Contemporary Mathematics

Gauss Sum Combinatorics and Metaplectic Eisenstein Series
Ben Brubaker, Daniel Bump, and Solomon Friedberg

It is a great pleasure to dedicate this paper to Steve Gelbart, who was one of thefirst to work in this area.
1. Introduction
1.1. Steve Gelbart, Whittaker models and the metaplectic group.
The metaplectic double cover of Sp(2r) and the Weil representation were introduced by Weil [37] in order to formulate results of Siegel on theta functions in theadelic setting. This was followed by two initially independent developments. First,
Shimura [32], [33] gave two extremely important constructions involving modular
forms of half-integral weight. Both Shimura integrals involved Rankin-Selberg convolutions with a theta function which, in the modern view, lives on the metaplectic
double cover of SL2 (or GL2 ). Second, the mainstream modern context of automorphic representations of adele groups emerged with Jacquet and Langlands [23]and Godement and Jacquet [22]. (Gelbart’s book [11] was important in making
this modern point of view accessible to a generation of workers in the field.) It
became clear that the metaplectic group and Shimura’s constructions needed to be
reworked in the modern language. Gelbart [12] was perhaps the first to talk about
automorphic forms of half-integral weight, particularly theta functions, in completely modern terms. The Shimura constructions were carried out on the adelegroup by Gelbart and Piatetski-Shapiro [17], [18] for the Shimura correspondence,and by Gelbart and Jacquet [14], [15] for the symmetric square. Shimura’s important constructions were thus extended, and as a particular important point thelifting from GL2 to GL3 was established.
Jacquet and Langlands [23] emphasized the uniqueness of Whittaker models
for representations of GL2 . Shalika [31] and Piatetski-Shapiro [29] showed thatuniqueness holds over nonarchimedean local fields. (See also Gelfand and Kazhdan [21] and Bernstein and Zelevinsky [1].)For metaplectic covers of these groups, the Whittaker models may or may not be
unique. Gelbart and Piatetski-Shapiro considered the representations of the double
cover of SL2 that have unique Whittaker models and found them to be associatedwith theta functions. On the other hand Gelbart, Howe and Piatetski-Shapiro [13]showed that representations of the double cover of GL2 have Whittaker models(in a slightly modified sense) that are unique. The failure of uniqueness of Whittaker models might seem a defect, but Waldspurger [36] showed that precisely thisc 0000 (copyright holder)1

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BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

lack of uniqueness is the source of an important phenomenon, in which the global
Whittaker models of modular forms of half-integral weight encode L-values of the
Shimura correspondent. Waldspurger’s phenomenon was clarifed by Gelbart andPiatetski-Shapiro [19]. And Gelbart and Soudry [20] gave examples of automorphic
forms on the double cover of SL2 that have local Whittaker models at all places
but no global Whittaker models.
Higher metaplectic covers were defined by Kubota [26] (for SL2 ) and Matsumoto [28] (for general simply-connected groups), and in general Whittaker models
are not unique. Kazhdan and Patterson [24] showed that on the n-fold cover of
GLr , the Whittaker coefficients of theta functions (residues of Eisenstein series)
have unique Whittaker models if r = n or n − 1. However for general automorphic
forms, including Eisenstein series, Whittaker models are not unique.
At first sight, Whittaker models of Eisenstein series on metaplectic groups
seem to be difficult to compute. When uniqueness of Whittaker models fails, thecoefficients can sometimes be computed, but they at first appear chaotic, and thismay account for the fact that their properties remained hidden for a very long time.
Nevertheless in [3]–[10] a theory of Weyl group multiple Dirichlet series has beendeveloped by Brubaker, Bump, Chinta, Friedberg, Gunnells and Hoffstein. These
Dirichlet series are conjectured to be (global) Whittaker coefficients of metaplectic
Eisenstein series, though recent progress in developing their properties has been
made possible by not emphasizing this connection in favor of other methods.
1.2. Main results and outline of the paper. This paper continues to develop the theory of Weyl group multiple Dirichlet series from the perspective of [6].
There, we presented a definition of multiple Dirichlet series attached to the rootsystem Ar , r ≥ 1. We recall the basic set-up.
Let F be a totally complex algebraic number field containing the group µ2n of
2n-th roots of unity. Let S be a finite set of places of F containing the archimedean
ones and those ramified over Q that is large enough that the ring oS of S-integers
in F is a principal ideal domain.
Then to any r-tuple of oS integers m = (m1 , · · · , mr ), we associate a multiple
Dirichlet series of type Ar of formZΨ (s; m) = ZΨ (s1 , · · · , sr ; m1 , · · · , mr )
(1)

X

=

HΨ(c1 , · · · , cr ; m1 , · · · , mr )Nc1−2s1 · · · Ncr−2srwhere the sum is over nonzero ideals ci of oS , and we are denoting s = (s1 , · · · , sr ) ∈Cr .
Remark 1.1. Here H and Ψ are defined when ci are nonzero elements of oS ,but their product is well-defined over ideals, since H and Ψ behave in a coordinated
way when ci is multiplied by a unit. Thus the sum is essentially over ideals ci oS .
However we will want to consider H independently of Ψ, so for each prime p of oS
we fix a generator p of p, and only consider ci and mi which are products of powersof these fixed p’s.
The function Ψ is chosen from a finite-dimensional vector space M of functions
on (FS× )n that is well-understood and defined in [5] and [4], but the function H
is more interesting. It has a twisted multiplicativity with respect to both the ci ,i = 1, . . . , r and m. That is, H can be decomposed into relatively prime pieces up
to an nth root of unity determined by the m and the ci . While this implies that ZΨ

GAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

3

is not an Euler product, the specification of its coefficients is nevertheless reduced
to the case where the ci and mi are powers of the same prime p. See [3], [5] or [6]for further details, including a precise definition of the twisted multiplicativity.
Thus to specify ZΨ we are reduced to defining H(pk1 , · · · , pkr ; pl1 , · · · , plr ) fora fixed prime p. These “local coefficients” are defined by weighted sums over strict
Gelfand-Tsetlin patterns with a certain fixed top row. Recall that a Gelfand-Tsetlin
pattern of rank r is an array of integers

(2)

T=


a00









a01
a11

a02

···

a12
..

a1r

.

.
arr

..


a0r 








where the rows interleave; that is, ai−1,j−1 > ai,j > ai−1,j . We will say that thepattern is strict if each row is strictly decreasing. The weighting factors attached toeach pattern are products of Gauss sums formed with nth power residue symbols.
A precise definition is reviewed in the next section.
In [6] we made two conjectures concerning these multiple Dirichlet series. The
first stated that the multiple Dirichlet series associated to Ar appear in the FourierWhittaker coefficients of certain minimal parabolic metaplectic Eisenstein series on
n-fold covers of GL(r + 1). The second conjectured that these series have analytic
continuation to meromorphic functions on the whole complex space Cr and possess
functional equations isomorphic to the Weyl group Sr+1 of the corresponding rootsystem Ar . This paper concerns the latter of these two conjectures.
Several special cases of the conjecture, depending on the initial data r, m andn, have been proved. For example, in [4] we proved this when n is sufficiently large(depending on r and m). In this case, there are fewer non-zero local coefficients,
so the proofs of functional equations are much simpler. The conjecture for generaln remains open.
The present paper reduces this conjecture to a single combinatorial identity, andthen outlines an approach to proving this resulting identity. More detailed piecesof this proof will appear in subsequent papers, but we feel that it is particularly
illuminating to see several examples of the sorts of combinatorial objects involved.
The key idea is that while the prime power contributions to H are in bijection with Gelfand-Tsetlin patterns of fixed top row, there is no canonical way ofidentifying the two sets. In fact, given one such map from patterns to H, we cancompose this with any involution of Gelfand-Tsetlin patterns preserving the top
row to get another parametrization of the local coefficients H. A distinguishedrole is played by an involution, originally defined by Schützenberger [30] for Youngtableaux, which has been translated into the language of Gelfand-Tsetlin patterns
by Berenstein and Kirillov [25].
In Theorem 2.2, we explain how our initial parametrization of H in terms ofpatterns from [6] can be used to inductively prove all but one functional equation associated to simple reflections in the Weyl group. A second parametrization,
obtained by composing the coordinate map of [6] with the Schützenberger involution, can be used to inductively prove a different collection of all but one ofthese functional equations. Thus, we reduce our conjecture to proving that the twoparametrizations of H by Gelfand-Tsetlin patterns are equal.

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BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

In Theorem 3.5, we further reduce the conjecture by decomposing the Schützenberger involution into involutions ti corresponding to simple reflections σi , i =1, . . . , r, and prove that one only needs the equivalence of two new parametrizationsof H by Gelfand-Tsetlin patterns which differ by a single tr . This allows us torestrict our attention, within Gelfand-Tsetlin patterns, to the top three rows, as tr
does not affect the remaining rows.
In the final section, we illustrate some of the combinatorial techniques used
to prove the reduction resulting from Theorem 3.5. These techniques prove therequired identity for all but one class of degenerate patterns, and we give an example
in the degenerate case to give a flavor of the remaining complexity.
We would like to thank Gautam Chinta, Paul Gunnells and the referee for
helpful comments. This work was supported by NSF FRG grants DMS-0354662
and DMS-0353964 and by NSA grant H98230-07-1-0015.

2. Definitions and a first reduction of the FE Conjecture
As noted above, in [6] we defined a Weyl group multiple Dirichlet series oftype Ar in terms of Gauss sums indexed by strict Gelfand-Tsetlin patterns. We
will repeat the definition now, with slightly different notation and give a second dual
definition. The two definitions are not known to be equivalent, and the remainderof this paper is devoted to progress toward this claim. To conclude the section, we
will explain how the equivalence of these two definitions would imply Conjecture 1
of [6], which asserts that the resulting multiple Dirichlet series possesses functional
equations isomorphic to the Weyl group Sr+1 of Ar . The conjecture will henceforth
be referred to as the FE conjecture. In the middle of the section, we also digress
to discuss the connections between these formulas and the representation theory of
SLr+1 (C).
2.1. Definitions for coefficients of Weyl group multiple Dirichlet series. Given a number field F and a set of
Qplaces S, as described in the introduction,let ψ be an additive character of FS = v∈S Fv that is trivial on oS but no larger
fractional ideal. If m, c ∈ oS with c 6= 0 let
X  a   am ψ,(3)g(m, c) =cca mod cwhere ac is the n-th power residue symbol. In much of the following discussion,
we fix a prime p, and use the following abbreviated notation when no confusion canarise:(4)g(a) = g(pa−1 , pa ),h(a) = g(pa , pa ),q = |o/po|.

If T is the Gelfand-Tsetlin pattern (2), define GR (T) = GL (T) = 0 if T is notstrict; assuming strictness, we define them as follows. Let 1 6 i 6 j 6 r and(5) Ri,j = Ri,j (T) =

r
X
k=j

(ai,k − ai−1,k ),Li,j = Li,j (T) =

j
X
k=i

(ai−1,k−1 − ai,k ).

GAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

5

We say that T is left-leaning at (i, j) if ai,j = ai−1,j−1 and that T is right-leaningat (i, j) if ai,j = ai−1,j . We define
 R
if T is right-leaning at (i, j); q i,jg(Ri,j ) if T is left-leaning at (i, j);(6)
GR (Ri,j ) = GR (Ri,j (T)) =
h(Ri,j ) otherwise.
Similarly, we define(7) L
if T is left-leaning at (i, j); q i,jg(Li,j ) if T is right-leaning at (i, j);
GL (Li,j ) = GL (Li,j (T)) =
h(Li,j ) otherwise.

Thus we have attached one of the number-theoretic quantities in (4) to each entryof the strict Gelfand-Tsetlin pattern.
LetYYGR (T) =GR (Ri,j ),GL (T) =GL (Li,j ).
16i6j6r

16i6j6r

If the dependence on p is to be emphasized, we will sometimes denote these by
GR (T; p) and GL (T; p). We further define two sets of “local coordinates”
kR (T) = (k1R , · · · , krR ),kL (T) = (k1L , · · · , krL ),bykiR = kiR (T) =

(8)

r
X
(ai,j − a0,j )j=i

andkiL = kiL (T) =

(9)

r
X

(a0,j−r−1+i − ar+1−i,j ).

j=r+1−iFinally, we define
HR (pk1 , · · · , pkr ; pl1 , · · · , plr ) =

(10)

X

GR (T)kR (T)=(k1 ,··· ,kr )and similarly(11)

HL (pk1 , · · · , pkr ; pl1 , · · · , plr ) =X

GL (T),kL (T)=(k1 ,··· ,kr )where, in each case, the sum runs over all Gelfand-Tsetlin patterns with top row(12)

λ + ρ = {l1 + . . . + lr + r, l2 + . . . + lr + r − 1, · · · , lr + 1, 0} ,satisfying the indicated condition beneath each sum. Hereλ = {l1 + . . . + lr , l2 + . . . + lr , · · · , lr , 0} ,and ρ = (r, r − 1, · · · , 0). As noted in the introduction, because we know how todecompose the multiple Dirichlet series in (1), via twisted multiplicativity in m
and the ci , i = 1, . . . , r., into prime power pieces, then either description (10) or(11) suffices to complete the definition of the series. For a precise description of thetwisted multiplicativity, see [6].

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BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

2.2. Gelfand-Tsetlin patterns and representation theory. We now explain how the above data defined from Gelfand-Tsetlin patterns relates to the representation theory of SLr+1 (C).
First we review the case where n = 1, which is explained more fully in [6].
When n = 1, the Weyl group multiple Dirichlet series are just Whittaker coefficients of Eisenstein series of minimal parabolic type on P GLr+1 . According to theformula of Shintani [34] and of Casselman-Shalika [7], these are Schur polynomials,
represented by the Weyl character formula. Specifically, the Satake isomorphism
associates with s and the prime p a conjugacy class A in the L-group SLr+1 (C),and the Shintani-Casselman-Shalika formula identifies the p-part of a Whittakercoefficient of the Eisenstein series with the character of an irreducible representation of SLr+1 (C) applied to A. Here the Whittaker coefficient is with respect toa character of the maximal unipotent whose p-part is determined by a partition λassociated to a highest weight vector in the representation.
The relevance of Gelfand-Tsetlin patterns can be seen from this – one may
give a description of the irreducible representations of SLr+1 (C) based on GelfandTsetlin patterns in which the patterns parametrize vectors in irreducible modules of
SLr+1 (C); in this parametrization, the top row of the pattern is the highest weightvector of the representation, and the row sums are data determining the weightspace in which the vector lies. We will not use this description – rather we will use
a variant due to Tokuyama in which the top row of the pattern is not λ but λ + ρas in (12), and only strict Gelfand-Tsetlin patterns are used. But we mention it
in order to inculcate the idea that the top row of the pattern corresponds to thehighest weight vector, and the row sums correspond to weights.
We return to the fact that the p-part of the multiple Dirichlet series when n = 1
is the character of the irreducible representation with highest weight vector λ applied to the conjugacy class A in SLr+1 (C). The Weyl character formula expresses
this character value as a ratio of two quantities. The numerator is an alternating
sum of |W | = (r + 1)! monomials (where W is the Weyl group). The denominatorin the Weyl character formula can be expressed as either an alternating sum or as a
product. Writing it as a product, it resembles the p-part of the normalizing factorof the Eisenstein series, which is a product of 12 r(r + 1) zeta functions. However,the zeta functions are at the wrong values; they are shifted.
But Tokuyama [35] gave a deformation of the Weyl character formula in which
both the numerator and the denominator are altered. The denominator remains a
product of 12 r(r + 1) factors, and these can be brought into agreement with the ppart of the normalizing factor. The numerator becomes a sum over Gelfand-Tsetlin
patterns. This deformation of the Weyl character formula can now be recognized
as the p-part of the multiple Dirichlet series as we have defined it. Specifically, HR
as we have defined it in (10) is the contribution of all terms in a particular weightspace of the irreducible representation to Tokuyama’s formula. The row sums of
the pattern, that is, the numbers kR , correspond to the weight to which the term
contributes in the formula for Tokuyama’s numerator. Summing over all kR gives
the p-part of the multiple Dirichlet series, which is in exact agreement with thenumerator in Tokuyama’s formula. The denominator in Tokuyama’s formula isthen the p-part in the normalizing factor of the Eisenstein series.
Thus when n = 1, there is a well-understood connection with the representation
theory of SLr+1 (C). When n > 1, we are presented with a further “deformation” ofGAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

7

Tokuyama’s formula involving Gauss sums. The connection apparently persists but
it is no longer with the representation theory of SLr+1 (C) but with its quantum
analog.
Before we can address this point, it will be useful explain the roles of kR andkL as defined in (8) and (9). The patterns T with fixed top row and fixed valueof either kR or kL parametrize basis vectors for a weight space given by the values(k1 , . . . , kr ). (Typically, the coordinates for weight spaces are expressed in termsof differences of row sums in the Gelfand-Tsetlin pattern, and so our choice of kR
or kL can be seen as a composition of these usual coordinates with an affine linear
map. The affine linear map has been chosen so that the support of the coordinates
has ki non-negative for all i = 1, . . . , r.) The two coordinate choices kR and kL arerelated by the Schützenberger involution on Gelfand-Tsetlin patterns, an involution
originally defined on semistandard Young tableaux by Schützenberger [30] in thecontext of jeu de taquin, which was translated into the language of Gelfand-Tsetlin
patterns by Kirillov and Berenstein [25].
Following [25], the Schützenberger involution is defined in terms of simpler involutions on Gelfand-Tsetlin patterns labelled t1 , . . . , tr . For any i with 1 ≤ i ≤ r,ti affects only the entries in the (i + 1)st row of the pattern. Using the indexing asin (2), we observe that ai,j is constrained (by the definition of Gelfand-Tsetlin patterns) to lie between max(ai−1,j , ai+1,j+1 ) and min(ai−1,j−1 , ai+1,j ). The involution
ti reflects it in this range, so that(13)ti (ai,j ) = min(ai−1,j−1 , ai+1,j ) + max(ai−1,j , ai+1,j+1 ) − ai,jfor j ∈ [i + 1, r − 1]. At the ends of the (i + 1)st row, we must modify this slightly:(14)

ti (ai,i )= ai+1,i + max(ai−1,i , ai+1,i+1 ) − ai,i ,ti (ai,r )=min(ai−1,r−1 , ai+1,r ) + ai−1,r − ai,rFrom these ti , we can build a collection of involutions as follows.
Let q0 be the identity map, and define recursively
qi = t1 t2 · · · ti qi−1 .

(15)

In particular, qr is the desired Schützenberger involution. (Note the operations
ti obviously have order 2. They do not satisfy the braid relations, so ti ti+1 ti 6=
ti+1 ti ti+1 . However they do satisfy ti tj = tj ti if |i − j| > 1, which implies that qi
has order 2.)
The Schützenberger involution interchanges the two weights kR and kL – see(29) below. It would be nice if we could assert that GR (T) = GL (qr T), and indeed,this is true if T is in some sense “in general position.” However, there are exceptions
to this, and careful bookkeeping is required. What appears to be actually true isthatXX
GR (T) =GL (qr T),
kR (T)=(k1 ,··· ,kr )kR (T)=(k1 ,··· ,kr )and the thrust of the proof of our later Theorem 3.5 is to reduce this statement
to a simpler combinatorial one using the involution and its components ti . Although only strict patterns have a nonzero contribution to (10) and (11), theSchützenberger involution does not perserve the property of strictness. This is
the reason that in (10) and (11) we defined the terms to be zero and summed overall patterns.

8

BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

Gautam Chinta and Paul Gunnells have called our attention to the fact that
the Rij and Lij defined in (5) are the numbers appearing in what we might call
Littelmann patterns. Littelmann [27] associated a sequence of integers with thefollowing data: first, a Gelfand-Tsetlin pattern (or, more generally, a vertex in a
crystal graph); and second, a reduced word representing the long element in the
Weyl group as a product of simple reflections. The sequence of integers is obtained,
as Littelmann explains in the introduction and in Section 5 of [27], by applying
rasing operators in an order determined by the reduced expression of the long Weylgroup element to the vertices of the crystal graph, and tabulating the number oftimes each raising operator can be applied. For one reduced word these numbers,
put into an array, coincide with our Rij (later defined as a “Γ array” in (21)). For
another reduced word, they coincide with our Lij (later defined as a “∆ array”
in (22)). The observation of Chinta and Gunnells regarding this connection will beof doubtless importance in the further development of this theory.
2.3. First Reduction of the FE Conjecture. The coefficients HR definedin (10) agree with the coefficients H defined in [6], though the notation differs
slightly. Their equivalence is easy to see if one bears in mind that, with notation
as defined in this section, g(pa , pb ) = h(b) whenever a > b. The coefficients HLdefined in (11) are introduced here for the first time.
Conjecture 2.1. With definitions as above, HR = HL .
Before exploring the proof of this conjecture in subsequent sections, we note animportant consequence.
Theorem 2.2. Conjecture 2.1 implies the FE conjecture (Conjecture 1 in [6]).
Proof. (Sketch) We will not make use of Conjecture 2.1 until quite late in the
proof, and so will take ZΨ to be defined by the sum (1) with H = HR until further
notice.
The functional equations that must be satisfied are formulated in [4]. In thecase of Ar , the dependence on m1 , · · · , mr can be made more explicit as follows.
LetY2
bij sjZΨ (s; m),Z̃Ψ (s; m) = Z̃Ψ (s; m; Ar ) =Nmir+1
i,j

where

bij =

i(r + 1 − j) if i 6 j,
(r + 1 − i)j if i > j,and let Z̃ ∗ be Z̃ multiplied by certain Gamma factors, and a product of 21 r(r + 1)
Dedekind zeta functions, which are given explicitly in [3]–[5].
Let A be the ring of (Dirichlet) polynomials in qv±2s1 , . . . , qv±2sr where v runs
through the finite set of nonarchimedean places in S, and qv denotes the cardinalityof the residue field. Let M = A⊗M. As in [4] we regard elements of M as functions
Ψ : Cr × (FS× )r −→ C such that for any fixed (s1 , . . . , sr ) ∈ Cr the function(C1 , C2 , . . . , Cr ) 7−→ Ψ(s1 , . . . , sr ; C1 , . . . , Cr )defines an element of M, while for any (C1 , . . . , Cr ) ∈ (FS× )r , the functions = (s1 , . . . , sr ) 7−→ Ψ(s1 , . . . , sr ; C1 , . . . , Cr )

GAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

9

is an element of A. Now the functional equation should have the following form:
there is an action of the Weyl group W ∼= Sr+1 on M such that∗∗Z̃wΨ(ws; m) = Z̃Ψ(s; m).

As noted in [4] and [5], it suffices to prove one functional equation for each simple
reflection in the Weyl group as these generate the entire group. Denoting the simple
reflections by σ1 , · · · , σr , we may argue inductively that the functional equations forAr−1 imply functional equations for σ2 , · · · , σr , leaving us only to prove a functional
equation for σ1 .
Indeed, the base case of this induction, r = 1, is handled by the paper [2],where a single functional equation is proved for the Dirichlet series with coefficients
H(c; m) = g(m, c) as defined in (3). One can easily confirm that the Gauss sum decomposes into prime-power pieces according to the twisted multiplicativity definedin [5] and [6] and that the resulting H(pk ; pl ) for each prime p with ordp (m) = l > 0
is given by


l+1
0
GR (T) = GR
= g(pl , pk ),kaccording to the recipe outlined in (10) above. Hence, the Weyl group multiple
Dirichlet series for A1 coincides with the so-called “Kubota” Dirichlet series definedin [2] and the functional equation follows.
We now sketch a proof of the induction step. Note thatY(16) H(c1 , · · · , cr ; m1 , · · · , mr ) = µc,mH(pk1 (p) , · · · , pkr (p) ; pl1 (p) , · · · , plr (p) ),pwhere ki (p) = ordp (ci ) and li (p) = ordp (mi ), and µc,m denotes a certain n-th
root of unity depending on c = (c1 , c2 , . . . , cr ) and m. More specifically, it may
be expressed in terms of the n-th power reciprocity law by the multiplicativity
properties of H that are set out in [3], [5] and [4]. The product is essentiallyfinite since unless p divides one of the ci or mi we have ki (p) = li (p) = 0 and thecorresponding factor equals 1 (as one may readily check from (6) and (8)).
Now using the definition H = HR , the coefficient H(pk1 , · · · , pkr ; pl1 , · · · , plr )is a sum of the GR (Tp ; p) over Gelfand-Tsetlin patterns, with top row (12) andwith the sum of the elements in the i + 1-st row being ki minus the sum of the lastr + 1 − i elements of the top row. Now if for each p we are given a Gelfand-Tsetlin
pattern
a00 (p)a01 (p)a02 (p)···a0r (p) 







a11 (p)a12 (p)a1r (p)
Tp =
.
.
..


..






arr (p)we may consider the pattern

a00




a11
T =





a01

a02

···

a12
..

a1r

.

.
arr

where
aij =

Y
p

paij (p) .

..


a0r 








,

10

BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

This is a “Gelfand-Tsetlin pattern” in which the elements are not rational integers,
but rather elements of oS , and the inequalities defining a Gelfand-Tsetlin pattern
have been replaced by divisibility conditions, namely ai−1,j |ai,j |ai−1,j−1 . We callT a global Gelfand-Tsetlin pattern. Now denoteY
GR T = µc,mGR (Tp ),pwhere µc,m is as in (16). We see that we may write HR (c; m) as a sum of GR T ,where now the sum is over global Gelfand-Tsetlin patterns. As explained in Remark 1.1 we are only considering global patterns in which the elements are productsof powers of a fixed set of generators of the primes; if we considered HΨ instead ofH, we could sum over ideals.
∗
Now Z̃Ψis a sum over global patterns with fixed top row depending on m =(m1 , · · · , mr ). We may break this up as follows. Let us fix the top two rows andconsider the resulting sum. If the second row of the global pattern isYYY
pd1 (p)+···+dr (p)+r−1 ,
pd2 (p)+···+dr (p)+r−2 , · · · ,pdr (p)pppthen letm0i =Y

pdi (p) ,(i = 1, · · · , r − 1).

p

One may check that the sum over the patterns with fixed top two rows is a Weylgroup multiple Dirichlet series attached to Ar−1 of the form
∗
0
0
Z̃Ψ
0 (s2 , · · · , sr ; m1 , · · · , mr−1 ; Ar−1 )times a product of Gamma functions and Dedekind zeta functions, and a product ofpowers of the norms of the mi that is invariant under the transformations σ2 , · · · , σr .
Thus one obtains functional equations for this subset of Weyl group generators for∗Z̃Ψ.
Now we may explain at last why Conjecture 2.1 implies the FE Conjecture. If
one defines ZΨ in terms of HL instead of HR , the argument proceeds as above, but
we obtain instead the functional equations for σ1 , · · · , σr−1 . Combining these withthe functional equations for the simple reflections that we had before, one obtains
functional equations with respect to the full Weyl group.

3. A second reduction to a combinatorial conjecture
Let q be either a complex number or an indeterminate, and let R be a Z[q]algebra with generators h(a) and g(a), one for each positive integer a, subject tothe following relations.
Relation (i). Suppose that a, b > 0. Thenh(b)h(a + b) = q b h(b)h(a),h(b)g(a + b) = q b h(b)g(a).

Relation (ii). Suppose that a, b > 0 and a + b > 0. Then(17)g(a + b)h(a)h(b) = h(a + b)g(a)g(b) + h(a + b)g(a + b).
Relation (iii). If a > 0 we have(18)

h(a)2 = g(a)h(a) + q a h(a).

We also define h(0) = 1, but g(0) will never appear.

GAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

11

Let g and h be as in the previous section. We reiterate that g(a) is only defined
if a > 0. Let q be the cardinality of o/po.
Lemma 3.1. If n|a and a > 0 thenh(a) = φ(pa ) = q a−1 (q − 1),g(a) = −q a−1 ,1while if n - a then h(a) = 0 and |g(a)| = q a− 2 . If n - a, b but n|a + b theng(a)g(b) = q a+b−1 .
If a ≡ b modulo n with a, b > 0, then h(a) = q a−b h(b) and g(a) = q a−b g(b).
These facts about Gauss sums are standard.
The Lemma implies that there is an algebra homomorphism from the algebra
R to the Z[q]-algebra generated by g(a) and h(a) for a > 0, for if one replacesg(a) by g(a) and h(a) by h(a) then Relations (i), (ii) and (iii) are still satisfied.
The Relations can be proved on a case-by-case basis depending on the divisibility
properties of a, b and a + b. Relation (i) follows since if h(b) = 0, both sides of both
equations vanish; otherwise n|b, and so Relation (i) follows from the last statementof Lemma 3.1. Both sides of (18) vanish unless n|a since h(a) appears in every
term; and if n|a, then by Lemma 3.1 we have h(a) = g(a) + q a , so (iii) follows. As
for (ii), equation (17) is harder than (18) but also follows from Lemma 3.1.
Remark 3.2. When one begins to investigate in this area, one encounters abewildering array of identities. One could base their combinatorial study on either
Lemma 3.1 or on Relations (i), (ii) and (iii). There are two advantages of the latterapproach. First, it avoids a descent into case-by-case considerations depending on
the divisibility of various parameters by n. A second very important advantage isthat it allows one to work in R; as we will illustrate later the relations that define
R are precisely those needed to prove the combinatorial Conjecture 3.4 below inthe rank one case, and it is our experience that no further identities will be neededin higher rank. There are also advantages to working directly with g and h, and we
may do so in subsequent papers, but to clarify the issues we formalize the relationsin this one.
By a short Gelfand-Tsetlin pattern (or short pattern) we mean an array


l2
l3
···
lr+1 
 l1
a1
a2
ar
(19)
t=
,


b1
···
br−1
where the rows are nonincreasing sequences of integers that interleave, that is,(20)

li > ai > li+1 ,ai > bi > ai+1 .

We will refer to l1 , · · · , lr+1 as the top row of t, a1 , · · · , ar as the middle row andb1 , · · · , br−1 as the bottom row .
Our aim is to define two R-valued functions GΓ and G∆ on the set of shortGelfand-Tsetlin patterns. Let us assume momentarily that t is strict. We willassociate with t two arrays


Γ1,1
Γ1,2
Γ1,r
Γ = Γ(t) =Γ2,1···
Γ2,r−1and
∆ = ∆(t) =

∆1,1

∆1,2
∆2,1

∆1,r
···

∆2,r−1,12

BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

where Γi,j = Γi,j (t) and ∆i,j = ∆i,j (t) are to be defined. If t is as in (19) then(21)

Γ1,j = Γ1,j (t) =rX(ak − lk+1 ),Γ2,j = Γ2,j (t) =

k=j

j
X

(ak − bk ),

k=1

and
(22)

∆1,j = ∆1,j (t) =jX(lk − ak ),∆2,j = ∆2,j (t) =

k=1

r−1
X
(bk − ak+1 ).
k=j

Let
 Γ
if aj = lj+1 ; q 1,jg(Γi,j ) if aj = lj ;Γ̃1,j = Γ̃1,j (t) =h(Γ1,j ) otherwise, Γ
if bj = aj ;
 q 2,jg(Γ2,j ) if bj = aj+1 ;Γ̃2,j = Γ̃2,j (t) =h(Γ2,j ) otherwise,

and also let ∆if aj = lj ;
 q 1,j˜g(∆1,j ) if aj = lj+1 ;∆1,j =
h(∆1,j ) otherwise,

 ∆if bj = aj+1
 q 2,j˜g(∆2,j ) if bj = aj ;∆2,j =
h(∆2,j ) otherwise.

Now define(23)GΓ (t) =YΓ̃i,j (t),G∆ (t) =Y

˜ i,j (t).
∆

i,j

i,j

The above definition assumed that t is strict. If t is not strict we define GΓ (t) =G∆ (t) = 0.
Remark 3.3. Observe the following important difference between how we define the products GR and GL for ordinary (r + 1-rowed) Gelfand-Tsetlin patterns
and the products GΓ and G∆ for short patterns. Referring to (5), if T is an ordinary Gelfand-Tsetlin pattern, we use a “right-leaning rule” to define Ri,j and
GR (Ri,j ) in every row; similarly we use a “left-leaning rule” to define the L array.
(The reader will understand the meaning of “left-leaning” and “right-leaning” inthis context after computing an example.) In contrast, for the short pattern, the Γarray is obtained by using a right-leaning rule in the middle row and a left-leaning
rule in the bottom row, while the ∆ array is obtained using a left-leaning rule inthe middle row and a right-leaning rule in the bottom row.
We define the weight k of t to be the sum of the ai . Finally, if t is a shortpattern we define another short pattern t0 as the image of t under t1 defined in (13)and (14). More explicitly,


l2
l3
···
lr+1 
 l1
a01
a02
a0r
(24)
t0 =
,


b1
···
br−1
where
(25)

a0i = min(li , bi−1 ) + max(li+1 , bi ) − ai ,(26)

a01 = l1 + max(l2 , b1 ) − a1 ,(i = 2, · · · , r − 1),

a0r = min(lr , br−1 ) + lr+1 − ar .

GAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

13

By a short pattern type S of rank r we mean a triple (l, b, k) specifying the following
data: a top row consisting of an integer sequence l = (l1 , · · · , lr+1 ), a bottom row
consisting of a sequence b = (b1 , · · · , br−1 ), and a positive integer k. It is assumedthat> . . . > lr+1 , that b1 > b2 > . . . > br−1 , that li > bi > li+2 , and that
P l1 > l2P
l i > k > bi .
We say that a short pattern t of rank r belongs to P
the type S if it has theprescribed top and bottom rows, and its weight is k (so i ai = k). By abuse ofnotation, we will use the notation t ∈ S to mean that t belongs to the type S.

(27)

Conjecture 3.4. We haveXGΓ (t) =XG∆ (t0 ).

t∈S

t∈S

Theorem 3.5. Conjecture 3.4 implies Conjecture 2.1.
Hence by Theorem 2.2 we see that Conjecture 3.4 implies the FE Conjecture.
Proof. We begin by observing that the operations t1 , · · · , tr on GelfandTsetlin patterns that were previously defined in (13) and (14) of our discussion of
the Schützenberger involution boil down to the operation t 7−→ t0 on short GelfandTsetlin patterns. Indeed, if 1 6 k 6 r, we extract from the Gelfand-Tsetlin pattern
T given by (2) the short pattern made with the (r−k)-, (r+1−k)- and (r+2−k)-throws of T. We apply the operation t 7−→ t0 to this short pattern and reinsert it
into T; the resulting pattern we call tk T. Note that only the (r + 1 − k)-th row
is changed by this procedure. If k = 1, the (r + 1)-th row of T is empty, so t1
should be interpreted as just replacing arr by ar−1,r−1 + ar−1,r − ar,r . We have−1
t1 · · · ti−1 = qi−1 qi−2
= qi−1 qi−2 and therefore(28)

qi = qi−1 qi−2 ti qi−1 .

Next let
P us consider the effect of the Schützenberger involution on the weights.
Let A0 = j ai,j be the sum of the i-th row of T. It may be checked that the row
sums of qr T are (in order)
A0 , A0 − Ar , A0 − Ar−1 , · · · , A0 − A1 .
From this it may easily be deduced that(29)

(qr T) = kL (T).

From this we see that Conjecture 2.1 will follow if we prove
X
X(30)
GR (T) =GL (qr T).
kR (T)=kkR (T)=k

We note that the sum is over all patterns with fixed top row and row sums. The
proof of will involve an induction on r, and so we introduce the temporary notations(r)
GR (T) = GR (T) and (r) GL (T) = GL (T).
Note that the sum in (29) is over all patterns with fixed top row and row sums.
The proof of will involve an induction on r, and so we introduce the temporary
notations (r) GR (T) = GR (T) and (r) GL (T) = GL (T).
If we discard the top row of T, we obtain a Gelfand-Tsetlin pattern Tr−1 ofrank r − 1; similarly let Tr−2 be the pattern obtained by discarding the top two

14

BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

rows. We will denote
GiR (T) =r
Y

GiL (T) =GR (Ri,j )(T),

j=i

r
Y

GL (Li,j )(T).

j=i

Let us show that if k = (k1 , · · · , kr ) is a fixed r-tuple then
X

(31)

X

GR (qr−1 T) =kR (T)=k

GrR (T)Gr−1
L (T)

r−2
Y

GiL (T)

i=1

kR (T)=k

and
X

(32)

GL (qr−1 qr−2 T) =kR (T)=kX

GrL (T)Gr−1
R (T)

r−2
Y

GiL (T).

i=1

kR (T)=k

To prove (31), the left-hand side equalsX

GrR (qr−1 T)

r−1Y

GiR (qr−1 T).

i=1kR (T)=k

We have GrR (qr−1 T) = GrR (T), because qr−1 does not affect the top two rows of T.
Also, let Tr−1 be the pattern of rank r − 1 obtained by discarding the top row of
T. Then we obtain
X
GR (qr−1 T) =kR (T)=kX

GrR (T) · (r−1) GR (qr−1 Tr−1 )kR (T)=kX=

GrR (T) · (r−1) GL (Tr−1 ),kR (T)=k

where we have used our induction hypothesis, and this is the right-hand side of (31).
This step must be understood as follows: fix the top row of Tr−1 (that is, the top
two rows) and then sum over all remaining rows, with the row sums fixed (dependingon k). In this summation GrR (T) is constant and may be pulled out of the innersummation (over all rows but the top). The inner sum may be treated using (30)with r replaced by r − 1. This proves (31).
Next we note that qr−1 qr−2 does not affect the top two rows of T, so GiL (qr−1 qr−2 T) =i
GL (T) when i = r. Thus the left-hand side of (32) equalsXX
GrL (qr−1 qr−2 T)·(r−1) GL (qr−1 qr−2 Tr−1 ) =GrL (T)·(r−1) GR (qr−2 Tr−1 ),kR (T)=kkR (T)=k

where we have used the induction hypothesis. Let Tr−2 be the pattern obtained byomitting the top two rows of T; we see that the left-hand side of (32) equalsX(r−2)
GrL (T)Gr−1
GR (qr−2 Tr−2 ).
R (qr−2 T) ·kR (T)=k

We note that qr−2 does not change the top three rows of T, and that by induction
we have (r−2) GR (qr−2 Tr−2 ) = (r−2) GL (Tr−2 ). Equation (32) follows.
Now Conjecture 3.4 implies that
(33)

X
kR (T)=k

GrR (T)Gr−1
L (T)

r−2
Y
i=1

GiL (T) =X
kR (T)=kr−1
GrL (tr T)GR
(tr T)r−2
Y
i=1

GiL (T).

GAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

15

Indeed, let t be the short pattern obtained by discarding all but the top three rowsof T. Thenr−1
GrL (tr T)GR
(tr T) = G∆ (t0 ).

GrR (T)Gr−1
L (T) = GΓ (t),So we may fix all rows but the second of T, and sum over all patternsQwith that row
r−2
allowed to vary, but with fixed row sum. In the inner summation i=1 GiL (T) is
constant, since it does not depend on the second row of T, so it may be pulled outof the inner summation. Applying Conjecture 3.4 to the inner sum proves (33).
Now (31), (32) and (33) show thatXX
GR (qr−1 T) =GL (qr−1 qr−2 tr T).
kR (T)=kkR (T)=k

Replacing T by qr−1 T (and changing k) gives
X
X
GR (T) =GL (qr−1 qr−2 tr qr−1 T).
kR (T)=kkR (T)=k

The theorem follows by (28).



4. Gauss sum combinatorics
With the reduction to Conjecture 3.4 we have entered into a very rich combinatorial landscape. We will only mention a few features.
A short pattern t is called superstrict if each defining inequality (20) is strict.
We call t nonresonant if li+1 6= bi for 1 6 i 6 r − 1. Finally, we call t stable if eachelement of the middle and bottom rows is equal to one of the two elements above it.
(Note that a short pattern may satisfy more than one of these conditions at once.)
Theorem 4.1. If t is superstrict, nonresonant or stable, thenGΓ (t) = G∆ (t0 ).

(34)

Thus in Conjecture 3.4 there is no need to sum over patterns of these threeclasses. The proof of Theorem 4.1 will be given in a subsequent paper. To givesome feeling for the combinatorial nature of this situation, we discuss briefly thesuperstrict case.
Lemma. There exists orderings of the Γij (t) and ∆ij (t0 ) such that0Γij (t)} = {γ1 , γ2 , · · · , γ2r−1 },∆ij (t0 )} = {δ10 , δ20 , · · · , δ2r−1}
with the following property. Extend the labelings by letting γ0 = γ2r = 0. Thenγk
if k is even,
(35)
δk0 =
γk + γk−1 − γk+1 if k is odd.
The proof of this somewhat tricky combinatorial Lemma will be given elsewhere, but let us do an example. Suppose that

t=


 45


37
40

28
30

34

14
15

20

5
7

10

3
6


0 


,

16

BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

t0 =


 45

37

28

42

32

10

The arrays Γ(t) and ∆(t0 ) are as follows:

/
8

 11
Γ(t) =

6
16∆(t0 ) =




9

20


,

3

5

19

34



14

8

6

6
21


13

22

5

4

,




/ 25 


22

6

.



5

17

8

2


0 

.




We have indicated the ordering of the entries with a pair of “snakes.” If we computethe ordered sets Γij and ∆ij defined above, we have:
k
γk
δk

0
0

1 2
11 8
3 8

3
6
8

4
6
6

5 6
16 5
17 5

7 8
21 22
4 22

9 10
3
0
25

we see that the Lemma is satisfied. We note that the snakes depend on the original
data, and we will give a recipe for finding them in a later paper, where the proof ofthis “Snake Lemma” is given. Assuming the lemma and proceeding with the proof,
we now assume that t is superstrict. We haveYYGΓ (t) =h(γi ),G∆ (t0 ) =h(δi ).
i

i

Assuming the Lemma, we rewrite the latter productYYYYh(γi )h(γi + γi−1 − γi+1 ) =h(γi )h(γi )q γi−1 −γi+1 ,i eveni even

i odd

i odd

where we have used Relation (i). The powers of q cancel, so (34) is satisfied.
This shows that the Lemma implies Theorem 4.1 in the superstrict case. The
nonresonant and stable cases require other ideas.
Although the patterns handled in Theorem 4.1 are in some sense most, still(34) is not true for all patterns. For example, it may be that t is non-strict whilet0 is strict. Then GΓ (t) = 0 but G∆ (t0 ) 6= 0. Thus (27) is not true without thesummation.
Resonance is a phenomenon that occurs when, in the notation (19), we haveli+1 = bi for one or several values of i. The terminology is suggested by quantum
chemistry, where some compounds such as benzene can occur with two or morestructures contributing to the wave function. In the context of small GelfandTsetlin patterns, patterns associated with a resonance can generally be grouped
into fairly small sets called packets such that if Π is a packet then all members ofΠ belong to the same type, andX
X(36)
GΓ (t) =G∆ (t0 ).
t∈Πt∈ΠGAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

17

The first example that occurs is with A2 . We consider (short) patterns in theresonant type


l2
l3 
 l1
a1
a2
,


l2
with a fixed row sum a1 + a2 = k. Among these patterns there will be two extremal
ones, one in which a1 is as large as possible (so either a1 = l1 or a2 = l3 or both),and one in which a1 is as small as possible (so a2 equals a1 or a2 or both).
Proposition 4.2. These two extremal patterns form a packet.
Proof. (Sketch) To prove this, there are several cases, and we take a typicalone. Suppose that the extremal patterns are




l2
l3 
l2
l3 
 l1
 l1
l1
x
l2
y
t1 =
,
t2 =
,




l2
l2
where x + l1 = y + l2 and l2 > x > l3 , l2 > y > l3 . Then



l2
l3 
 l1
 l1
l2
l2 + l3 − x
l1
t01 =
,
t02 =



l2

l2
l2 + l3 − y
l2


l3 

.



We find thatGΓ (t1 ) =GΓ (t01 ) =


a


 c
b



 b



,


c

a

GΓ (t2 ) =








 c



,

G∆ (t02 ) =

?>=<
89:;
0


 ?>=<
89:;
0



c



c
c

,




,



where a = x − l3 , b = l1 − l2 , c = a + b = y − l3 = l1 + x − l2 − l3 , and our convention
is that we circle an entry w if it contributes q w ; box it if it contributes g(w), andleave it unboxed and uncircled if it contributes h(w) in the definitions (23). Thus
if Π = {t1 , t2 }, then (36) boils down to the identity
g(c)h(a)h(b) + g(c)h(c) = g(b)h(c)g(a) + h(c)2 ,c = a + b,

which the reader will easily deduce from our relations. Even for A2 there are several
more cases, and the reader may treat them as an exercise.
For higher resonances, in which li+1 = bi for several consecutive i, many fascinating phenomena occur. We will give just one example, cautioning the reader
that in some ways it is not entirely typical. We now consider short patterns in theresonant type


l2
l3
l4 
 l1
x
y
z
t=


l2
l3
where l1 > l2 > l3 > l4 and x + y + z = k.
We will assume first that
2l2 > l1 + l3 ,max(l1 + l2 + l4 , l1 + 2l3 ) < k < 2l2 + l3 .

18

BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

For example, one could take the top row to be (63, 47, 25, 0) and the row sum to be
117. Let u = k − l2 − l3 − l4 = α1 + α2 = β1 + β2 = γ1 + γ2 = δ1 + δ2 , where
α1 = l1 − l2 ,β1 = l1 − l3 ,γ1 = l2 − l3 ,

δ1 = l3 − l4 .

We will also denotegα = g(α1 )g(α2 ),hα = h(α1 )h(α2 ),and similarly define gβ , gγ , gδ and hβ , hγ , hδ .
It is convenient to visualize the patterns in a type by mapping them into Zr bymeans of the second row. Since the row sum is fixed, one element is redundant, andwe may vizualize the type as the set of lattice points in some polytope. In this casewith r = 3, we embed the patterns into Z2 by mapping this t to (y, z). Assuming(4) the patterns form a trapezoid, which we can diagram thus:
a??
b??
??
??
??
??
??
??
??
c
??
??
??
e??
??
??
??
d
The labeled spots are the following five short patterns:
t

 l1

l2

l3


l4 

k − l1 − l2


l
l
2
3


l2
l3
l4 
 l1
l2
l2
k − 2l2
b=


l2
l3


l2
l3
l4 
 l1
l2
k − l2 − l3
l3
c=


l
l
2
3


l2
l3
l4 
 l1
l1
k − l1 − l3
l3
d=


l
l
2
3


l2
l3
l4 
 l1
l1
2l2 − l1
k − 2l2
e=


l2
l3
a=

l1

l2

GΓ (t)G∆ (t0 )g(u)gα hβh(u)gα gβ0h(u)2 gγ

q u h(u)gδ

g(u)h(u)hδg(u)hα gδ

g(u)gα hδ

g(u)hα hγh(u)gα hγ

Any pattern in the interior of the trapezoid is superstrict, hence consists of a singleton packet. On the other hand, the patterns on the boundary must be grouped
into packets of size 2, 3 and 5.
We claim that {a, b, c, d, e} is a packet. Indeed, from the above table this meansg(u)gα hβ + q u h(u)gδ + g(u)hα gδ + g(u)hα hγ
2h(u)gα gβ + h(u) gγ + g(u)h(u)hδ + g(u)gα hδ + h(u)gα hγ .

=GAUSS SUM COMBINATORICS AND METAPLECTIC EISENSTEIN SERIES

19

By Relation (iii) it is enough to prove the vanishing ofg(u)gα hβ + h(u)2 gδ − h(u)g(u)gδ + g(u)hα gδ + g(u)hα hγ−h(u)gα gβ − h(u)2 gγ − g(u)h(u)hδ − g(u)gα hδ − h(u)gα hγ .
Indeed, this may be rewritten+(hγ + gδ )[g(u)hα − h(u)gα − h(u)g(u)]+h(u)[g(u)hγ − h(u)gγ − h(u)g(u)]−(h(u) + gα )[g(u)hδ − h(u)gδ − h(u)g(u)]+gα [g(u)hβ − h(u)gβ − h(u)g(u)],which vanishes by Relation (ii).
The patterns on the interior of the segments from c to b and from e to d areequal in number, and they can be grouped into packets of order two by combining
two patterns that lie on the same vertical line. To see this, let k − 2l2 < a 6 l3 . Let
u = k − l2 − l3 − l4 as before, and write u = α1 + α2 = ε1 + ε2 , where α1 = l1 − l2and ε1 = a − l4 . As before we will denote gα = g(α1 )g(α2 ), hα = h(α1 )h(α2 ) andsimilarly for gε and hε . Consider the following two patterns.
tt[d,e] (a) =t[c,b] (a) =


 l1


 l1

l2

l3
k − l1 − a

l1

a

l2
l2

l3
l3
k − l2 − a

l2

a

l2



l3


l4 


l4 

GΓ (t)G∆ (t)

g(u)hα hεh(u)hε gα

q u g(u)hεh(u)2 hεThe notation indicates which segment each pattern lies in. It is straightforward tocheck that the relations imply
g(u)hα + q u g(u)=h(u)gα + h(u)2 .

Thus {t[d,e] (a), t[c,b] (a)} is a packet.
On the other hand, the three segments from a to b, from a to e and from c
to d each have the same number of patterns, and these can be grouped together
in packets of order 3. To see this, denote again u = k − l2 − l3 − l4 = α1 + α2 =ε1 + ε2 = θ1 + θ2 = δ1 + δ2 where α1 = l1 − l2 , ε1 = a − l4 , θ1 = a + l2 − l3 − l4and δ1 = l3 − l4 . Consider the following three patterns, one from each of thesesegments.
tt[a,e] (a) =t[a,b] (a) =


 l1


 l1

l2

l3
k − l1 − a

l1
l2

l2
k − l2 − at[d,c] (a) =


 l1


l2
l2



a
l3
l3
a

GΓ (t)G∆ (t0 )g(u)hα hεh(u)gα hεh(u)gθ hεh(u)hθ gεh(u)hθ gδ

g(u)hθ hδl3

a + l2 − l3
l2



l4 

l3

l2
k − l2 − a


l4 

l3
l3


l4 


20

BEN BRUBAKER, DANIEL BUMP, AND SOLOMON FRIEDBERG

To see that these three form a packet, one must proveg(u)hα hε + h(u)gθ hε + h(u)hθ gδ

=h(u)gα hε + h(u)hθ gε + g(u)hθ hδ .
This may be deduced from the relations as follows. The right-hand side minus theleft-hand side equalshε [g(u)hα − h(u)gα ] + h(u)[gθ hε − hθ gε ] + hθ [h(u)gδ − g(u)hδ ]=hε h(u)g(u) + h(u)[gθ hε − hθ gε ] − h(u)g(u)hθ=hε [h(u)g(u) + h(u)gθ ] − hθ [h(u)gε + g(u)hθ ]=g(u)hθ hε − g(u)hθ hε=0.

The above considerations verify Conjecture 3.4 for this particular resonant type.
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Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 021394307, email: brubaker@math.mit.edu
Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USAemail: bump@math.stanford.edu
Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA.
email: friedber@bc.edu