Eisenstein Series, Crystals and Ice
Benjamin Brubaker, Daniel Bump, and Solomon Friedberg∗
July 31, 2011

1

Introduction

Automorphic forms are generalizations of periodic functions; they are functions on a group thatare invariant under a discrete subgroup. A natural way to arrange this invariance is by averaging.
Eisenstein series are an important class of functions obtained in this way. It is possible to give
explicit formulas for their Fourier coefficients. Such formulas can provide clues to deep connections
with other fields. As an example, Langlands’ study of Eisenstein series inspired his far-reaching
conjectures that dictate the role of automorphic forms in modern number theory.
In this article, we present two new explicit formulas for the Fourier coefficients of (certain)
Eisenstein series, each given in terms of a combinatorial model: crystal graphs and square ice.
Crystal graphs encode important data associated to Lie group representations while ice models
arise in the study of statistical mechanics. Both will be described from scratch in subsequent
sections.
We were led to these surprising combinatorial connections by studying Eisenstein series not juston a group, but more generally on a family of covers of the group. We will present formulas for their
Fourier coefficients which hold even in this generality. In the simplest case, the Fourier coefficientsof Eisenstein series are described in terms of symmetric functions known as Schur polynomials, sothat is where our story begins.

2

Schur polynomials

The symmetric group on n letters, Sn , acts on the ring of polynomials Z[x1 , . . . , xn ] by permuting
the variables. A polynomial is symmetric if it is invariant under this action. Classical examples are
the familiar elementary symmetric functionsXej =x i1 · · · x ij .
1≤i1 <···<ij ≤nSince the property of being symmetric is preserved by sums and products, the symmetric polynomials make up a subring Λn of Z[x1 , . . . , xn ]. The ej , 1 ≤ j ≤ n, generate this subring.
Since Λn is also an abelian group under polynomial addition, it is natural to seek a set thatgenerates Λn as an abelian group. One such set is given by the Schur polynomials (first considered∗ This work was supported by NSF grants DMS-0652609, DMS-0652817, DMS-0652529, DMS-0702438, DMS0844185, DMS-1001079, and DMS-1001326, and NSA grants H98230-07-1-0015 and H98230-10-1-0183.

1

by Jacobi), which are indexed by partitions. A partition of a positive
Pinteger k is a non-increasing
sequence of non-negative integers λ = (λ1 , λ2 , . . .) such that k =λi ; necessarily only a finite
number of terms in the sequence are nonzero. Partitions are added componentwise. If λ = (λi ) isa partition with λi = 0 for i > n, let ρ = (n − 1, n − 2, . . . , 0, . . .), and let
λ +n−jaλ+ρ = det(xi j)1≤i,j≤n .

Then aρ divides aλ+ρ and the quotient sλ := aλ+ρ /aρ is the Schur polynomial. It is a homogeneous,
symmetric polynomial of degree k. For example, we haves(k,0) (x1 , x2 )s(2,1,0) (x1 , x2 , x3 )= xk1 + xk−1
x2 + · · · + x1 xk−1
+ xk2
1
2
=

x21 x2 + x21 x3 + x1 x22 + 2x1 x2 x3 + x1 x23 + x22 x3 + x2 x23 .

(1)
(2)

The sλ , running over all partitions λ with λi = 0 for i > n, form a basis for Λn . Schur showedthat these polynomials describe the characters of representations of the symmetric and generallinear groups. (See Macdonald [16] for more details.) As we will see in subsequent sections, thesecharacters are connected to the Fourier coefficients of Eisenstein series.

3

Eisenstein series on SL(2)Let H = {z = x + iy ∈ C | y > 0} denote the complex upper half plane. The group SL2 (R) acts on
H by linear fractional transformation:
az + ba b,where γ =∈ SL2 (R).
γ(z) =c d
cz + dIt is of interest to find functions that are automorphic—invariant under the action of a discrete
subgroup of SL2 (R). The modular group Γ = SL2 (Z) is of particular importance. One may createa family of automorphic functions on Γ by averaging. To this end for each s ∈ C with Re(s) > 1,
define the unnormalized Eisenstein series



X
1 n
s
n ∈ Z .1
E(z, s) =Im(γ(z)) , where Γ∞ =0 1γ∈Γ∞ \ΓNote that we must quotient out by the subgroup Γ∞ since this is an infinite group that stabilizes
the imaginary part of z. The definition makes clear that the Eisenstein series is automorphic –E(γ(z), s) = E(z, s) for all γ ∈ Γ. Using the identity Im(γz) = y/|cz + d|2 , we may reparametrize
the sum in terms of integer pairs (c, d). Indeed each pair of relatively prime integers (c, d) is thebottom row of a matrix in Γ and two matrices γ1 and γ2 ∈ Γ have the same bottom row if and only
if γ1 γ2−1 ∈ Γ∞ . Thus the Eisenstein series may be expressed in the formE(z, s) =X(c,d)∈Z2
gcd(c,d)=1

ys,|cz + d|2s

(3)

1 In the first line of this article, we described automorphic forms as functions on groups, but here we’ve defined E(z, s) as a function on the upperplane coshalf H. The resolution of this apparent discrepancy is thatθ sin θH ' SL2 (R)/SO2 (R) where SO2 (R) =θ ∈ [0, 2π) . Indeed, SL2 (R) acts transitively on the point i− sin θ cos θby linear fractional transformation with stabilizer SO2 (R).

2

from which one may deduce that the series converges absolutely for Re(s) > 1.
The series E(z, s) has many spectacular analytic properties. To describe them, define thenormalized Eisenstein series,
E ∗ (z, s) = 21 π −s Γ(s)ζ(2s)E(z, s),(4)where ζ(s) is the Riemann zeta function and Γ(s) is the Gamma function. One can show thatE ∗ (z, s) has analytic continuation to a meromorphic function for s ∈ C and satisfies the functional
equation E ∗ (z, s) = E ∗ (z, 1 − s). This may be proved by spectral methods, as E(z, s) is aneigenfunction of the Laplace-Beltrami operator on H.
This fact has far reaching consequences for the theory of automorphic forms. As an illustrationin our present case, observe that the invariance under γ = ( 10 11 ) implies that E ∗ (z +1, s) = E ∗ (z, s).
Hence the Eisenstein series admits a Fourier series with respect to the real variable x as follows:
Z 1∞
X
E ∗ (z, s) =E ∗ (x + iy, s)e−2πirx dx.
a(r, y, s)e2πinx , where a(r, y, s) =0r=−∞

In the special case r = 0, one can show thata(0, y, s) = y s ξ(2s) + y 1−s ξ(2 − 2s),where ξ(s) = 2 π −s Γ(s)ζ(2s). Because a(0, y, s) inherits the analytic properties of the Fourier series,
the analytic continuation and functional equation of the Riemann zeta function follow.
What about the remaining Fourier coefficients? A calculation (see for example [6], Section 1.6)
shows that if r 6= 0 thenXa(r, y, s) = 2|r|s−1/2 σ1−2s (|r|)y 1/2 Ks−1/2 (2π|r|y) where σ1−2s (r) =
m1−2s
m|r

and K denotes a K-Bessel function.
Let us shift s to s+ 12 and examine the “arithmetic parts” of the non-constant Fourier coefficientsof E ∗ (z, s + 12 ):defa(r) = |r|s σ−2s (|r|).
They are multiplicative. That is, if gcd(r1 , r2 ) = 1, then a(r1 r2 ) = a(r1 )a(r2 ). Thus they arecompletely determined by their values at prime powers r = pk . Moreover,a(pk ) = pks + p(k−2)s + . . . + p−ks .
A fundamental theme of automorphic forms identifies these coefficients with values of charactersof a representation. Let V denote the standard representation of SL2 (C) and let ∨k V denotethe k-th symmetric power. Thus if A ∈ SL2 (C) has eigenvalues α, β then ∨k A has eigenvalues
αk , αk−1 β, . . . , αβ k−1 , β k . The character χk of the representation ∨k V is given byX
χk (A) = tr(∨k (A)) =αk1 β k2 .
k1 +k2 =kComparing with our earlier expression for the arithmetic piece a(pk ), we find
 s

p
k
.
a(p ) = χk
p−s
3

(5)

Notice that a(pk ) is thus the Schur polynomial in (1) evaluated at (x1 , x2 ) = (ps , p−s ):a(pk ) = s(k,0) (ps , p−s ).

(6)This identity has substantial generalizations. Indeed, one can define Eisenstein series analogous
to E(z, s) for any reductive group G. In this generality, the notion of Fourier coefficient is replaced by that of Whittaker coefficient. The Casselman-Shalika formula [7], first proved for GL(n)by Shintani [17], states that the values on prime powers of these coefficients may be captured bycharacters of a representation. For GL(n), these characters are expressed in terms of Schur polynomials. For more general groups, the representation is not of the complex points of G, but rather
a representation of the Langlands dual group2 of G.
These generalizations are usually stated in a different language. The coefficients a(pk ) above
are expressible as integrals on groups over p-adic fields known as p-adic Whittaker functions. The
local version of the Eisenstein series is an induced representation, and the Whittaker function isa p-adic integral evaluated on a canonical vector in the representation space. Similarly, one may
study the Whittaker functions attached to more general Eisenstein series, corresponding to moregeneral induced representations. These may be shown to be products of Langlands L-functions,and this observation is important in the study of those L-functions.
These constructions have been known for many years. The goal of this article is to put themin a new context, by considering a group together with its covers. When we do this, we find thatthe formula (5) and its generalizations may be reinterpreted in terms of crystal graphs, which are
combinatorial structures introduced by Kashiwara in the context of representations of quantum
groups. We begin by illustrating this for covers of SL(2) before discussing higher rank.

4

Eisenstein series on covers of SL(2)The classical metaplectic group is a two-sheeted cover of a symplectic group over either the reals or ap-adic field. This group was introduced by Weil and explains the transformation formulas for theta
functions. More generally, Kubota and Matsumoto, working over a number field L containing a fullset of n-th roots of unity, defined a family of n-sheeted covers of SL(2) (or any simply connectedgroup) for each n > 1. Informally, we may think of such a cover as follows: it is an n-sheeted cover,where the sheets are indexed by the n-th roots of unity. The group law requires moving between
the sheets, and the n-th root of unity that arises in taking the product of two group elements iscomputed using the arithmetic of the field L.3 In fact, the existence of this group is closely related
to the n-th power reciprocity law.
For these groups, one may define an Eisenstein series En (z, s) as an average, similar to (3). The
definition is modified by adding an extra factor in the average that keeps track of the sheets of thecover. The Fourier coefficients of En (z, s) turn out to be of great interest: they are Dirichlet series
made with Gauss sums.
R∞
A Gauss sum is a discrete analogue of the Gamma integral Γ(s) = 0 y s e−y dy
y – a productof multiplicative and additive characters summed over the invertible elements of a finite ring. For
2 In fact, the dual group enters subtly into the computation above. The Eisenstein series E(z, s) may be regarded
as a function on PGL2 and the Langlands dual of this group is SL2 (C).
3 In more detail, let A denote the adèles, an appropriately restricted product over all completions of L. Then nL
fold metaplectic group is a central extension of SL2 (AL ) by µn : 1 −→ µn −→ G̃ −→ SL2 (AL ) −→ 1. This extension
is described by means of a two-cocycle which is constructed using the n-power Hilbert symbols of completions of L.
(It is not the adelic points of an algebraic group). See [3] for further information.

4

example if the cover degree is n = 3, we may take L = Q(e2πi/3 ) with ring of integers oL = Z[e2πi/3 ].
Let e(·) be an additive character of L which is trivial on oL but no larger fractional ideal. Then for
integers m, c ∈ oL with c 6= 0, letX t
g3 (m, c) =e(mt/c),(7)c 3t(mod c)gcd(t,c)=1

where the sum is over t ∈ oL that are invertible mod c and (−)3 is the cubic residue symbol.4 For
general n and L, we may define a Gauss sum gn (m, c) made with n-th power residue symbols. To
do so, we must pass from the ring of integers oL to a localization oL,S where denominators areallowed at a finite set of places S, and some additional technicalities result.
Kubota computed the Fourier expansion of En (z, s), whose m-th coefficient is a K-Bessel function times an arithmetic part a(m). In the special case n = 3,
Xa(m) = kmks−1/2c∈oLc≡1 (mod 3)

g3 (m, c)
,
kck2s

(8)

where k k denotes the absolute norm. The form for general n is much the same with an arithmetic
part involving gn (m, c) in place of g3 . The series is easily seen to converge absolute for <(s) > 3/4,
and since E(z, s) has analytic continuation and functional equation, a(m) inherits these properties
as well. This series (and its generalizations) are basic objects of interest.
Let us recall two properties of Gauss sums valid for any n ≥ 1. Using the Chinese Remainder
Theorem, one may show that if gcd(c1 , c2 ) = 1, then   c1
c2
gn (m, c1 c2 ) =gn (m, c1 )gn (m, c2 )(9)
c2 n c1 n
and if gcd(m1 , c) = 1 then an easy change of variables shows that for any integer m2
gn (m1 m2 , c) = m −11cngn (m2 , c).

In particular, (9) shows that the Dirichlet series in (8) is not expressible as an Euler product – aproduct over primes – when n > 2. This is quite different from the situation for n = 1, 2 and, moregenerally, for Langlands L-functions. Instead, we see that to combine contributions from relatively
prime c1 and c2 , we must introduce n-th roots of unity depending on arithmetic. For these reasons,
we call series with such a property twisted Euler products. See [11] for more information and further
examples.
Though not strictly multiplicative, these two properties allow one to reconstruct gn (m, c) fromits values at prime powers gn (pa , pb ) for non-negative integers a, b. Thus we may restrict to these
simpler cases in describing the Fourier coefficients.
Let us consider the coefficients gn (pa , pb ) at a given prime p. Here a is fixed (it is the order of mat p) and b is varying. These coefficients come in three flavors. First, there is the case b = 0, where
4 Thus (t/c)

3 is a cube root of unity and is 1 if c is a cube, and (t1 t2 /c)3 = (t1 /c)3 (t2 /c)3 .

5the coefficient is simply 1 = pb . Second, there are the coefficients for 1 ≤ b ≤ a. The inequalityb ≤ a makes the additive character in (7) trivial, and so this coefficient is the functionφ(pb ) if n|b,hn (b) =0
otherwise,
where φ(pb ) = |(oL,S /pb oL,S )× | is the Euler phi function for oL,S . Finally, there is the case b = a+1.
In this case, the Gauss sum is always nonzero and it is not possible to evaluate it in closed form
except in special cases. We write this sum simply as gn (a + 1) for short. For b ≥ a + 2, the Gausssum is zero (which follows from expressing the sum in terms of a nontrivial character over a group).
Hence the entire contribution to the pa -th Fourier coefficient can be summarized in the following
diagram.
b=0

1

a

2

a+1

···
g=1

hn (1)hn (2)(10)hn (a)

gn (a + 1)

We have circled the location b = 0 and boxed the location b = a + 1 since the contributions are
special at these locations, while at b such that 1 ≤ b ≤ a, the contribution is hn (b). This is themost common situation. Notice that the diagram is the same for any n, it is only the functions gn
and hn that depend on n.
For the non-metaplectic Eisenstein series (the special case n = 1), we saw in Section 3 that thecoefficients at powers of p may also be described in terms of Schur polynomials. The connection to
the sums of Gauss sums presented here is as follows. We work over Q for convenience. The residue
symbol (t/c)1 is trivial, andX
h1 (a) = φ(pa ), g1 (a + 1) =e(t/p) = −pa .
(11)t(mod pa+1 )gcd(t,p)=1

Thus the arithmetic part a(pk ) of the pk -th Fourier coefficient described in this Section has theformpk(s−1/2) 1 + φ(p)p−2s + · · · + φ(pk )p−2ks − pk p−2(k+1)s = (1 − p−2s )s(k,0) (ps−1/2 , p−(s−1/2) ).
After sending s 7→ s + 1/2 as before, this coincides with the formula (6) above. Note that the
Eisenstein series defined in (4) was normalized by a zeta function which explains the extra factor(1 − p−2s ) here.
Returning to the case of general n, the description of the Fourier coefficient as a sum of Gauss
sums governed by (10) above has broad generalizations. Indeed the underlying graph in (10) may
be viewed as a crystal graph associated to a highest weight representation of SL2 . In the next
section, we will discuss crystal graphs in more detail and explain how they may be used to give ageneralization to covers of SLr+1 for any r and any cover degree n.

5Eisenstein series on covers of SLr+1 and crystal graphs

We continue to work over a number field L containing n n-th roots of unity. One can define an
n-fold cover of (the adelic points of) SLr+1 for any r, and a corresponding Eisenstein series En
6

for this group. It is an average of a suitable function, this time a function of r complex variables
s1 , . . . , sr , over a discrete subgroup.5
Fourier coefficients generalize to Whittaker coefficients. These are defined by integrating En
against a character of U , the subgroup of upper triangular unipotent matrices of SLr+1 (AL ). The
characters of U are indexed by r-tuples m = (m1 , . . . , mr ) of elements of oL . Indeed, a character ofU depends only on the r locations just above the main diagonal since everything else is in [U, U ].
Then the Whittaker coefficients are defined by integration against this character.
The main theorem of [3] expresses the arithmetic part a(m) of these Whittaker coefficients ass −1/2s −1/2kmk11· · · kmkrr
times the multiple Dirichlet series
X Hn (m; c1 , . . . , cr ).
kc1 k2s1 · · · kcr k2sr
c ,...,c1rThis is a generalization of (8). The coefficients Hn are once again twisted multiplicative, and thisallows one to reduce to their study to that of the coefficients Hn (p`1 , . . . , p`r ; pk1 , . . . , pkr ) attached
to a given prime p of oL . Here the `i and ki are non-negative integers. The coefficients H turn out
to be built out of the functions gn , hn , and powers of kpk, that already appeared in the previous
section for the n-fold cover of SL(2). However, the exact description is considerably more subtle.
It involves the theory of crystal graphs.
To explain further, we briefly recall several important facts about finite dimensional representations of Lie groups and their crystal graphs. A weight of GLr+1 is a rational character of the
diagonal torus T of GLr+1 . The weights
be identified with elements of the lattice Λ = Zr+1 :Q µmayµiif µ = (µ1 , · · · , µr+1 ) ∈ Λ then t := ti with t = diag(t1 , . . . , tr+1 ) ∈ T is such a character. A
weight for a representation V of the associated Lie algebra glr+1 (C) is a weight µ such that there
exists a non-zero vector in V that transforms under the torus by µ; it is highest if no larger weightsatisfies this property.6 Cartan’s Theorem of the Highest Weight states that every finite-dimensional
irreducible complex representation of glr+1 (C) (or any complex semisimple finite-dimensional Lie
algebra) has a unique highest weight vector (up to scalars) and that the highest weight classifies
the representation.
The quantum group Uq (glr+1 (C)) is a deformation of the universal enveloping algebra of glr+1 (C)that is obtained when a parameter q is introduced into the relations that describe the universal
enveloping algebra. (See Hong and Kang [13].) Finite-dimensional representations are once again
classified by highest weight. Let λ be a dominant weight (that is, λ1 > λ2 > · · · > λr+1 ). Then
Kashiwara [14] associates with λ a crystal graph Bλ , a directed graph whose vertices correspondto basis vectors for the representation of Uq (glr+1 (C)) with highest weight λ. The edges of thisgraph are colored with one color for each simple root, and describe the action of the unipotents
in the Lie algebra on this basis as q → 0. The crystal graph Bλ comes endowed with a map “wt”
to the weight lattice Λ which is compatible with the graph structure. Walking one step along anedge of Bλ in the direction of the highest weight vector (resp. lowest weight vector) corresponds toincreasing (resp. decreasing) the weight of the vertex by the simple root with which it is labeled.7
5 There are more general Eisenstein series built from automorphic forms on lower rank groups, but we do notconsider them here.
6 Recall that the weights are partially ordered as follows: λ > µ if λ − µ is a non-negative linear combination of
simple roots. In terms of coordinates, λi = µi + hi − hi+1 for each i, where the hi are non-negative integers andh0 = hr+2 = 0.
P
7 The map “wt” is such thatwt(v) is the character of an irreducible representation of GLr+1 (C) whosev∈Bλ tassociated Lie algebra representation has highest weight λ.

7

λ

v

w0 λ

Figure 1: The crystal graph with highest weight λ = (4, 2, 0)
Figure 1 depicts a gl3 crystal with highest weight λ = (4, 2, 0) and lowest weight w0 λ = (0, 2, 4).
It is drawn so that vertices of the same weight are clustered together diagonally.
Berenstein and Zelevinsky [2] and Littelmann [15] associate paths to each vertex in Bλ .8 To dothis, choose a reduced factorization of the long element w0 of the Weyl group into simple reflections(i.e., one of minimal length). Walk the graph toward the highest weight vector in the order thatthe simple reflections appear in the factorization, going as far in a given direction as the graph willallow before changing to the next color. It turns out that such a factorization always leads to apath to the highest weight vector λ. The sequence BZL(v) of path lengths so obtained parametrizes
the vertex v of Bλ . (Alternatively, we could record path lengths toward the lowest weight vector
w0 λ from v.)
For example, in Figure 1 we have indicated a walk from a vertex v to the highest weight vector
λ. It corresponds to the factorization of the long element w0 = s1 s2 s1 of the symmetric group S3 ,the Weyl group of GL3 . Thus we walk along the graph in order s1 , s2 , s1 (=blue,red,blue). The
lengths of the corresponding paths are 1, 3, 2, respectively, so BZL(v) = (1, 3, 2).
The main theorem of [3] computes the coefficients Hn (p`1 , · · · , p`r ; pk1 , · · · , pkr ) by attachingproducts of Gauss sums to BZL sequences. Let λr+1 = 0 and λi = `i + λi+1 when i ≤ r, and let
λ be the dominant weight λ = (λ1 , λ2 , . . . , λr+1 ).9 Let ρ denote the Weyl vector, that is, half thesum of the positive roots, or in coordinates (r, r − 1, . . . , 1, 0).
Theorem 1 The coefficient Hn is given byHn (p`1 , · · · , p`r ; pk1 , · · · , pkr ) =X

Gn (v),(12)v∈Bλ+ρ
wt(v)=µwhere the function
Pr Gn (v) is described below and µ is the weight related to (k1 , · · · , kr ) by thecondition that i=1 ki αi = (λ + ρ) − µ where αi are the simple roots.
8 Berenstein and Zelevinsky refer to these paths as “strings.”
9 By fixing λr+1 = 0, we parametrize representations of SLr+1 (C), the Langlands dual group of PGLr+1 .

8The definition of Gn (v) depends on a recipe for walking the graph, so it depends on the choice ofa reduced expression for w0 in the symmetric group Sr+1 . We will work with two different choices;
these give rise to two different functions Gn (v). In terms of the simple reflections si (recorded bytheir index i ∈ [1, r]), let us fix eitherΣ = Σ1 := (r, r − 1, r, r − 2, r − 1, r, · · · , 1, 2, 3, · · · , r)(13)

Σ = Σ2 := (1, 2, 1, 3, 2, 1, · · · , r, r − 1, · · · , 3, 2, 1)

(14)

or
and take the associated path lengths BZL(v) = (b1 , . . . , bN ) to the highest weight vector. (Wesuppress the dependence on Σ.) We then decorate the entries bi as follows. The length bi is boxed if
the i-th leg of the path is maximal (i.e. contains the entire root string). In Figure 1, with Σ = Σ2 ,BZL(v) = (1, 3, 2), both the 1 and 2 are boxed while the 3 is not (since it is part of an s2 root stringof length 4). An entry bi is circled if the (`(w0 ) − i)-th leg of the path to the lowest weight vector
is maximal (here `(w0 ) denotes the length of w0 ). Thus in Figure 1, the path lengths to the lowest
weight vector are (0, 1, 1), none of which are maximal. Hence none of the entries in the decorated
BZL sequence ( 1 , 3, 2 ) are circled.
Then we prove thatbi
if bi is circled (but not boxed), kpk
Y  gn (bi ) if bi is boxed (but not circled),(15)
Gn (v) = Gn,Σ (v) = hn (bi ) if neither,bi ∈BZL(v) 0if both,where gn (b) and hn (b) are the Gauss sum and degenerate Gauss sum, respectively, described in theprevious section. Notice that this definition exactly matches the description given in Section 4 andpictured in (10) in the special case of SL2 .
In Figure 1, the vertex v belongs to a weight space with multiplicity two. Again using Σ =(1, 2, 1), the other vertex in the weight space containing v has decorated BZL sequence (2, 3 , 1).
Thus applying Theorem 1 with Gn (v) as in (15), we haveHn (p2 , p; p3 , p3 ) = Gn 1 , 3, 2 + Gn 2, 3 , 1= gn (1)hn (3)gn (2) + hn (2)gn (3)hn (1).

(16)

Since hn (b) = 0 unless n divides b, this term is non-zero only for the cover degrees n = 1 or 3. It isnoteworthy that expressions like (16) for the function Hn in terms of Gauss sums are uniform in n.
Because we may use either Σ1 or Σ2 to define Gn (v), these are two explicit formulas for the
Whittaker coefficient. The equality of the expression in (12) for Σ1 and Σ2 is not formal, and isestablished directly in [2] by an elaborate blend of number-theoretic and combinatorial arguments.
It is an open problem to give a definition of Gn (v) obtaining the Whittaker coefficient for an
arbitrary reduced decomposition of the long element w0 of the Weyl group.
In closing this section, we mention that there are not one but two distinct generalizations of
the Casselman-Shalika formula to the metaplectic case. Chinta and Gunnells [8] and Chinta andOffen [9] show that the p-parts of the Whittaker coefficients of metaplectic Eisenstein series oncovers of SLr+1 can also be expressed as quotients of sums over the Weyl group, directly analogous
to the Weyl character formula.
9

6

The case n = 1: Tokuyama’s deformation formula

When n = 1, we are concerned with Eisenstein series on an algebraic group and not a cover. In
that case, the Whittaker coefficients may be computed in two different ways. First, Theorem 1
provides an answer in terms of crystal graphs. This result holds for any n ≥ 1. Second, theformula of Shintani [17] and Casselman and Shalika [7] (which holds only for n = 1) expresses theWhittaker coefficients of normalized Eisenstein series as the values of the characters of irreducible
representations of SLr+1 (C). These characters are given by Schur polynomials sλ , as described in
Section 2.
These two expressions for the Whittaker coefficients are related by the following result (cf. [2]).
Theorem 2 Let z = (z1 , . . . , zr+1 ) and let q = kpk. For any dominant weight λ,YX (zi − q −1 zj ) sλ (z) =G1 (v)q −hλ+ρ−wt(v),ρi z wt(v) ,i<jv∈Bρ+λwhere the G1 (v) are computed as in (15) using the reduced word Σ2 .
We illustrate Theorem 2 with λ = (2, 1, 0), so that λ + ρ = (4, 2, 0) and Bλ+ρ is the crystal
pictured earlier. Let us compare the monomials z1 z22 z33 appearing on both sides of the theorem
for this choice of λ. The coefficient of this monomial appearing on the right-hand side is (up to
a power of q) just the value of Hn (p2 , p; p3 , p3 ) computed in (16) in the special case n = 1. Aftersimplification using (11),H1 (p2 , p; p3 , p3 ) = qφ(p3 ) − q 2 φ(p2 )φ(p) = −q 5 + 3q 4 − 2q 3 .
Since hλ + ρ − wt(v), ρi = 6, these terms should be multiplied by q −6 to obtain the complete
contribution to the monomial z1 z22 z33 on the right-hand side.
The left-hand side is just (z1 − q −1 z2 )(z1 − q −1 z3 )(z2 − q −1 z3 )s(2,1,0) (z1 , z2 , z3 ) where s(2,1,0) (z)is given in (2). Expanding, we see that the coefficients of z1 z22 z33 indeed match. For example, termswith q −3 on the left can only come from taking the term 2z1 z2 z3 in the Schur polynomial andmultiplying by q −3 z2 z32 from the product.
In general, after taking into account the normalizing factors that appear in the CasselmanShalika formula, Theorem 2 shows that the Casselman-Shalika formula and Theorem 1 in the casen = 1 are equivalent.
Theorem 2 is equivalent to an earlier result of Tokuyama [18], and may be viewed as a deformation of the Weyl character formula (which results from setting q = 1). Tokuyama’s formulation uses
combinatorial arrays called Gelfand-Tsetlin patterns. We highlight that the character with highest
weight λ is expressed as a combinatorial sum over basis vectors of a crystal of highest weight λ + ρ.

7

Ice models for Whittaker coefficients

In this final section, we describe another combinatorial representation of the p-parts of Whittakercoefficients. These can be described using square ice, a particular example of a two-dimensionallattice model . We describe these in detail when n = 1; that is, when the Whittaker coefficients at
the prime p are given in terms of Schur polynomials. An ice model description for arbitrary covers
is presented in [5].
10Two-dimensional lattice models arise in statistical mechanics, where they can be used to represent thin sheets of matter such as ice. Any such model has a certain set of admissible configurations
called states and each state is assigned a value known as a Boltzmann weight. A primary goal is tounderstand the behavior of the partition function of the model, the sum of the Boltzmann weights
over all states.10 Lattice models for which the partition function may be explicitly evaluated arecalled exactly solved and are of particular interest. See Baxter [1]. The study of ice models wasadvanced by ideas of representation theory and ultimately led to the discovery of quantum groups.
See Faddeev [10] for a history.
For the application to Whittaker functions, a lattice model is given for any partition λ =(λ1 , . . . , λr+1 ) having λr+1 = 0 as follows. Form a rectangular array of lattice points with r + 1
rows (numbered r + 1 to 1 from top to bottom) and λ1 + r + 1 columns numbered 0 to λ1 + r fromright to left. Add vertical and horizontal edges from each lattice point, so the points are embedded
in a rectangular array of line segments.
Each boundary edge of this configuration is labeled with a fixed “spin” + or −. The left andbottom edges are all assigned a + spin and the right edge spins are all −. The spins along the top
edges correspond to λ + ρ = (λ1 + r, λ2 + r − 1, · · · , λr + 1, 0) as follows: place a − spin at the topof a column numbered λi + (r + 1 − i) for i ∈ [1, r + 1] and place a + spin above the remainingcolumns. The figure below illustrates these boundary conditions associated to λ = (2, 1, 0) so thatλ + ρ = (4, 2, 0), our running example.
4

3

2

1

0

3

3

3

3

3

2

2

2

2

2

1

1

1

1

1

(17)

The states of this model will be assignments of spins to the internal edges, pictured with open
circles above. The only requirement on these spins is that each vertex in the grid has adjacent spins
matching one of the following six configurations:
Ice
Configuration

i

i

i

i

i

i

A model with this restriction is often called a six-vertex model or square ice.11 Given the boundary
10 The term ‘partition function’ should not be confused with our earlier use of ‘partition’ of a positive integer.
11 We may think of the vertices in the grid as oxygen atoms, and the six possible choices of adjacent spins are the4ways of arranging two nearby hydrogen atoms on adjacent edges.
211conditions as above, here is one such admissible filling (i.e. a state).
4

3

2

1

0

3

3

3

3

3

2

2

2

2

2

1

1

1

1

1

(18)

To describe the Boltzmann weight for a state, we first assign a weight to each of the six types ofvertices (which is allowed to vary depending on the row in which it appears). Then the Boltzmann
weight of the state is the product of all weights of vertices appearing in the grid. Summing theBoltzmann weight over all states with fixed boundary conditions gives the partition function forthe model.
For example, choose weights for the vertices as follows:
Ice
Configuration
Weight

i

i

i

i

i

i
(19)

1

ti zi

1

zi

zi (ti + 1)

1

where the ti and zi are arbitrary parameters corresponding to the row number i.12 Then the
Boltzmann weight of the state (18) is:t3 (1 + t3 )z33 · t2 z22 · z1 .
Setting t2 = t3 = −1/q, this is precisely equal to G1 (v) q −hλ+ρ−wt(v),ρi z wt(v) , which appears in theright-hand side of Theorem 2, where v is the vertex pictured in the crystal graph of Section 5. This
is no accident. There is a bijection between vertices v of the crystal Bλ+ρ having G1 (v) 6= 0 andstates of the model with boundary conditions corresponding to λ + ρ as above. See [4] for details.
Hamel and King [12] evaluated the partition function of an equivalent model and choice ofBoltzmann weights by means of tableaux combinatorics, and showed that it exactly equals the lefthand side of Tokuyama’s theorem. In [4], we show that as long as the Boltzmann weights satisfy
a single algebraic relation (which includes the case of Hamel and King), the resulting partition
function may be given in terms of a Schur polynomial. We also give a different approach to theseresults which we now sketch.
Let Sλ denote the set of states for the model above with boundary conditions corresponding to
λ+ρ. Let Z(Sλ ) be the partition function of the model with Boltzmann weights assigned according
to the table (19). We prove in [4] thatY
Z(Sλ ) =(tj zj + zi )sλ (z1 , · · · , zn ),(20)i<j
12 Note that these Boltzmann weights are not symmetric under the interchange of + and −, in contrast to the‘field-free’ situation that is often considered in the literature.

12where the right-hand side has alreadyQ appeared in the statement of Theorem 2. The critical stepof the proof is to show that Z(Sλ ) i<j (tj zj + zi )−1 is symmetric in the sense that it is unchanged
if the same permutation is applied to both (z1 , . . . , zr+1 ) and (t1 , . . . , tr+1 ). Once this is known,
it is possible to show that it is a polynomial in the zi and ti , and by comparing degrees that it
is independent of the ti ; finally, taking ti = −1 one may invoke the Weyl character formula andconclude that it is equal to the Schur polynomial.
In order to prove the desired symmetry property we use an instance of the Yang-Baxter equation.
In the context of a lattice model, given three fixed sets of weights R, S, and T , the Yang-Baxter
equation is the identity of partition functions


τ


Z


σ


S

R





β



β


τ



=Z



T ρ 
σ




T

α

α



θ 




θ 


R
S


,


ρ 


(21)

for all choices of boundary spins ± for α, β, σ, τ, θ, ρ. Here the R vertices have been rotated by45◦ for ease in drawing the diagram. Note that both sides of this identity are sums over all states
resulting from choices of the three internal edge spins indicated by empty circles above.13 Baxter
first employed the Yang-Baxter equation as a method for obtaining exactly solved models.
In the application at hand, S and T are weights given in (19) for two rows. It may be shown(cf. [4]) that there exists a choice of weights R such that the Yang-Baxter equation holds. Attach
this vertex between the S and T rows thus:

S

S

S

−→
T

S

(22)

R

T

T

T

This multiplies the partition function by a weight of R, which happens to be one of the linear
factors in (20). Then applying the Yang-Baxter equation several times, this R-vertex may be
13 This may be reformulated algebraically by regarding the Boltzmann weights R, S, T as giving endomorphisms of
V ⊗ V for an abstract two-dimensional vector space V . See [4] for an exposition. Then the Yang-Baxter equation is
the identity
R12 S13 T23 = T23 S13 R12 ,where the notation Rij is the endomorphism of V ⊗ V ⊗ V in which R is applied to the i-th and j-th copies of V
and the identity map to the k-th copy, where {i, j, k} = {1, 2, 3}.

13

moved rightward, leaving the partition function invariant. Picking up from (22), this looks like:

T

−→

S

T

−→

R
S

T

T

R
S

S

Then discarding the R-vertex divides by another Boltzmann weight of R, which is another one of
the linear factors in (20). Note that S and T are interchanged, reflecting the symmetry of the Schurfunction in (20), and leading to a proof of that equation.
The Yang-Baxter equation can also be used to directly establish the equivalence of the two
descriptions in Theorem 1 obtained from the reduced decompositions (13) and (14) when n = 1.
See Chapter 19 of [2].
The Langlands program describes the role of automorphic forms on reductive groups in number
theory. Automorphic forms on covering groups have been used to prove cases of the Langlands
conjectures, but they themselves do not strictly fit into its usual formulations. Studying automorphic forms on covers reveals connections with crystals and lattice models, which are mathematicalobjects that first appeared in other contexts—quantum groups and mathematical physics. The
exploration of these exciting connections is only beginning.

References[1]

R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, London, 1982.

[2]

A. Berenstein and A. Zelevinsky. Tensor product multiplicities, canonical bases and totally
positive varieties. Invent. Math, 143: 77–128, 2001.

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NJ, 2011.

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B. Brubaker, D. Bump, and S. Friedberg. Weyl group multiple Dirichlet series, Eisensteinseries and crystal bases. Annals of Math., 173(2):1081–1120, 2011.

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B. Brubaker, D. Bump, and S. Friedberg. Schur polynomials and the Yang-Baxter equation.
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B. Brubaker, D. Bump, G. Chinta, S. Friedberg and P. Gunnells. Metaplectic Ice. Preprint,
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D. Bump. Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics, Vol. 55. Cambridge University Press, Cambridge, 1997.

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W. Casselman and J. Shalika. The unramified principal series of p-adic groups. II. The Whittaker function. Compositio Math., 41(2):207–231, 1980.
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[10] L. Faddeev. History and perspectives of quantum groups. Milan J. Math., 74:279–294, 2006.
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