Multiple Dirichlet Series
Daniel Bump
Dept. of Mathematics, Stanford University, Stanford CA 94305-2125 email:bump@math.stanford.edu

Summary. This introductory article aims to provide a roadmap to many of theinterrelated papers in this volume and to a portion of the field of multiple Dirichlet
series, particularly emerging new ideas. It is both a survey of the recent literature,
and an introduction to the combinatorial aspects of Weyl group multiple Dirichlet
series, a class of multiple Dirichlet series that are not Euler products, but which may
nevertheless be reconstructed from their p-parts. These p-parts are combinatoriallyinteresting, and may often be identified with p-adic Whittaker functions.

This survey article is intended to help orient the reader to certain topics inmultiple Dirichlet series. There are several other expository articles that thereader might also want to consult, though we do not assume any familiarity
with them. The article [20] which appeared in 1996 contained many of theideas in an early, undeveloped form. The articles [9] and [26], which appearedin 2006, also survey the field from different points of view, and it is hoped thatthese papers will be complementary to this one. Further expository material
may be found in some of the chapters of [14].
Since approximately 2003 there has been an intensive development of thesubject into areas related to combinatorics, representation theory, statisticalmechanics and other areas. These are scarcely touched on in [20], [9] and [26],and indeed are topics that have largely developed during the last few years.
However these combinatorial developments are discussed in [14] as well as thisintroductory paper and other papers in this volume.
Preparation of this paper was supported in part by NSF grant DMS1001079. We would like to thank Ben Brubaker, Solomon Friedberg and Kohji
Matsumoto for helpful comments.

1 Moments of L-functions
The subject of multiple Dirichlet series originated in analytic number theory.
If {an } is a sequence of real or complex numbers, then a typical Tauberian2Daniel Bump

theoremP draws conclusions about the an from the behavior of the Dirichlet
series n an n−s . If the an are themselves L-functions or other Dirichlet series,
this is then a multiple Dirichlet series.
One may try to study moments of L-functions this way. For example Goldfeld and Hoffstein [37] considered a pair of Dirichlet series whose coefficientsareX
1
Z± (w, s) =Ad (s)|d| 2 −2w ,
(1)
±d>0

where the coefficients Ad (s) are essentially quadratic L-functions. More precisely, if d is squarefree

andAdk2 (s)=Ad (s)Ad (s) =L2 (2s − 12 , χd ),
ζ2 (4s − 1)Xχd (d3 )µ(d3 )d2−4s+3/2 −2s+1/2
d3

d1 d2 d3 = k
d2 , d3 odd

where µ is the Möbius function and χd is the quadratic character χd (c) = dc
in terms of the Kronecker symbol. The subscript 2 applied to the L-function
and zeta function ζ means that the 2-parts have been removed.
Goldfeld and Hoffstein applied the theory of Eisenstein series of halfintegral weight to obtain the meromorphic continuation and functional equations of Z± . They showed that there are poles at w = 43 and 54 − s, then useda Tauberian argument to obtain estimates for the mean values of L-functions.
For example, they showed 19 X
1
L, χd = c1 x log(x) + c2 x + O x 32 +ε2
1 < ±d < x
d squarefreewith known constants c1 and c2 .
Note that Z± is a double Dirichlet series (in s, w) since if we substitutethe expression for the L-function L(w, χc ) we have “essentially”X X d 111
1−2w
Z± (s, w) =L 2s − , χd |d| 2=(2)|c| 2 −2s |d| 2 −2w .
2
c
d

c,d

Equation (2) gives two heuristic expressions representing the multiple Dirichlet series with the intention of explaining as simply as possible what we expect
to be true, and what form the generalizations must be. Such a heuristic form
ignores a number of details, such as the fact that the coefficients are only
described correctly if d is squarefree (both expressions) and that c and d
are coprime (second expression). Later we will first generalize the heuristic
form by attaching a multiple Dirichlet series to an arbitrary root system. The

Multiple Dirichlet Series

3heuristic version will have predictive value, but will still ignore importantdetails, so we will then have to consider how to make a rigorous definition.
To give an immediate heuristic generalization of (2), let us consider, withcomplex parameters s1 , · · · , sk and w a multiple Dirichlet series



X 
1
1
1
(3)
L 2s1 − , χd · · · L 2sk − , χd |d| 2 −2w
22c
for k = 0, 1, 2, 3, · · · . If one could prove meromorphic continuation of thisDirichlet series to all s with wi = 12 , the Lindelöf hypothesis in the quadratic
aspect would follow from Tauberian arguments. A similar approach to the
Lindelöf hypothesis in the t aspect would consider insteadZ ∞ζ (σ1 ± it) · · · ζ (σk ± it) t−2w dt(4)
1

where for each zeta function we choose a sign ±; if k is even, we may choose
half of them positive and the other half negative. This is equivalent to theusual momentsZTζ (σ1 ± it) · · · ζ (σk ± it) dt

0which have been studied since the work in the 1920’s of Hardy and Littlewood,
Ingham, Titchmarsh and others. It is possible regard (4) as a multiple Dirichlet
series, and indeed both (3) and (4) are treated together in Diaconu, Goldfeld
and Hoffstein [32]. See [31] in this volume for a discussion of the sixth integral
moment and its connection with the spectral theory of Eisenstein series on
GL3 .
Returning to (3), there are two problems: to make a correct definitionof the multiple Dirichlet series, and to determine its analytic properties. If
k = 1, 2 or 3, these can both be solved, and the multiple Dirichlet series has
global analytic continuation. In these cases, the multiple Dirichlet series was
initially constructed by applying a Rankin-Selberg construction to Eisensteinseries (“of half-integral weight”) on the metaplectic double covers of the groups
Sp(2k) for k = 1, 2, 3. No corresponding constructions could be found fork > 3, but since Rankin-Selberg constructions are often tricky, that did notconstitute a proof that such constructions may not exist undiscovered.
In [20] a different approach was taken. If k > 3, then it may be possible
to write down a correct definition of the multiple Dirichlet series, and indeed
this has essentially been done in the very interesting special case k = 4.
See Bucur and Diaconu [19]. Nevertheless, the approach taken in [20], which
we will next explain, shows that the multiple Dirichlet series cannot havemeromorphic continuation to all si and w if k > 3.
The analog for (3) of the second expression in (2) would have the formX d d
1
1
1
(5)
···
|c1 | 2 −2s1 · · · |ck | 2 −2sk |d| 2 −2w .
c1
ck
d,c1 ,··· ,ck4Daniel Bump

It will be helpful to associate with this Dirichlet series a graph whose vertices
are the variables d, c1 , · · · , ck . We connect two vertices if a quadratic symbol is attached to them. Our point of view (which is justified when rigorous
foundations are supplied) is that due
 to the quadratic reciprocity law we do
not have to distinguish between dc and dc , Thus for heuristic purposes, thegraph determines the Dirichlet series. If k = 1, 2 or 3, the graph looks like
this:
c3
d

c1

c1

d

c2

c1

d

(6)
c2

We could clearly associate a multiple Dirichlet series with a more generalgraph, at least in this imprecise heuristic form. The interesting cases will bewhen the diagram is a Dynkin diagram.
Here we will only consider cases where the diagram is a “simply-laced”
Dynkin diagram, that is, the diagram of a root system of Cartan type A,
D or E. A simply-laced root system is one in which all roots have the samelength, and these are their Cartan types. More general Dynkin diagrams arealso associated with multiple Dirichlet series, and we will come to these below.
We recall that a Coxeter group is a group with generators σ1 , · · · , σr , eachof order two, such that the relations
σi2 = 1,(σi σj )n(i,j) = 1

give a presentation of the group, where n(i, j) is the order of σi σj . We may
associate with the Coxeter group a graph, which consists of one node for eachgenerator σi , with the following conditions. If n(i, j) = 2, so that σi and σjcommute, there is no edge connecting i and j. Otherwise, there is an edge.
If n(i, j) = 3, it is not necessary to label the edge, but if n(i, j) > 3 it islabeled with n(i, j). (In Dynkin diagrams it is usual to interpret these labels
as double or triple bonds.) We will consider only the case where n(i, j) = 2 or
3. In these cases, the Coxeter group is finite if and only if the diagram is thefinite union of the Dynkin diagrams of finite Weyl groups of types Ar , Dr or
Er .
As we will now explain, the group of functional equations of a multiple
Dirichlet series such as (5) is expected to be the Coxeter group of its Dynkin
diagram. For example, consider (5) when k = 3. We collect the coefficientsof c1 :X  d  d  d 1
1
11|c1 | 2 −2s1 |c2 | 2 −2s3 |c3 | 2 −2s3 |d| 2 −2w =c1
c2
c3
d,c1 ,c2 ,c3
"
#

 
X
X d 
d
1
1
1
1
d
−2s
−2s
−2w
−2s
2
3
1
.
|c3 | 2|d| 2
|c2 | 2
|c1 | 2c2
c3
c1
cd,c2 ,c3

1

Multiple Dirichlet Series

5This equals
X  d  d d,c2 ,c3

c2

c3



1
1
1
1
|c2 | 2 −2s2 |c3 | 2 −2s3 |d| 2 −2w L 2s1 − , χd .
2

The functional equation for the Dirichlet L-function has the form




1
1
1−2s1
L 2s1 − , χd = (∗) |d|L 2(1 − s1 ) − , χd22where (∗) is a ratio of Gamma functions and powers of π. This factor isindependent of d, so we see that the functional equation has the effect(s1 , s2 , s3 , w) 7→ (1 − s1 , s2 , s3 , w + s1 − 12 ) .
In the general case, let the variables be s1 , · · · , sr . Thus if we are considering(5) then r = k + 1 and sr = w, but we now have in mind a general graph suchas a Dynkin diagram, and r is the number of nodes. We have a functional
equation which sends si to 1 − si . If j 6= i, thensj
if i, j are not connected by an edge;sj 7−→(7)
si + sj − 12 if i, j are connected by an edge.
These functional equations generate the Coxeter group associated with thediagram, and its group of functional equations is the geometric realization ofthat group as a group generated by reflections.
We may now see why the Dirichlet series (3) is expected to have meromorphic continuation to all si and w when k 6 3 but not in general. These are
the cases where the graph is the diagram of a finite Weyl group, of types A2 ,(1)
A3 or D4 . If k = 4, the graph is the diagram of the affine Weyl group D4 ,and the corresponding Coxeter group is infinite. The meromorphic continuation in (s1 , s2 , s3 , s4 , w) cannot be to all of C5 since the known set of polar
hyperplanes will have accumulation points.
We have now given heuristically a large family of multiple Dirichlet series,
one for each simply-laced Dynkin diagram. (The simply-laced assumption may
be eliminated, as we will explain later.) Only three of them, for Cartan types
A2 , A3 and D4 , are related to moments of L-functions. The case k = 3, related
to D4 , was applied in [32] to the third moment after the combinatorics needed
to precisely define the Dirichlet series were established in [21].
Although only these three examples are related to quadratic moments of Lfunctions, others in this family have applications to analytic number theory.
Chinta gave a remarkable example in [25], where the A5 multiple Dirichlet
series is used to study the distribution of central values of biquadratic Lfunctions. The distribution of n-th order twists of an L-function were studied
by Friedberg, Hoffstein and Lieman [36], and it was shown in Brubaker andBump [7] that these could be related to n-th order Weyl group multiple Dirichlet series of order n. (In this survey, the Dirichlet series we have considered inthis section correspond to n = 2 but we will come to general n below.)6

Daniel Bump

One may also consider the Dirichlet series that are (heuristically) of theform L(w, π, χd ) |d|−w where π is an automorphic representation of GLk . If
k = 2, then there is considerable literature of the case where n = 2; seefor example [6] and the references therein. If k = 2 and n = 3, there isa remarkable theory in [17]; there is a finite group of functional equations,transform the Dirichlet series into various different ones. If k = 3 and n = 2,
then there is also a finite group of functional equations; see [21]. The papers
cited in this paragraph predate the recent development of the combinatorial
theory, but the combinatorics of multiple Dirichlet series involving GLk cusp
forms is under investigation by Brubaker and Friedberg.
A double Dirichlet similar
was considered by Reznikov [53].
P to thatn in [36]This is the Dirichlet series
L(s, χ )|n|−w , where χ is a Hecke character of
infinite order for Q(i). Using a method of Bernstein, he proved the meromorphic continuation of this multiple Dirichlet series and determined the poles.
Despite the similarity of this multiple Dirichlet series to that of [36], this series does not fit the same way in the theory of Weyl group multiple Dirichlet
series.
While the origins of our subject are in analytic number theory, our emphasis in this paper will not be such applications, but rather on emerging
connections with areas of combinatorics, including quantum groups and mathematical physics, and the theory of Whittaker functions. As we will see, theproblem of giving precise definitions of the multiple Dirichlet series, even when
the general nature of the Dirichlet series is known, is a daunting combinatorial
one. Early investigations, such as [25] and [21] took an ad hoc approach substituting computer algebra or brute force computation for real insight. Thisis sufficient for applications on a case-by-case basis but also unsatisfactory. In
recent years, the combinatorial theory has been examined more closely, andits study may turn out to be as interesting as the original problem.

2 A method of analytic continuation
Let us consider a double Dirichlet series which might be writtenXZΨ∗ (s1 , s2 ) = (∗)ZΨ (s1 , s2 ),ZΨ (s1 , s2 ) =AΨ (n, m)n−s1 m−s2 .
n,mHere (∗) denotes some Gamma functions and powers of π. The Dirichlet series is allowed to depend on a parameter Ψ drawn from a finite-dimensional
vector space Ω. It is assumed convergent in some region C such as the one
in the following figure, which shows the region for (2). We have graphed the
projection onto R2 obtained by taking the real parts of s1 and s2 .

Multiple Dirichlet Series7

s1 = 21

C( 34 , 43 )( 12 , 12 )s2 = 21

Collecting the coefficients of m−s2 for each m gives a collection of Dirichlet
series in one variable s1 which have functional equations. These may be withrespect to some transformation such as the following, which is a functional
equation of (2).
1.
σ1 : (s1 , s2 ) 7−→ 1 − s1 , s1 + s2 −2More precisely, there may be an action of σ1 on Ω, or more properly on M⊗Ω,
where M is the field of meromorphic functions in s1 and s2 , such that thefunctional equation has the form
1
= ZΨ∗ (s1 , s2 ).
Zσ∗1 Ψ 1 − s1 , s1 + s2 −2Thus Ψ 7−→ σ1 Ψ is a linear transformation of the vector space Ω which, when
written out as a matrix, could involve meromorphic functions of s1 and s2 .
This is the scattering matrix . In some cases these meromorphic functions areholomorphic, or even just Dirichlet polynomials in a finite number of integers.
For the example (2) we would take polynomials in 2−s1 and 2−s2 .
This gives the meromorphic continuation to the convex hull of C ∪ σ1 C.

8Daniel Bump

hull(C ∪ σ1 C)( 34 , 34 )( 12 , 21 )Similarly (we assume) that collecting the coefficients of the other variablegives another functional equation, which in the example (2) is1
σ2 : (s1 , s2 ) 7−→ s1 + s2 − , 1 − s2 .
2
The functional equations may be iterated, so we get analytic continuation to
the the union of hull(C ∪ σ1 C) with hull(C ∪ σ2 C) and σ1 hull(C ∪ σ2 C).

( 41 , 1)( 34 , 43 )(0, 34 )(1, 41 )(8)
At this point, there are two ways of proceeding, one better than the other.
We could continue to iterate the functional equations until we obtained mero-Multiple Dirichlet Series

9morphic continuation to a region such as this:

There are two problems with this. One is that we have not obtained themeromorphic continuation in the area near the origin. The other is that
we have obtained two different meromorphic continuations to the region
σ1 σ2 σ1 C = σ2 σ1 σ2 C that is darkly shaded. We do not know that these twomeromorphic continuations agree. This agreement, the braid relation, shouldbe true in a suitable sense but in fact since there is a scattering matrix involved we must be careful in formulating it. We want an action of W on Ψ
extending the one already mentioned for σ1 such that ZΨ∗ (s1 , s2 ) satisfies∗∗
Zw(Ψ) (w(s1 , s2 )) = ZΨ (s1 , s2 ),(9)and the braid relation means that σ1 σ2 σ1 (Ψ ) = σ2 σ1 σ2 (Ψ ).
A better procedure is to use a theorem in complex variables, Bochner’s
convexity theorem [5] to assert meromorphic continuation once one has obtained meromorphic continuation to a region such as (8) whose convex hullis C2 . Bochner’s theorem is as follows: let U be an open subset of Cr wherer > 2 that is the preimage of an open subset of Rr under the projection
map; such a set is called a tube domain. Then any holomorphic function ona tube domain has analytic continuation to its convex hull. In our case, wehave a meromorphic function, but the polar divisor is a set of hyperplanes,
and the theorem is easily extended to this case. Hence once we have meromorphic continuation to (8) we obtain meromorphicity on C2 . The braid relation
σ1 σ2 σ1 (Ψ ) = σ2 σ1 σ2 (Ψ ) is then a consequence. See [9] and [11] for further
details.
Now we come to the fundamental combinatorial question. Once one has
decided roughly what the Dirichlet series is to look like, the exact coefficients10

Daniel Bump

are still not precisely defined. How can the coefficients be determined in sucha way that the functional equations (9) are true for both σ1 and σ2 ? For the
Dirichlet series (3), this is not too hard when k = 1, but when k = 3, thecombinatorics are rather daunting. They were treated in [21] using difficult
manipulations that were the only way before the combinatorial properties of
Weyl group multiple Dirichlet series began to be established. Similarly in theexample of Chinta [25], the method of solving the combinatorial problem wasto use a computer program to find a Dirichlet series with very special combinatorial properties. There has been a great deal of progress in the basic
combinatorial problem since these early papers, and this progress has implications beyond the original practical problem of giving a proper definition ofa multiple Dirichlet series with a group of functional equations.

3 Kubota Dirichlet series
Let n be a positive integer: we will define some Dirichlet series related to then-th power reciprocity law, so n = 2 in Section 1. Let F be a number fieldcontaining the group µn of n-th roots of unity. We will further assume that
F contains the group µ2n of 2n-th roots of unity, that is, that −1 is an n-th
power in F . The assumption that µn ⊂ F is essential; the assumption that
µ2n ⊂ F is only a matter of convenience. We will make use of the n-th powerreciprocity law.
We will define a family of Dirichlet series with analytic continuation andfunctional equations, called Kubota Dirichlet series. If n = 2, these are thequadratic L-functions L(2s − 21 , χd ). If n = 1, these Dirichlet series are divisor
sums, actually finite Dirichlet polynomials. For general n, they are generating
functions of n-th order Gauss sums.
Let S be a finite set of places of F containing all places dividing n andall archimedean ones. If v is a place of F , let Fv be the completion at v.
If v is nonarchimedean, let ov be the ring of integers in Fv . Let oS be thering of S-integersQin F , that is, those elements x ∈ F such that x ∈ ov for allv∈/ S. Let FS = v∈S Fv . We may embed oS in FS diagonally. It is a discrete,cocompact subgroup. We may choose
Q S so large that oS is a principal ideal
domain. If a ∈ FS let |a| denote v∈S |a|v . It is the Jacobian of the map
x 7−→ ax. If a ∈ oS then |a| is a nonnegative rational integer.
We recall the n-th order reciprocity law and n-th order Gauss sums. See
Neukirch [51] for proofs. Properties of the reciprocity symbol and Gauss sums
are more systematically summarized in [11].
The n-th order Hilbert symbol ( , )v is a skew-symmetric pairing of Fv× ××
Fv into µn . Define a pairing ( , ) on FS× by
Y( x, y ) =(xv , yv )v ,x, y ∈ FS× .
v∈SMultiple Dirichlet Series

11Then the n-th power residues symbol dc , defined for nonzero elements c, d ∈o, satisfies the n-th power reciprocity law
 
c
d
= (d, c).
(10)dcLet ψ be an additive character of FS that is trivial on oS but no larger
fractional ideal. Let
X  c   mc g(m, d) =ψ.
ddc mod dThe sum is well defined since both factors only depend on c modulo d. It has
the twisted multiplicativity properties:  ′dd′g(m, d) g(m, d′ ) if gcd(d, d′ ) = 1,g(m, dd ) =d′dg(cm, d) = c −1g(m, d)if c, d are coprime.
dand the absolute value, for p prime in oS :p|g(m, p)| = |p| if gcd(m, p) = 1.

(11)

× n
Let Ψ be a function on FS× such that Ψ (εc) = (ε, c)Ψ (c) when ε ∈ o×
S (FS ) .
The vector space of such functions is nonzero but finite-dimensional. LetX
DΨ (s; m) =Ψ (c)g(m, c) |c|−2s .
cThis has a functional equation under s 7−→ 1 − s. To state it, let
Gn (s) = (2π)−(n−1)(2s−1)Γ (n(2s − 1)).
Γ (2s − 1)

DefineDΨ∗ (s, m) = G(s)N ζF (2ns − n + 1) DΨ (s, m)where N is the number of archimedean places (all complex) and ζF is the
Dedekind zeta function of F . Then Kubota [46] proved a functional equation
for this, as a consequence of the functional equations of Eisenstein series on
the n-fold metaplectic covers of SL2 , which he developed for this purpose.
In the form we need it, this is proved by the same method in Brubaker andBump [18], and a similar result is in Patterson and Eckhardt [34].
To state these functional equations, there exists a family of Dirichlet polynomials Pη indexed by η in FS× /(FS× )n such thatXDΨ∗ (s, m) =(12)
|m|1−2s Pmη (s) DΨ̃∗ η (1 − s, m),η∈FS× /(FS× )n12Daniel BumpwhereΨ̃η (c) = (η, c)Ψ (c−1 η −1 ).
The polynomial Pη is actually a polynomial in qv−s where v runs through thefinite places in S, and qv is the cardinality of the residue field. It is important
for applications that Pη is independent of m.

4 A more general heuristic form
−1 p|c| times a
If n = 2, and m and c are coprime then g(m, c) equals mcfactor which may be combined with Ψ and ignored for heuristic purposes. ThusDΨ (s; m) is essentially L(2s− 21 , χm ). We may now give the following heuristic
generalization of the Dynkin diagram multiple Dirichlet series described in
Section 1. Let us start with a Dynkin diagram, which we will at first assume
is simply-laced (Type A, D or E). As in Section
heuristic purposes 1, for purely
it is not necessary to distinguish between dc and dc since by the reciprocity
law they differ by a factor (d, c) which may also be combined with Ψ , and may
be ignored for heuristic purposes. Ultimately such factors must eventually bekept track of, but at the moment they are unimportant.
The nodes i = 1, · · · , r of the Dynkin diagram are in bijection with thesimple roots of some root system Φ. We choose one complex parameter si foreach i, and one “twisting parameter” mi , which is a nonzero integer in oS .
The multiple Dirichlet series then has the heuristic form


 −1
X
Y
d
i
 g(mi , di )|di |−2si .
dj
d1 ,··· ,dri, j adjacent

The form of the coefficient is only correct if di are squarefree and coprime,and even then there is a caveat, but this heuristic form is sufficient for extrapolating the expected properties of the multiple Dirichlet series. Whereas
before, on expanding in powers of one of the si parameters, we obtained aquadratic L-function, now we obtain a Kubota Dirichlet series.
If the Dynkin diagram is not simply-laced, there are long roots and shortroots. In this case there is also a heuristic form, which we will not discuss here.
For each distict pair of simple roots αi and αj let r(αi , αj ) be the numberof bonds connecting the nodes connecting αi and αj in the Dynkin diagram.
Thus if θ is the angle between αi and αj let
0 if αi , αj are orthogonal,
1 if θ = 2π
3 ,r(αi , αj ) =
3π
2
if
θ
=

4 ,


3 if θ = 5π
6 .
Normalize the roots so short roots have length 1; thus every long root α has||αk2 = 1, 2 or 3, the last case occurring only with G2 . Let

Multiple Dirichlet Series

X  c t  mc 
gα (m, d) =
,
ψ
d
d

13

t = kαk2 .

c mod dThen the heuristic form of the multiple Dirichlet series is"#XY  di −r(αi ,αj ) Y
−2s
i


.
gαi (mi , di )|di |
dj
i
i,j

(13)

d1 ,··· ,dr

5 Foundations and the Combinatorial Problem
The first set of foundations for Weyl group multiple Dirichlet series were given
by Fisher and Friedberg [35], and these were used in all earlier papers. Anotherset of foundations were explained in [9] and [11] and these have been used forthe most part in subsequent papers. We recall them in this section.
Let V be the ambient vector space of Φ. Let h , i be a W -invariant innerproduct on V such that the short roots have length 1. Let B : V ⊗ Cr P−→ C
be the bilinear map that sends (αi , s) to si , where s = (s1 , · · · , sr ) toki si
and αi is the i-th simple root. Let ρ∨ denote the vector (1, · · · , 1) ∈ Cr .
The reason for this notation is explained in [11]. The Weyl group action ons, corresponding to the group of functional equations, may be expressed interms of B: we require that




1
1
B wα, w(s) − ρ∨ = B α, s − ρ∨22
for w ∈ W .
We fix an ordering of simple roots of Φ, so that in order they are{α1 , · · · , αr }. Some of the formulas depend on this ordering, but in an inessential way. Following [11] let us define M to be the nonzero but finite dimensional
space of functions on Ψ : (FS× )r −→ C that satisfy


r
Y

Y
2
2hα ,α ikα k(εi , Ci )S iΨ (ε1 C1 , · · · , εr Cr ) =(εi , Cj )S i j
Ψ (C1 , · · · , Cr )


i=1

i<j

(14)
× n
×
when ε1 , · · · , εr ∈ o×
(F
)
and
C
∈
F
.
i
S
S
S
We seek a function H(C1 , · · · , Cr ; m1 , · · · , mr ) defined if the Ci and mi
are nonzero elements of oS with the following properties. There is the multiplicativity condition
H(C1 C1′ , · · · , Cr Cr′ ; m1 , · · · , mr )=H(C1 , · · · , Cr ; m1 , · · · , mr ) H(C1′ , · · · , Cr′ ; m1 , · · · , mr )!2hαi ,αj i  kαi k2  ′ kαi k2 Yr 
2hαi ,αj i
Y
Ci
Ci
Ci′
Ci
.
Ci′
Ci
Cj′
Cj
i=1
i<j

(15)

14

Daniel Bump

There is another multiplicativity condition which, unlike (15), does involvethe mi . If gcd(m′1 · · · m′r , C1 · · · Cr ) = 1 we require:


m′1
C1

−||α1 ||

2

···



H(C1 , · · · , Cr ; m1 m′1 , · · · , mr m′r ) =m′rCr−||αr ||2

H(C1 , · · · , Cr ; m1 , · · · , mr ).

(16)

The conditions (14) and (15) together imply that if C1 , · · · , Cr is each
multiplied by a unit, then the value of Ψ H(C1 , · · · , Cr ) is unchanged. SinceoS is a principal ideal ring, we see that Ψ H(C1 , · · · , Cr ) is really a functionof ideals. Let
XZΨ (s1 , · · · , sr ; m1 , · · · , mr ) =Ψ (C1 , · · · , Cr )H(C1 , · · · , Cr ; m1 , · · · , mr )|C1 |−2s1 · · · |Cr |−2sr(17)where the summation is over ideals (Ci ). Also let"#Y∗ZΨ (s1 , · · · , sr ; m1 , · · · , mr ) =ζα (s)Gα (s) ZΨ (s1 , · · · , sr ; m1 , · · · , mr ),
α∈Φ+

(18)

where, if α is a positive root1
1 + 2n(α) α, s − ρ∨
,
2



1
1
+ α, s − ρ∨Gα (s) =Gn(α)
22ζα (s) =ζFwithn if α is a short root, n if α is a long root and Φ 6= G2 , and n is oddn(α) = n2 if α is a long root and Φ 6= G2 , and n is even
n if α is a long root and Φ = G2 , and 3 ∤ nn
3 if α is a long root and Φ = G2 , and 3|n.

(19)

The Kubota Dirichlet series DΨ (s; m) is the special case if ZΨ where theroot system is of type A1 .
We still have not fully described H, so we haven’t given a proper definitionof ZΨ . The multiplicativities (15) and (16) together imply that the function
H is determined by its values on prime powers. In other words if we specify
H(pk1 , · · · , pkr ; pl1 , · · · , plr ) for prime elements p, the function is determined.
The fundamental combinatorial problem is this: given a global field F in
which −1 is an n-th power and a root system, give a correct definition of themultiple Dirichlet series extrapolating the heuristic one, such that expandingin powers of every si gives a sum of Kubota Dirichlet series all having thesame functional equations. Naturally this must be made more precise. We
will write s = (s1 , · · · , sr ) and m = (m1 , · · · , mr ).

Multiple Dirichlet Series

15

Fundamental Combinatorial Problem. Define H(pk1 , · · · , pkr ; pl1 , · · · , plr )in such a way that for each index i the series ZΨ (s; m) has an expansion
X(20)DΨi (si , M ) PM (s)M

for some Ψi , where PM is a Dirichlet polynomial, such that for each i we havePM (σi s) = |M |1−2si PM (s).

(21)

If this can be done, then we have a functional equationZΨ (s; m) = ZΨ ′ (σi s; m)

(22)

′

for some Ψ . Here σi is the simple reflection in the Weyl group action on
the parameters s; if the root system is simply-laced it is the action (7), orsee [11] for the general case. The method of analytic continuation describedin Section 2 is applicable. This yields both the meromorphic continuation andthe scattering matrix, which we recall from Section 2 amounts to an action of
W on Ψ such that in (22) we have Ψ ′ = σi Ψ and more generally∗ZwΨ(ws; m) = ZΨ∗ (s; m).

The normalizing factor in (18) works out as follows: the factor ζα Gα with
α = αi is needed to normalize the Kubota Dirichlet series in (20). The othersuch factors are simply permuted by s 7−→ σi (s).
Let us consider briefly how this works in the case of Type A2 . See [9] fora complete discussion and detailed proof for this case. We have noted abovethat specifying the coefficients H(pk1 , pk2 ; pl1 , pl2 ) completely determines thefunction H. In this example, let us take m1 = m2 = 1 so l1 = l2 = 0 for all p.
The coefficients to be described are given by the following table
Let the nonzero values of H(pk1 , pk2 ; 1, 1) be given by the following table:
k1
0
1
2
01
g(1, p)k2
1 g(1, p) g(p, p)g(1, p) g(p, p2 )g(1, p)
2g(p, p2 )g(1, p) g(p, p2 )g(1, p)2
Then, collecting terms with equal powers of |p|−s2 we have a decomposition(20) where the summation includes terms of the following type:DΨ ′ (s1 ; 1),g(1, p)|p|−2s2 DΨ ′′ (s1 ; p),g(1, p)g(p, p2 )|p|−2s1 −4s2 DΨ ′′′ (s1 ; 1),for suitable Ψ ′ , Ψ ′′ and Ψ ′′′ . We recognize the p-parts of these Kubota Dirichlet
series from the tabulated values by collecting the terms in each column of thetable.
Early papers in this subject gave ad hoc solutions to the combinatorial
problem. Such direct verifications become fairly difficult, for example in [25]and [21].

16

Daniel Bump

6 p-parts
Let us define the p-part of Z to be the Dirichlet series∞X

H(pk1 , · · · , pkr ; pl1 , · · · , plr )|p|−2k1 s1 −...−2kr sr .

(23)

ki =0

We fix the representative p of the prime and ignore Ψ . By twisted multiplicativity, if the p-parts are known for all p, the multiple Dirichlet series isdetermined.
Returning to (17), let us consider the effect of the parameters m1 , · · · , mr .
These are called twisting parameters, and the term “twisting”
is supposedP
to evoke the usual twisting of L-functions: if L(s, f ) = Pan n−s is some Lfunction and χ is a Dirichlet character then L(s, f, χ) =χ(n)an n−s . The
term “twisting” in the present context is both apt and in a way misleading,as we will now explain.
First suppose that m1 , · · · , mr are coprime to C1 , · · · , Cr . Then by (16)
we have
H(C1 , · · · , Cr ;m1 , · · · , mr ) =
−||α1 ||2
−||αr ||2

m1
mr
···
H(C1 , · · · , Cr ; 1, · · · , 1) (24)
C1
Cr
Thus these coefficients are indeed simply multiplied by an n-th order character, as the term “twisting” suggests.
On the other hand, if the mi are not coprime to the Ci , then the effect ofrthe mi is much more profound. For example in H(pk1 , · · · , pkP; p l1 , · · · , p lr )it is important to think of (l1 , · · · , lr ) as indexing a weight,li ̟i , where̟1 , · · · , ̟r are the fundamental dominant weights of the root system Φ. Then
we may think of the p-part (23) as being something related to the character ofan irreducible representation of the associated Lie group, times a deformationof the Weyl denominator, but with the weight multiplicities replaced by sumsof products of Gauss sums. In particular, varying mi = pli affects the p-partin a profound way, no simple twisting.
With mi general, their meaning may be explained as follows: specifying
m1 , · · · , mr is equivalent to specifying, for each p, dominant weight λp such′li ′that λPp = 0 for almost all p. Indeed, factor mi = p mi where p ∤ mi and take
λp = li ̟i .
Now
P let the prime p and the exponents l1 , · · · , lr be fixed, and let λ =λp = li ̟i be the corresponding dominant weight. Also let
ρ=

r
X
i=1

̟i =

1 X
α
2
+
α∈Φ

be the Weyl vector. Let W be the Weyl group of Φ. If w ∈ W , let k(w)Pbe ther-tuple of nonnegative integers (k1 , · · · , kr ) such that ρ+λ−w(ρ+λ) = ki αi .

Multiple Dirichlet Series

17

The coefficients H(pk1 , · · · , pkr ; pl1 , · · · , plr ) in general do not admit aneasy description, but if (k1 , · · · , kr ) = k(w) for some w, then it is a productof l(w) Gauss sums, where l(w) is the length of w. To make this explicit, letΦw be the set of all positive roots α such that w(α) is a negative root, so|Φw | = l(w). Then (see [12]) we haveY
H(pk1 , . . . , pkr ; pl1 , . . . , plr ) =gkαk2 (phλ+ρ,αi−1 , phλ+ρ,αi ).
(25)
α∈Φw

Let Supp(λ) be the support of H(pk1 , · · · , pkr ; pl1 , · · · , plr ), that is, the set ofk = (k1 , · · · , kr ) such that H(pk1 , · · · , pkr ; pl1 , · · · , plr ) 6= 0. Then by (25),{k(w)|w ∈ W } is contained in Supp(λ).
Most importantly, the |W | points k(w) are the extremal values of the
support. That is, Supp(λ) is contained in the convex hull of k(w). These |W |extremal points are called stable in [11] and [12] for the following reason. If
n is sufficiently large, then Supp(λ) = {k(w)|w ∈ W }, and in this case, thevalues (25) are the only nonzero values of H(pk1 , . . . , pkr ; pl1 , . . . , plr ). So thesevalues are “stable” and the combinatorial problem is solved by (25).
For arbitrary n, Supp(λ) is at least contained in the convex hull of{k(w)|w ∈ W }. But for interior points of this convex polytope, the descriptionof H(pk1 , . . . , pkr ; pl1 , . . . , plr ) is much more difficult. We will next look at thevarious approaches.

7 Multiple Dirichlet Series and Combinatorics
In this section we will introduce the modern combinatorial theory of the pparts of Weyl group multiple Dirichlet series. There are several different methods of representing the p-parts of multiple Dirichlet series to be considered,each with its own individual combinatorial flavor. The combinatorial theory
has only taken shape in the last few years. We will state things most fully inthe “nonmetaplectic” case n = 1, leaving the reader hopefully oriented andready to explore the general cases in the literature.
We see that correctly specifying the p-part of the function H produces
a Dirichlet series ZΨ with global meromorphic continuation. These functions
turn out to be extremely interesting. Several definitions of H emerged, andproving their equivalence proved to be nontrivial. Moreover, as the functions
H were intensively studied, various clues seemed to suggest connections withthe theory of quantum groups. We will discuss these points in this section.
The following main classes of definitions were found.
••••

Definition by the “Averaging method,” sometimes known as the ChintaGunnells method.
Definition as spherical p-adic Whittaker functions.
Definition as sums over crystal bases.
Definition as partition functions of statistical-mechanical lattice models.

18

Daniel Bump

The first and second definitions give uniform descriptions for all root systems
and all n. The third and fourth definitions are on a case-by-case basis andhave not been carried out for all n. Nevertheless they are very interesting,and it is the latter two approaches that suggest connections with quantum
groups.
The equivalence of these different definitions is by no means clear or easy.
However it is now mostly proved by the following scheme.
Chinta-Gunnells
Whittaker
CrystalStatistical←→←→←→methodfunction
description
model
The equivalence of the averaging method with the Whittaker definition wasproved by Chinta and Offen [29] for Type A by generalizing the original proof
of Casselman and Shalika. This was extended by McNamara [49] to arbitrary
Cartan types. Two results which both assert that the Whittaker definitionis equivalent to the crystal definition (in Type A) are Brubaker, Bump andFriedberg [13] and McNamara [50]. The first paper directly computes theWhittaker coefficients of Eisenstein series, and the second paper proceeds
locally by partitioning the unipotent integration into cells that contributethe individual terms in the sum over the crystal. The relationship between
the statistical model scheme and the crystal description must be done on acase-by-case basis, but we will discuss these below.
Yet another possibility has appeared on the horizon within the last few
months.
•Approach p-adic Whittaker functions by means of Demazure-Lusztig operators and “metaplectic” generalizations of them.

The above remarks concern mainly what is, in the language of Whittakermodels, the spherical Whittaker function. However it is useful to consider alarger class of Whittaker functions, namely the Iwahori fixed vectors in the
Whittaker model. When these are considered, the Demazure-Lusztig operatorsand their metaplectic analogs appear.
There are other objects in mathematics that may be related to these.
•••

Some examples of zeta functions of prehomogeneous vector spaces seem tobe specializations of Weyl group multiple Dirichlet series. These connections are under investigation by Chinta and Taniguchi.
Jacquet conjectured that an O(r) period of an automorphic form on GLr
is related to a Whittaker coefficent of the Shimura correspondent on the
double cover of GLr . Applying this to Eisenstein series, this would meanthat orthogonal periods of Eisenstein series on GLr are related to TypeAr−1 . When r − 1 = 2, this was investigated by Chinta and Offen [30].
Zeta functions of prehomogeneous vector spaces as well as the Witten
zeta functions studied by Komori, Matsumoto and Tsumura [45] in thisvolume are both special cases of Shintani zeta functions. It is by no means
clear that the Witten zeta functions can be related to Weyl group multiple
Dirichlet series but potentially there are undiscovered connections.

Multiple Dirichlet Series

19

Let us begin with p-adic Whittaker functions. Casselman and Shalika [24]showed that the values of the spherical Whittaker function are expressible asvalues of the characters of irreducible representations of the L-group, times adeformation of the Weyl denominator. We begin by reviewing this important
formula.
Let G be a split Chevalley group, or more generally a split reductive groupdefined over Z. Let F be a nonarchimedean local field with residue field o/p =Fq , where o is the ring of integers and p its maximal ideal. Let B = T N bea Borel subgroup, where T is a maximal split torus, and N is the unipotent
radical. The root system lives in the group X ∗ (T ) of rational characters of T
and the roots so that N is the subgroup generated by the root groups of thepositive roots.
We may take the algebraic groups G, T, B, N to be defined over o. Then
G(o) is a special maximal compact subgroup. If w is an element of the Weylgroup W , we will choose a representative for it in G(o), which, by abuse ofnotation, we will also denote as w.
Let Ĝ be the (connected) Langlands L-group. It is an algebraic groupdefined over C. Then G and Ĝ contain split maximal tori T and T̂ respectively;
T we have already chosen. Then T̂ (C) is isomorphic to the group of unramified
characters of T (F ), that is, the characters that are trivial on T (o). If z ∈ T̂ (C),let τz denote the corresponding unramified character.
Let Λ be the weight lattice of T̂ , that is, the group of rational characters.
Then Λ is isomorphic to T (F )/T (o). The isomorphism may be chosen sothat if λ is a weight and aλ is a representative of the corresponding coset in
T (F )/T (o), thenτz (aλ ) = z λ .
(26)
There are now two root systems to be considered: the root system of G relative
to T , and the root system of Ĝ relative to T̂ . The latter is more important forus, so we will denote it as Φ. Thus Φ is contained in the Euclidean vector spaceR ⊗ Λ. If α ∈ Φ then the corresponding coroot α∨ is a root of G with respect
to T , and we will denote by iα : SL2 −→ G the corresponding Chevalley
embedding.
For example, let G = GLr+1 . Then Ĝ = GLr+1 . We take T and T̂ to be
the diagonal tori. We may identify the weight lattice Λ of T̂ with Zr+1 in sucha way that λ = (λ1 , · · · , λr+1 ) ∈ Zr+1 corresponds to the rational character


z1
Y


z =  ...
ziλi .
 7−→
i
zr+1

If p is a generator of p we may take
 λ
p 1

..
aλ = 
.

pλr+1




.

20

Daniel Bump

Then (26) is satisfied with


τz 



t1
..

.
tr+1

 Y ordp (ti )zi.
=Returning to the general case, let z ∈ T̂ (C). We may induce τz to G(F ),by considering the vector space Vz of functions f : G(F ) −→ C that satisfy
f (bg) = (δ 1/2 τz )(b)f (g),b ∈ B(F ).

(27)

The group G(F ) acts on Vz by right translation. If z is in general position, thisrepresentation is irreducible, and unchanged if z is replaced by any conjugate
by an element of the Weyl group. If it is not irreducible, at least its set of
irreducible constitutents are unchanged if z is conjugated.
Let ψ : N (F ) −→ C be a character.
We will assume that if α is a simple
1xof F is trivial on o but no largerroot, then the character x 7−→ iα1fractional ideal. If f ∈ Vz and g ∈ G(F ), define the Whittaker function on
G(F ) associated with f byZ
Wf (g) =f (w0 ng) ψ(n) dn,(28)
N (F )

where w0 is a representative in G(o) of the long Weyl group element. The
integral is convergent if |z α | < 1 for positive roots α; for other z, it may be
extended by analytic continuation. The space Vz has a distinguished spherical
vector f ◦ characterized by the assumption that f ◦ (g) = 1 for g ∈ G(o). Let
W ◦ = Wf ◦ .
Theorem 1. (Casselman-Shalika [24]) Let λ ∈ Λ. Then

 Q
−1 α
z ) χλ (z) if λ is dominant,+ (1 − q
−1/2
◦
α∈Φ
δ
(aλ ) W (aλ ) =0otherwise.

(29)

Here with λ dominant, χλ is the character of the finite-dimensional irreducible representation of Ĝ having highest weight λ. Note that the producton the right-hand side is a deformation of the Weyl denominator. Thus if ρ
is half the sum of the positive roots, on specializing q −1 to 1, the right-handside of (29) becomes
"
#
Y
X
α
(1 − z ) χλ (z) = z ρ(−1)l(w) z w(λ+ρ) ,
(30)
α∈Φ+

w∈W

where we have used the Weyl character formula to rewrite the specialization
as a sum over the Weyl group.

Multiple Dirichlet Series

21

We next consider how expressions such as the character χλ may be interpreted as the p-parts of multiple Dirichlet series.
Let q be a power of a rational
prime. Let λ be a dominant weight. We
Pconsider an expression E = µ m(µ) z µ , where the sum is over weights µ,
and m(µ) is a complex number that is nonzero for only finitely many µ. Moreprecisely, we assume m(µ) = 0 unless µ is in the convex hull of the polytope
spanned by the W -orbit of λ, and moreover λ − µ is in the root lattice, whichis the lattice in Λ spanned by Φ.1 We will also assume that m does not vanishon the W -orbit of λ, though it may vanish for roots in the interior of thepolytope. We will call E a λ-expression. For example, χλ is a λ-expression,and the numerator in the Weyl character formula, in other words (30), is a
(λ + ρ)-expression.
Given a λ-expression E, let us show how to obtain a Dirichlet polynomial,that is, a polynomial in q −2s1 , · · · , q −2sr where r is the rank of Ĝ. Given µ,
there
exist nonnegative integers (k1 , · · · , kr ) = (k1 (µ), · · · , kr (µ)) such thatPki αi = µ − w0 (λ), where w0 is the long Weyl group element. Then we callXm(µ)q −2k1 (µ)s1 −...−2kr (µ)srµ

the Dirichlet polynomial associated with the λ-expression E. The p-parts ofthe multiple Dirichlet series that we are considering are all of this type. If
n = 1, the (λ + ρ)-expression producing the p-part is
"
#
Y
−1 α
(1 − q z ) χλ (z)
(31)
α∈Φ+

which, we observe, differs from (30) by the insertion of q −1 . Thus the p-part isa deformation of (30). Comparing with the Casselman-Shalika formula (29),
we see that this is essentially a value of the spherical Whittaker function.
Similarly, the p-part (23) with q = |p| is derived from a certain (λ + ρ)expression, and these (λ + ρ)-expressions turn out to be values of spherical
Whittaker functions on metaplectic covers of G. The integer li in (23) is theinner product of λ with the coroot α∨
i . These (λ + ρ)-expressions might beregarded as analogs of (31) in which the integers m(µ) have been replacedby sums of products of Gauss sums. As we will explain, they are extremelyinteresting objects from a purely combinatorial point of view.
The averaging method of Chinta-Gunnells expresses the p-part of the multiple Dirichlet series as a ratio in which the numerator is a sum over the Weylgroup, and in the case n = 1, it reduces to the right-hand side of (30). Whenn > 1, the Weyl group action on functions is non-obvious; the simple reflections involve Gauss sums and congruence conditions, and verifying the braid
relations is not a simple matter. See Chinta and Gunnells [27] for this action,
1

If Ĝ is semisimple then the root lattice has finite index in Λ.

22

Daniel Bump

and Patterson [52] for a meditation on the relationship between the method
and the intertwining operators for principal series representations.
We turn next to the crystal description. Crystals arose from the representation theory of quantum groups, that is, quantized enveloping algebras. Letĝ be a complex Lie algebra, which for us will be the Lie algebra of Ĝ. Thenthe quantized enveloping algebra Uq (ĝ) is a Hopf algebra that is a deformation of the usual universal enveloping algebra U (ĝ) to which it reduces whenq = 1. The representations of Ĝ(C) correspond bijectively to characters of ĝ,and hence to U (ĝ); they extend naturally to representations of Uq (ĝ).
Suppose that λ is a dominant weight, which is the highest weight vector ofan irreducible representation of of Ĝ and hence of Uq (ĝ). This module Uq (ĝ)
has a distinguished basis, Kashiwara’s global crystal basis, which is closely
related to Lusztig’s canonical basis. Let us denote it as Bλ . Let 0 be the zeroelement Let
 of α1 , · · · , αr be the simple roots. If α = αi , then the module.
00
01
are in a certain sense approximated by maps ei
and diαdiα
1000
and fi from Bλ to Bλ ∪ {0}, the Kashiwara operators; each such operator
applied to v ∈ Bλ either gives 0 or another element of the basis.
Also, every element of the crystal basis lies in a well-defined weight space
so there is a map wt : Bλ −→ Λ mapping each element to its weight. We haveXχλ (z) =
zλ.
(32)
v∈Bλ

The maps ei and fi shift the weights: if x, y ∈ Bλ then fi (x) = y if and only ifei (y) = x and in this case, wt(y) = wt(x) − α. When this is true, we draw aniarrow x −→ y, and the resulting directed graph, with edges labeled by indicesi is the crystal graph.
Let us consider an example. Take G = GL3 . Its Cartan Type is A2 . In thiscase, the weight lattice Λ may be identified with Z3 . If λ is a weight, thenwith this identification λ = (λ1 , λ2 , λ3 ) where z λ = z1λ1 z2λ2 z3λ3 . The weight
is dominant if λ1 > λ2 > λ3 . Assume that λ is dominant, and furthermore
λ3 > 0, so that λ is a partition. The elements of Bλ may be identified withsemistandard Young tableaux with shape λ. So if λ = (3, 1, 0) and r = 2,
the crystal graph of Bλ is shown in Figure 1. The weight of a tableau T is
wt(T ) = (µ1 , µ2 , µ3 ) where µi is the number of entries in T equal to i.
If w ∈ W then Bλ has a unique element vwλ of weight wλ. We will callthese elements extremal. Consider a function f on Bλ , which we assume does
not vanish on the extremal elements. Then we may considerXf (v) z wt(v) .
Ef =
v∈Bλ

This is a λ-expression. For example if f (v) = 1 for all v, then Ef is thecharacter of the irreducible representation with highest weight λ.

Multiple Dirichlet Series

23

We call a weight strongly dominant if it is of the form λ + ρ with λ dominant. A dominant weight is strongly dominant if and only if W acts freely on
its orbit, or equalently, if it is in the interior of the positive Weyl chamber.
Let λ + ρ be a strongly dominant weight. We recall that the numerator (30)in the Weyl character formula, as are the p-parts of the Weyl group multiple
Dirichlet series, are (λ + ρ)-expressions. So we may hope to find a function fon Bλ+ρ such that Ef equals this numerator or p-part. In some cases, a deformation of the Weyl character formula exists in which the numerator is a sum
over Bλ+ρ . The formula will express χλ as a ratio of this sum to
Qa denominatorthat is a deformation of the Weyl denominator, of the form α∈Φ+ (1 − tz α )where t is a deformation parameter. Taking t = 1, only the extremal elementsof Bλ+ρ will make a nonzero contribution, and the numerator will reduce tothe numerator in the Weyl character formula. Taking t = 0, the only terms
that contribute will be those in the image of a map Bλ −→ Bλ+ρ , and thesum reduces to a sum over Bλ . Thus when t = 0, the formula reduces to (32).
Most importantly for us, taking t = q −1 and comparing with (29), we see thatthe sum over Bλ+ρ is exactly δ 1/2 (aλ )W ◦ (aλ ).
2

1 1 1
3

1 1 1
2
1

1

1 1 3
3
1

2

1 1 3
2
1
1 1 2
3

2

1 1 2
2
1

1

1 3 3
3
1

2

1 3 3
2
2
1 2 3
1
3
2 3 3
3

2

2

1 2 3
2
1 2 2
1
3
2 2 3
3

2

2

1 2 2
2

2 2 2
3

Fig. 1. The Crystal B(3,1,0) with an element of weight (1, 2, 1) highlighted.

Moreover such formulas exist for metaplectic Whittaker functions. In otherwords, there is often a way of summing over Bλ+ρ and obtaining a Whittaker
function on the n-fold cover of some group. There should actually be one such

24

Daniel Bump

formula for every reduced decomposition of the long Weyl group element into
a product of simple reflections, and in some sense this is true. But in practice
only certain such decompositions give clean and elegant formulas. Here is alist of cases where nice formulas are known, rigorously or conjecturally.
•
•

•

•

Cartan Type Ar , any n. See [14] and [13]. These produce Whittaker functions on the n-fold covers of GLr+1 . In this case proofs are complete.
Cartan Type Br , n even. These would produce Whittaker functions oneven covers of Sp2r . For general even n, the representation is conjectural,and even in the n = 2 case there is still work to be done. See [10], thisvolume, for the case n = 2.
Cartan Type Cr , n odd. See [3], this volume and Ivanov [40] for a discussionof the Yang-Baxter equation. These will produce Whittaker functions on
the n-covers of Sp2r+1 . Only the case n = 1 is proved.
Cartan Type Dr , any n. See [28], this volume. This case is still largely
conjectural.

Let us explain how this works for Type Ar for arbitrary r, so G = GLr+1(or SLr+1 ). If n = 1, the formula in question is Tokuyama’s formula.
Tokuyama [54] expressed his formula as a sum over strict Gelfand-Tsetlin
patterns, but it may be reformulated in terms of tableaux, or crystals. Using
crystals, Tokuyama’s formula may be written
"
#
X
Y
α
G♭ (v)z wt(v)−w0 ρ ,(33)(1 − tz ) χλ (z) =α∈Φ+v∈Bλ+ρ

where w0 is the long Weyl group element, and the function G♭ (v) will be
described below. Tokuyama’s formula is a deformation of the Weyl character formula, which is obtained by specializing t −→ 1; the formula (32) isrecovered by specializing t −→ 0. Taking t = q −1 to be the cardinality of
the residue field for a nonarchimedean local field, and combining Tokuyama’s
formula with the Casselman-Shalika formula (29), it gives a formula for thep-adic Whittaker function, and similar formulas give the p-part of the multiple
Dirichlet series. Since this is the case we are concerned with, we will write q −1
instead of t for the deformation parameter, even if occasionally we want tothink of it as an indeterminate.
To define G♭ (v), we first associate with v a BZL-pattern (for Berenstein,
Zelevinsky [4] and Littelmann [47]). We choose a “long word” by which wemean a decomposition of w0 = σω1 · · · σωN into a product of simple reflectionsσωi (1 6 i 6 r), where N is the number of positive roots. Let b1 be thenumber of times we may apply fω1 to v, that is, the largest integer such thatfωb11 (v) 6= 0. Then we we let b2 be the largest integer such that fωb22 fωb11 (v) 6= 0.
Continuing this way, we define b1 , · · · , bN . We may characterize BZL(v) =(b1 , b2 , · · · , bωN ) as the unique sequence of N nonnegative integers such thatfωbNN · · · fωb11 (v) is the unique vector vw0 λ with lowest weight w0 λ.

Multiple Dirichlet Series

25

We should be able to give a suitable definition of G♭ for any Cartan typeand any long word, but in practice we only know how to give precise combinatorial definitions in certain cases. We will choose this word:(ω1 , · · · , ωN ) = (1, 2, 1, 3, 2, 1, · · · )corresponding to the decomposition into simple reflections:w0 = σω1 · · · σωN = σ1 σ2 σ1 σ3 σ2 σ1 · · · σr σr−1 · · · σ1 ,where σi is the i-th simple reflection in W . We will write BZL(v) in a tabular
array:


..
..
. 
 .

b4 b5 b6 
(34)
BZL(v) = b2 b3 b1

This has the significance that each column corresponds to a single root operator fi where i = 1 for the rightmost column, i = 2 for the next column, andso forth.
Now we will decorate the pattern by drawing boxes or circles around various bi according to certain rules that we will now discuss. We describe thecircling rule first. It may be proved that bi satisfies the inequality bi > bi+1 ,except in the case that i is a triangular number, so that bi is the last entry in
its row; in the latter case, we only have the inequality bi > 0. In either caseWe circle bi if it’s inequality is an equality. In other words we circle bi if (inthe first case) bi = bi+1 or (in the second case) bi = 0.
For the boxing rule, we box bi if eωi+1 fωbii · · · fωb11 (v) = 0.
Let us consider an example. We take (ω1 , ω2 , ω1 ) = (2, 1, 2) and v to be122
the elementin the crystal B(3,1,0) that is highlighted in Figure 1. Then3
it is easy to see that
11
BZL(v) =.
1
We decorate this as follows. Since b2 = b3 , we circle b2 . Moreover eω1 (v) =0, since (referring to the crystal graph) there is no way to move in the e1
direction. Thus b1 is boxed and the decorated BZL pattern looks like this:


89:;
?>=<
1 1

BZL(v) = 
1

The boxing and circling rules may seem artificial, but are actually natural,
for they have the following interpretation. Kashiwara defined, in addition to
the crystals Bλ corresponding to the finite-dimensional irreducible representations, a crystal B∞ which is a crystal basis of the quantized universal eveloping

26

Daniel Bump

algebra of the lower unipotent part of the Lie algebra of G. There is anothercrystal Tλ with precisely one element having weight λ. Then B∞ ⊗ Tλ+ρ is a
“universal” crystal with a highest weight vector having weight λ + ρ. There isthen a morphism of crystals Bλ+ρ −→ B∞ ⊗ Tλ+ρ . See Kashiwara [41]. Thismorphism may be made explicit by adopting a viewpoint similar to Littelmann [47]. Indeed, the notion of BZL patterns makes sense for B∞ ⊗ Tλ+ρ ,and the set of BZL patterns is precisely the cone of patterns (34) such thatb1 > 0,b2 > b3 > 0,

b4 > b5 > b6 > 0,··· .

The morphism Bλ+ρ −→ B∞ ⊗ Tλ+ρ is then characterized by the condition
that corresponding elements of the two crystals have the same BZL pattern.
See Figure 1 of Bump and Nakasuji [23] for a picture of this embedding.
Now the circling rule may be explained as follows: embedding v ∈ Bλ+ρinto B∞ ⊗ Tλ+ρ , an entry in the BZL pattern is circled if and only if BZL(v)lies on the boundary of the cone. Similarly there is a crystal B−∞ ⊗ Tw0 (λ+ρ)
with a unique lowest weight vector having weight w0 (λ + ρ), and we may
embed Bλ+ρ into this crystal by matching up the lowest weight vectors, andan entry is boxed if and only if BZL(v) lies on the boundary of this opposite
cone.
Returning to Tokuyama’s formula, define1
if bi is circled but not boxed;




N

Y  −q −1 if bi is boxed but not circled;♭
G (v) =−1
 1 − q if bi is neither circled nor boxed; 

i=1 


0
if bi is both circled and boxed.

This was generalized to the n > 1 case as follows. There is, in this generality, a Whittaker function on the metaplectic group, and as in the case n = 1,
we haveXδ −1/2 (aλ )W ◦ (aλ ) =G♭ (v)z wt(v)−w0 ρ .
(35)v∈Bλ+ρ

Now the definition of G♭ must be slightly changed. Let a be a positive integer.
Define g(a) = q −a g(pa−1 , pa ) and h(a) = q −a g(pa , pa ) in terms of the Gausssum discussed in the last section. These depend only on a modulo n. Then we
have


1
if bi is circled but not boxed;




N 

Y
g(bi ) if bi is boxed but not circled;♭
G (v) =
 h(gi ) if bi is neither circled nor boxed; 

i=1 


0
if bi is both circled and boxed.
If n = 1, this reduces to our previous definition.
Tokuyama’s formula may be given another interpretation, as evaluating
the partition function of a statistical mechanical system of the free-fermionicsix-vertex model. For this, see [15], Chapter 19 of [14], and the paper [8] inthis volume. Since more details may be found in these references, we will beMultiple Dirichlet Series27

brief. In a statistical mechanical system, there is given a collection of (many)states, and each state is assigned a Boltzmann weight which is a measure of
how energetic the state is; more more highly energetic states are less probable.
For example, the two-dimensional Ising model consists of a collection ofsites, each of which may be assigned a spin + or −. These sites might represent
atoms in a ferromagnetic substance, and in the two dimensional model theylie in a plane. A state of the system consists of an assignment of spins to everysite. At each site, there is a local Boltzmann weight, depending on spin at thesite and at its nearest neighbors. This system was analyzed by Onsager, whofound, surprisingly, that the partition function could be evaluated explicitly.
Later investigators considered models in which the spins are assigned not
to the sites themselves, but to edges in a grid connecting the sites. The Boltzmann weight at the vertex depends on the configuration of spins on the fouredges adjacent to the vertex. Thus if the site is x, we label the four adjacentedges with εi = + or − by the following scheme:

ε1

ε2
xε

3

ε4

(36)

Of particular interest to us is the six-vertex or two-dimensional ice model
which was solved by Lieb and Sutherland in the 1960’s, though it is thetreatment of Baxter [1] that is most important to us. There are six admissible
configurations. These are given by the following table.
x

Boltzmann
a1 = a1 (x) a2
weight

x

x

b1

x

b2

x

c1

x

c2

The set of Boltzmann weights used may vary from site to site, so if (as in thetable) the site is x we may write a1 (x) to indicate this dependence.
If a1 = a2 , b1 = b2 and c1 = c2 then the site is called field-free. If a1 a2 +b1 b2 = c1 c2 , the site is called free-fermionic. The term comes from physics: inthe free-fermionic case, the row transfer matrices (see [8]) are differentiatedversions of the Hamiltonians of a quantum mechanical system, the XXZ model,
and the quanta for this model are particles of spin 1/2, called fermions.
Hamel and King [38] and Brubaker, Bump and Friedberg [15] showed that
one may exhibit a free-fermionic six-vertex model whose partition function isexactly the Tokuyama expression (33). In [8] in this volume, this is generalized
to a system whose partition function is the metaplectic spherical Whittaker
function (35). The explanation for this is as follows: there is a map from the28Daniel Bump

set of states of the model to the Bλ+ρ crystal. The map is not surjective, but
its image is precisely the set of v ∈ Bλ+ρ such that G♭ (v) 6= 0.
Thus using this bijection of the set of states with those v ∈ Bλ+ρ suchthat G♭ (v) 6= 0 means that Tokuyama’s theorem may be formulated either
as the evaluation of a sum over a crystal or as the partition function of astatistical-mechanical system. But there is a subtle and important difference
between these two setups. For example, different tools are available. Thereis a an automorphism of the crystal graph, the Schützenberger involution,that takes a vertex of weight µ to one of weight w0 µ, where w0 is the long
Weyl group element; this is sometimes useful in proofs. The set of states ofthe statistical-mechanical system has no such involution, yet another, more
powerful tool becomes available: the Yang-Baxter equation.
To describe it, let us associate a matrix with the Boltzmann weights at asite as follows. Let V be a two-dimensional vector space with basis v+ andv− . (In the metaplectic case, the scheme proposed in [8] gives its dimension as2n.) Associate with each site an endomorphism of V ⊗ V as follows. With thevertices labeled as in (36), let R be the linear transformation such that thecoefficient of vε3 ⊗ vε4 in R(vε1 ⊗ vε2 ) is the Boltzmann weight corresponding
to the four spins ε1 , ε2 , ε3 , ε4 . Thus the only nonzero entries in the matrix of
R with respect to the basis v± ⊗ v± are a1 , a2 , b1 , b2 , c1 , c2 . We will call anendomorphism R of V ⊗ V (or its matrix) an R-matrix . We then denote by
R12 , R13 and R23 endomorphisms of V ⊗ V ⊗ V in which Rij acts on thei-th and j-th components, and the identity 1V acts on the remaining one. For
example R12 = R ⊗ 1V .
We are interested in endomorphisms R, S and T of V ⊗ V such thatR12 S13 T23 = T23 S13 R12 .
It is not hard to check that this is equivalent to (25) in [8]. This was called
the star-triangle relation by Baxter, and the Yang-Baxter equation by others,
particularly in the case where R, S, T are either all equal, or drawn from thesame parametrized family. In particular let Γ be a group, and g 7−→ R(g) a
map from Γ into the set of R-matrices such that
R12 (g)R13 (gh)R23 (h) = R23 (h)R13 (gh)R12 (g).

(37)

Then (37) is called a parametrized Yang-Baxter equation.
In Section 9.6 of [1], Baxter essentially found parametrized Yang-Baxter
equations in the field-free case, where a1 = a2 = a, b1 = b2 = b and c1 =c2 = c. Fix a complex number ∆. Then his construction gives a parametrized
Yang-Baxter equation, with parameter group C× , such that the image of R
consists of endomorphisms of V ⊗ V with corresponding to such field-free Rmatrices with (a2 +b2 −c2 )/2ab = ∆. This construction led to the developmentof quantum groups. In the formulation of Drinfeld [33], this instance of theb 2 ). The parameter group
Yang-Baxter equation is related to Hopf algebra Uq (sl1indexes modules of this Hopf algebra with ∆ = 2 (q + q −1 ), and Yang-Baxter

Multiple Dirichlet Series

29b 2 ), or aequation is a consequence of a property (quasitriangularity) of Uq (slsuitable completion.
A parametrized Yang-Baxter equation with parameter group SL(2, C) was
given in Korepin, Boguliubov and Izergin [44], p. 126. The parametrized Rmatrices are contained within the free-fermionic six-vertex model. Scalar Rmatrices may be added trivially, so the actual group is SL(2, C)×GL1 (C). This
nonabelian parametrized Yang-Baxter equation was rediscovered in slightly
greater generality by Brubaker, Bump and Friedberg [15], who found aparametrized Yang-Baxter equation for the entire set of R-matrices in thefree-fermionic six-vertex model, with parameter group GL2 (C) × GL1 (C). It
is an interesting question how to formulate this in terms of a Hopf algebra,
analogous to the field-free case.
Brubaker, Bump and Friedberg [15] showed that a system may be found,with free-fermionic Boltzmann weights, whose partition function is precisely(33). This fact was generalized Bump, McNamara and Nakasuji [22], whoshowed that one may replace the character on the left-hand side by a factorial
Schur function. Then the parametrized free-fermionic Yang-Baxter equation
can be used to prove Tokuyama’s formula (or its generalization to factorial
Schur functions). Moreover, in [8] a different generalization is given, in which
the partition function represents the metaplectic Whittaker function. In thelatter case, however, no Yang-Baxter equation is known if n > 1.
When n = 1, various facts about Whittaker functions may be proved using the free-fermionic Yang-Baxter equation. One fact that may be checkedis that the partition function representing (33), divided by the product on
the left-hand side of the equation, is symmetric, in other words invariant under permuting the eigenvalues of z. This is a step in a proof of Tokuyama’s
theorem. As explained in [8], this fact has a generalization to partition functions representing metaplectic Whittaker functions, and seems amenable to
the Yang-Baxter equation, but no Yang-Baxter equation is known in this case.
In (35), the definition of G♭ depends on the choice of a reduced word
representing the long Weyl group element. Two particular long words are
considered, and the Yang-Baxter equation is used to show that both representations give the same result. If n > 1, this remains true, but again the
Yang-Baxter equation is unavailable. Consequently different proofs, based on
the Schützenberger involution of the crystal Bλ+ρ are given. However thesearguments require extremely difficult combinatorial arguments, and it would
be good to have an alternative approach based on the Yang-Baxter equation.
See [10] another application of the free-fermionic Yang-Baxter equation tometaplectic Whittaker functions, this time on the double cover of Sp(2r).

8 Demazure operators
Let (π, V ) be a principal series representation of G(F ), where G is a split
semisimple Lie group. The theory described above, including the Casselman-

30

Daniel Bump

Shalika formula (and its metaplectic generalizations) is for the spherical Whittaker function, that is, the K-fixed vector in the Whittaker model, where
K = G(o).
Let J be the Iwahori subgroup, which is the inverse image of B(Fq ) underthe map G(o) −→ G(Fq ) that is reduction mod p. We may consider moregenerally the space V J of J-fixed vectors in the Whittaker model. These play
an important role in the proof of the Casselman-Shalika formula which, wehave seen, is a key result in the above discussion.
Until 2011, the investigations that we have been discussing in the above
pages concentrated on the unique (up to scalar) K-fixed vector, rather thanelements of V J , though the Iwahori fixed vectors appeared in the work of
Chinta and Offen [29] and McNamara [49] generalizing Casselman and Shalika.
Still, the essence of the Casselman-Shalika proof is to finesse as much as
possible in order to avoid getting involved with direct calculations of Iwahori
Whittaker functions. But it turns out that there is an elegant calculus of
Iwahori Whittaker functions, and this is likely to be a key to the relationshipbetween the theory of Whittaker functions and combinatorics.
If w ∈ W , the Demazure operator ∂w acts on the ring O(T̂ ) of rational
functions on T̂ . To define it, first consider the case where w = σi is a simple
reflection. Then if f is a rational function on T̂ (C),∂σi f (z) =f (z) − z −αi f (σi z).
1 − z −αi

The numerator is divisible by the denominator, so this is again a rational
function. The definition of ∂w is completed by the requirement that if l(ww′ ) =l(w) + l(w′ ), where l is the length function on W , then ∂ww′ = ∂w ∂w′ .
If λ is a dominant weight then ∂w0 z λ is the character χλ (z), and forgeneral w we will call ∂w z λ a Demazure character . These first arose in thecohomology of line bundles over Schubert varieties, and they have proved tobe quite important in combinatorics. As Littelmann and Kashiwara showed,
they may be interpreted as operators on functions on crystals. As we will
explain, Demazure operators, and the related Demazure-Lusztig operators
arise naturally in the theory of Whittaker functions.
Iwahori and Matsumoto observed that V J is naturally a module for the
convolution ring of compactly supported J-biinvariant functions, and they determined the structure of this ring. Later Bernstein, Zelevinsky and Lusztig
gave a different presentation of this ring. It is the (extended) affine Hecke
algebra H̃q , and it has also turned out to be a key object in combinatorics
independent of its origin in the representation theory of p-adic groups. Restricting ourselves to the semisimple case for simplicity, this algebra may be
defined as follows. It contains a |W |-dimensional subalgebra Hq with generators T1 , · · · , Tr subject to the quadratic relations
Ti2 = (q − 1)Ti + q
together with the braid relations: when i 6= j,

Multiple Dirichlet Series

31

Ti Tj Ti · · · = Tj Ti Tj · · ·where the number of terms on either side is the order of σi σj , and as before
σi is the i-th simple reflection.
The algebra H̃q is the amalgam of Hq with an abelian subalgebra ζ Λisomorphic to the weight lattice Λ. If λ ∈ Λ let ζ λ be the corresponding
e we have the relation
element of ζ Λ . To complete the presentation of H


v−1
λ
σi λ
λ
σi λ
(ζ λ − ζ σi λ ),(38)
Ti ζ − ζ Ti = ζ Ti − Ti ζ=
1 − ζ −αisometimes known as the Bernstein relation.
Though historically it first appeared in the representation theory of p-adic
groups, the affine Hecke algebra appears in other contexts. For example, theinvestigation of Kazhdan and Lusztig [42], motivated by Springer’s work on
the representation theory of Weyl groups, used H̃q in a fundamental way, andled to applications in different areas of mathematics, such as the topology offlag varieties and the structure of Verma modules. Significantly for the present
discussion, Lusztig [48] showed that H̃v (with v an indeterminate) may berealized as a ring acting on the equivariant K-theory of the flag manifold ofĜ, and Kazhdan and Lusztig [48], [43] then applied this back to the local
Langlands correspondence by constructing the irreducible representations ofG(F ) having an Iwahori fixed vector.
The equivariant K-theory of Ĝ may be described as follows. Let O(T̂ ) be
the ring of rational functions on T̂ (C). In our previous notation, it is simply
the group algebra of the weight lattice Λ. If X is the flag variety of Ĝ then
KĜ (X) ∼= O(T̂ ). Better still, let M = Ĝ × GL1 , where the GL1 acts triviallyon X. Then KM (X) ∼= C[v, v −1 ] ⊗ O(T̂ ) where v is a parameter.
The starting point of the investigations of Kazhdan and Lusztig is a representation of H̃v on this ring. In this representation on C[v, v −1 ] ⊗ O(T̂ ) thegenerators of Hv act by certain operators called Demazure-Lusztig operators,
while the commutative subalgebra ζ Λ acts by multiplication. We will call thisrepresentation of H̃v on C[v, v −1 ] ⊗ O(T̂ ) the Lusztig representation.
The same representation of H̃ appears in another way, independent of
Lusztig’s cohomological interpretation. There are two versions of this.
••Ion [39] observed such a representation in the space of Iwahori fixed vectorsof the spherical model of an unramified principal series representation. He
concluded that these matrix coefficients are expressed in terms of the nonsymmetric Macdonald polynomials. His methods are based on the double
affine Hecke algebra.
Brubaker, Bump and Licata [16] found a representation equivalent to theLusztig representation acting on Whittaker functions. Their method could
also be used in the setting of [39].

After Brubaker, Bump and Licata mentioned the connection between
Whittaker functions and Demazure characters, Chinta and Gunnells began

32

Daniel Bump

looking at the metaplectic case. They found “metaplectic Demazure operators” involving Gauss sums that are related to the Chinta-Gunnells representation. Also, with A. Schilling, Brubaker, Bump and Licata looked at thepossibility that the results of [16] could be reinterpreted in terms of the crystal graph, similarly to the crystal interpretation of Tokuyama’s formula. This
seems to be a promising line of investigation.
Let us briefly recall the results of [16]. Let V = Vz be as in (27). Let Ω
be one of the following two linear functionals on V : it is either the Whittaker
functionalZΩ(f ) =f (wn) ψ(n) dn,
N (F )

R
where ψ is as in (28), or the spherical functional Ω(f ) = K f (k) dk. If w ∈ W
let Φw be the element of V J defined as follows. Every element of G(F ) may
be written as bw′ k with b ∈ B(F ), w′ ∈ W and k ∈ J. Then, with τz as in
(26),
 1/2
δ τz (b) if w = w′ ,
Φw (bw′ k) =0otherwise.
The |W | functions Φw are a basis of the space of J-fixed vectors in V . (Theaction π : G(F ) → End(V ) is by right translation.) We also define
Ww (g) = Ω(π(g)Φw ).
X
X
Φ̃w =
Φu ,
W̃w =
Wu
u>w

u>w

where u > w is with respect to the Bruhat order.
Let λ be a weight; if Ω is the Whittaker functional, we require λ to be
dominant. We may regard Ww (aλ ) as an element of C[q, q −1 ] ⊗ O(T̂ ). Then
there exist operators Ti on C[q, q −1 ] ⊗ O(T̂ ) such that
Ti2 = (q −1 − 1)Ti + q −1
and which also satisfy the braid relations. Therefore we obtain a representationof Hq−1 on C[q, q −1 ] ⊗ O(T̂ ). It may be extended to an action of H̃q−1 . Now
if the simple reflection σi is a left descent of w ∈ W , that is, l(σi w) < l(w),then
Wσi w (aλ ) = Ti Ww (aλ ).
(See [16]). The operators Ti are slightly different in the two cases (Ω the
Whittaker or Spherical functional.) In both cases they are essentially the
Demazure-Lusztig operators. For definiteness, we will describe them when Ωis the Whittaker functional. If f is a function on T̂ (C), define∂i′ f (z) =f (σi z) − z −αi f (z)f (z) − z αi f (σi z)
=
.
α
1−z i
1 − z −αiMultiple Dirichlet Series

33

This is the usual Demazure operator conjugated by the map z 7→ −z. Thenthe operators Ti are given by
D′i = (1 − q −1 z α )∂α′ ,T′i = D′i − 1.

The Whittaker functions Ww can thus be obtained from Ww0 by applying the
Ti . Moreover Ww0 has a particularly simple form:
 1/2
δ (aλ )z w0 λ if λ is dominant,Ww0 (aλ ) =0otherwise.
In conclusion, the Lusztig representation arises naturally in the theory ofWhittaker functions or, in Ion’s setup, K, J-biinvariant matrix coefficients. It
gives a calculus whereby the Whittaker functions may be computed recursively
from the simplest one Ww0 .
It is also important to consider W̃w . For example, W̃1 is the spherical
Whittaker function which we have discussed at length in the previous sections.
In the theory of multiple Dirichlet series it might be useful to substitute W̃w for
the p-part at a finite number of places. In the study of the W̃w the remarkablecombinatorics of the Bruhat order begins to play an important role. See [16]for further information. An important issue is to extend the theory of theprevious sections to the theory of the W̃w , and to carry out this unified theory
in the metaplectic context.

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Daniel Bump

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