtheabovetable,thesizeofV depends strongly on the choice of representative roots. The smallest choice is obtained in the fourth row.

The explicit Fourier expansions of η min and η ntm

We will now illustrate Theorems B, E,andF in the case of E 8 . According to theorems B and E, the general structure of the expansions of automorphic forms η min and η ntm attached to the minimal and next-to-minimal representation of E 8 are given by where the notation and the definitions of the individual terms are given in sections 1.4 and 1.7.

To illustrate this more explicitly, we now pick the Bourbaki enumeration as in Theorem F that is quasi-abelian for E 8 .L e tP = LU be the Heisenberg parabolic of , where all coefficients are evaluated for the automorphic form η = η ntm .Theelements g 8 and γ 8 are defined in Section 1.7 and Section 1.4,respectively . As discussed in Section 1.1, the expansion can be separated into an abelian contribution and a nonabelian contribution. The form of the expansion given above reflects this structure, as we now explain in more detail. We focus on the next-to-minimal case as this is the more complicated case.

Let ψ U be a unitary character on U(A),trivialonU(K). It is supported only on the abelianization U ab = C/U.Theabelian contribution to the Fourier expansion is then given by the constant term with respect to the center of the Heisenberg group

η ntm (zg)dz, (5.15) which can be expanded into a Fourier sum of the form ∑ ψU where we sum over all characters ψ U . The first term in the expansion F Sα 8 ,0 [η ntm ](g) is the constant term of η ntm with respect to U, i.e., corresponding to the contribution with trivial character ψ U . The abelian part, corresponding to terms labeled A,o fth en o n tri v i a l Fourier coefficients is made up of the second, third, and fourth terms on the right-the Fourier expansion for the next-to-minimal spherical Eisenstein series on E 8 was given in [BP17,equation (3.15)] that we reproduce here for convenience

e 2πi⟨ ,a⟩ (5.17) Here, explicit coordinates on E 8 /(Spin 16 /Z 2 ) adapted to the E 7 parabolic are used. Specifically, R is a coordinate for the GL 1 factor in the Levi and a denotes (axionic) coordinates on the 56-dimensional abelian part of the unipotent. L α is a lattice in this 56-dimensional representation of E 7 and the coordinates on the E 7 factor of the Levi enter implicitly through the functions Z( ) and Δ( ). We do not require their precise form for the present comparison. K s denotes the modified Bessel function and η E6 min a spherical vector in the minimal representation of E 6 .

We now est ablish t hat (5.17)and(5.14) are compatible. The Fourier expansion in (5.17)i sw r i t t e ni nt e r m so fs u m so v e rc h a r g e s in the integral lattice L α in the 56-dimensional unipotent and thus resembles structurally (5.14)a bo v ea sth es pa c e (g * ) Sα -2 represents the space of characters on this unipotent. The Fourier mode for a "charge" is given by e 2πi⟨ ,a⟩ and is the character on (g) Sα 2 . Besides the constant term F Sα ,0 [η], there is a sum over characters in the minimal and next-to-minimal orbits within (g * ) Sα -2 ;t h el a s tt e r mi no u r( 5.14)i san o n a b e l i a nt e r mt h a tw a sn o t determined in [BP17].

Minimal characters correspond to charges such that they satisfy the (rank-one) condition × = 0 in the notation of [BP17]andlookingat(5.17), we see that there are two contributions from such charges. These correspond exactly to the two terms A 8 and A 8 j in the first line of our (5.14): The first term A 8 represents the purely minimal charges, while the second term A 8 j in our equation is the second line of (5.17)wherea minimal charge is combined with a minimal automorphic form on E 6 . Expanding this minimal automorphic form on E 6 leads to Whittaker coefficients of the form W φ+ψ as they are given in the third term of the of the first line in (5.14), i.e., corresponding to A 8 j .Thesumsoverj, j-1 ,andg × -β j in our expression correspond to the E 7 orbits of such charges .ThetermA 88 in our formula (5.14) contains a noncompact integral over Whittaker coefficient W φ+ψ and corresponds to the last line in (5.17)wh e r ea similar integrated Whittaker coefficient B 5/2,3/2 appears. The nonabelian terms with B-labels in the last line of (5.14) have not been determined in [BP17]andaregivenby the ellipses in (5.17).