1 Introduction

Langlands L-functions, which play a fundamental role in modern number theory, are func-tions of one complex variable, with Euler products and functional equations associated withautomorphic forms on reductive algebraic groups. In prior work, the principal investigatorshave studyied Weyl group multiple Dirichlet series. Like L-functions, these are Dirichletseries with multiplicative coefficients and functional equations. But they are functions ofseveral complex variables whose group of functional equations is a Weyl group. Moreover,the multiplicativity of the coefficients is “twisted,” so they are not Euler products. That is,the prime power supported coefficients (henceforth “p-parts”) combine by means of rootsof unity coming from the metaplectic cocycle, whose existence is related to the reciprocitylaws of number theory.

Weyl group multiple Dirichlet series arise naturally as the Fourier-Whittaker coefficientsof Eisenstein series on metaplectic covers of reductive groups. Their p-parts, which are keyobjects of study, are metaplectic Whittaker functions over a local field. They may beviewed as generalizations of characters of finite-dimensional irreducible representations ofLie groups in which the weights of the representation are modified by nth order Gausssums, where n is the degree of the metaplectic cover. Indeed in the non-metaplectic case(n = 1) the Casselman-Shalika formula identifies these p-parts with such characters.

The remarkable Casselman-Shalika formula is of key importance in automorphic forms,being central in both the Rankin-Selberg method and the Langlands-Shahidi method. Re-cently the Casselman-Shalika formula has appeared in a crucial way in the geometric Lang-lands program and in connection with mirror symmetry. One expects that the Whittakerfunctions of metaplectic automorphic forms will play a similar vital role, but until recentlythese functions and the Fourier-Whittaker coefficients have been very mysterious.

Our recent work has uncovered surprising relations between the p-parts of Weyl groupmultiple Dirichlet series and other mathematical objects. For instance, the p-parts may berealized as functions on Kashiwara crystals, which are important combinatorial structureson bases of representations of quantum groups. The p-parts can also be constructed by anaveraging procedure in the spirit of the Weyl character formula. They can be related toMirkovic-Vilonen cycles in the affine Grassmannian and to periods of automorphic forms.Finally, they can even be described in terms of certain two-dimensional lattice modelsoriginating in statistical mechanics. These recent developments and others are evidencethat there should exist far-reaching connections between the p-parts of multiple Dirichletseries, or more generally of Whittaker coefficients of automorphic forms, and quantumgroups. Such connections are not predicted by the classical theory of automorphic formsand L-functions, and establishing them has the potential to significantly transform our un-derstanding of automorphic forms on metaplectic groups. This is the focus of the proposedwork.

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2 Results of Prior NSF Support

The prior FRG grant “FRG Collaborative Research: Combinatorial representation theory,multiple Dirichlet series and moments of L-functions” (July 1, 2007 to June 30, 2010)supported 9 principal investigators and 1 postdoctoral researcher. We report on the priorresults from the 4 principal investigators, Brubaker, Bump, Chinta and Friedberg, who aresubmitting this new proposal. This research includes many common papers with Gunnells,who is a fifth principal investigator on this proposal. We give a somewhat lengthy treat-ment of parts of this prior work since it will help put the new work and the potential forestablishing a connection to quantum groups in context.

Our first body of results concerns Whittaker functions, Weyl group multiple Dirichletseries, and their p-parts. In their 2007 paper [19], Brubaker, Bump, Friedberg and Hoffsteinstudied the Whittaker coefficients of minimal parabolic Eisenstein series on an n-fold coverof SL3. More precisely, let F be a totally complex number field containing the group µnof nth roots of unity, with ring of integers o. Define the metaplectic Eisenstein seriesE(g, (s1, s2)) := E(g) by

E(g) =∑

γ∈Γ∞(f)\Γ(f)

κ(γ)f(γg), where f

y1 ∗ ∗y2 ∗

y3

g

= |y1|2s2 |y3|−2s1f(g)

is a smooth function on SL3(F∞). Further, f is a suitably chosen integral ideal so thatthe Kubota map κ : Γ(f)→ µn is a homomorphism on the principal congruence subgroupΓ(f) in GL3(o). (See Section 1 of [19] for details.) Then given elements m1,m2 ∈ o, theyconsider the (m1,m2)th Whittaker coefficient given by

∫(f\C)3

E

w0

1 x1 x3

1 x2

1

g

ψ(−m1x1 −m2x2)dx1dx2dx3, (1)

where w0 is the long element of the Weyl group and ψ is a non-trivial additive character ofconductor o. They showed that these Whittaker coefficients were multiple Dirichlet seriesin two complex variables whose p-parts could be described combinatorially: each p-part isa weighted sum over strict Gelfand-Tsetlin (GT) patterns, where the weights are productsof nth order Gauss sums. Recall that GT patterns are triangular arrays of integers thatweakly decrease in the rows and such that consecutive rows interleave; strict patterns arethose that have strictly decreasing rows. The set of all (rather than strict) GT patterns withfixed top row is in bijection with a set of basis vectors for a finite-dimensional irreduciblerepresentation of SL3(C) whose highest weight is encoded in the top row. In our contextthis top row is determined by the p-adic valuations of m1 and m2 in (1).

Based on this evidence, they made two conjectures in [19]. First, they defined a fam-ily of multiple Dirichlet series in r complex variables associated to SLr+1 whose p-parts

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have a similar combinatorial description; these series were conjectured to have analyticcontinuation and a group of functional equations isomorphic to the Weyl group of SLr+1.The definition of this p-part they gave is equivalent to the crystal graph definition belowin (5). Second, these multiple Dirichlet series were conjectured to match the Whittakercoefficients of minimal parabolic Eisenstein series on the n-fold cover of SLr+1.

During the period of prior support, principal investigators Brubaker, Bump and Fried-berg have proven both these conjectures. In the book [16] (which makes use of [15]), wegive a direct proof of the conjectured functional equations. In [17], we prove the connectionto minimal parabolic Eisenstein series. Both these works are the first steps in new projectsbelow. We now briefly outline the methods of proof.

The proof of functional equations in [15, 16] argues by induction on the rank r, wherethe case r = 1 is essentially work of Kubota [48], tailored to our situation in [10]. Theinductive step is achieved by comparing two definitions of the p-part of a multiple Dirichletseries in terms of Gelfand-Tsetlin patterns and proving their equivalence. The proof of theequivalence of these definitions is quite subtle and requires combinatorial ideas as well asnumber-theoretic information. Later we will see that it is also the starting point of ourattempt to connect this work with quantum groups through the Yang-Baxter equation.

The method of proof has two important consequences. First, the two descriptionsof the p-part on GT patterns may be reinterpreted as functions on a crystal graph forthe associated highest weight representation of the universal enveloping algebra Uq(slr+1),each associated to a different reduced decomposition of the long element of the Weylgroup. Many aspects of the weighting function on patterns and methods of proof havemore natural definitions in terms of the crystal. A large portion of [16] is devoted tothese connections. Second, the equality of the two descriptions can be reformulated inthe language of statistical mechanics as the commutativity of two transfer matrices for a2-dimensional lattice model. At least when n = 1, the method of Baxter [3] can be usedto give an alternate proof in terms of the Yang-Baxter equation. We discuss these twoapproaches and programs of research built around them in the project description.

The second conjecture, identifying the Weyl group multiple Dirichlet series with meta-plectic Whittaker functions, is proved in [17]. The proof is based on induction in stages—that is, realizing the metaplectic SLr+1 Eisenstein series as a maximal parabolic Eisensteinseries whose inducing data is a metaplectic SLr Eisenstein series. The coset representa-tives may then be chosen in a particular way that leads to a formula compatible with acorresponding inductive property of the Gelfand-Tsetlin description.

Peter McNamara, a student of Brubaker, has provided further insight into the rela-tionship between Whittaker functions and crystals in [54]. Working over a local field, hedemonstrates that the Whittaker integral attached to a spherical vector in an unramifiedprincipal series representation can be computed via an explicit Iwasawa decomposition thatdecomposes a unipotent radical of the Borel into “cells.” This decomposition applies toany simply connected Chevalley group. McNamara provides an identification between thesecells and Lusztig’s canonical bases [51], as well as to a geometric realization of crystals—

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Mirkovic-Vilonen cycles in the affine Grassmannian (cf. [2]). The decomposition of theunipotent again depends on a reduced expression for the long element of the Weyl group,and McNamara has worked this out explicitly for w0 = s1s2s1s3s2s1 · · · of Ar, where hedemonstrates that the p-adic Whittaker function exactly matches the p-part of the multipleDirichlet series, whose crystal definition will be presented in (5) in the Project Descriptionbelow. For arbitrary reduced expressions of the long word, it seems quite difficult to givean explicit formula for the cell decomposition.

We now turn from type A to a general Cartan-Killing type. Let Φ be a reduced rootsystem of rank r and (m1, . . . ,mr) a fixed tuple of nonzero elements of the ring of integerso. Given this data, one can heuristically define a Weyl group multiple Dirichlet seriesgeneralizing the previously discussed series for type A. This series is a Dirichlet series inthe r variables s1, . . . , sr with a group of functional equations isomorphic to the Weyl groupW of Φ. The Eisenstein conjecture of Brubaker, Bump, and Friedberg [13] predicts thatsuch a series should be the Fourier-Whittaker coefficient of a minimal parabolic Eisensteinseries on a metaplectic cover of a simply connected reductive group G whose L-group hasroot system Φ.

A general technique for building Weyl group multiple Dirichlet series has been devel-oped by Chinta and Gunnells, and involves a deformation of the Weyl character formula.Rather than constructing the p-part directly, Chinta-Gunnells translate the desired globalfunctional equations into a W -action on the field of rational functions C(x1, . . . , xr). Theythen construct an invariant rational function fλ by averaging over W . Here λ is a highestweight vector for Φ, determined by the p-adic valuations of the mi. The rational func-tion fλ can be written as the ratio of two polynomials Nλ/D, where the denominatoris independent of λ and is a deformation of the usual Weyl denominator. If one writesNλ =

∑Hλ(pk1 , . . . , pkr)xk11 · · ·xkrr , then the p-part is determined by the coefficients Hλ.

This method has been pursued by Chinta and Gunnells in [25,28–30] and in [27] togetherwith Friedberg. In [28] the authors treat simply-laced root systems, quadratic symbols(n = 2), and λ trivial. This is extended to nontrivial λ in [27], again for simply-laced rootsystems and n = 2, where combinatorics of the p-part are investigated and agreement isproved with [13] for the coefficients attached to the vertices of the Newton polytope ofNλ. In [30] the authors study the case of n ≥ 2 for A2 and trivial λ; the arguments theregeneralize to the simply-laced case and trivial λ. Finally in [29] Chinta-Gunnells treat thecase of general Weyl groups, general λ, and general n.

To summarize, there are at present two methods to constructing the p-parts of Weylgroup multiple Dirichlet series. Both are related to characters of representations of thesemisimple complex Lie algebra attached to Φ. Let λ be a strictly dominant weight for Φ.

• The Gelfand-Tsetlin or crystal approach [14, 16, 19], which works for Φ = Ar, givesformulas for the coefficients Hλ(pk1 , . . . , pkr). These formulas are written in terms ofGauss sums and statistics extracted from Gelfand-Tsetlin patterns for the represen-tation of slr+1(C) of lowest weight −λ.

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• The averaging approach [27–30], which works for all Φ, uses a “metaplectic” defor-mation of the Weyl character formula to construct a rational function with knowndenominator, whose numerator is then taken to define Nλ.

Both approaches have their advantages and limitations. The Gelfand-Tsetlin construc-tion gives explicit formulas for coefficients that are uniform in n and that lead to a directconnection with the global Fourier coefficients of Borel Eisenstein series on the n-fold coverof SLr+1 [17]. However it suffers from the obvious disadvantage that the exact correctdefinition has only been found for certain cases, for example types A or B (the latter foreven n). The averaging approach, on the other hand, works for all Φ, quickly leads tothe definition of the multiple Dirichlet series, yet has the drawback that it seems difficultto get similarly explicit formulas for the coefficients of Nλ. By combining recent work ofChinta-Offen [34] and McNamara [54], we know that in type A the two definitions of Hλ

coincide, although we would still like a direct combinatorial proof.

We have detailed some of the work over the period of prior support most directlyrelevant to to projects proposed in the next section. The following 21 citations representa complete list of papers written or co-authored by the principal investigators during theperiod of prior support (2007–present), many of which are closely connected to the workproposed:

[1, 4, 5, 14–20,23,26–34,40]

Training and dissemination efforts. The grant supported one postdoctoral researcher,Bucur, who has accepted a tenure-track position at UCSD. Two PIs are advising doctoralstudents (Brubaker is advising Lennon, McNamara and Tabony, Bump is advising Ivanov),many of whom are working on projects closely related to the proposal. Friedberg will be-gin advising students now that Boston College has approved its new doctoral program,and Chinta and Gunnells have both just had students finish (Mohler and Boland, respec-tively). Bump also worked extensively with postdoctoral researcher Nakasuji. The PIshelped organize and run a 5-day workshop on “Multiple Dirichlet series and applicationsto automorphic forms” at the International Centre for Mathematical Sciences, Edinburgh,United Kingdom in August 2008. This was attended by over 60 researchers from 12 coun-tries and many graduate students and included a detailed series of introductory lecturesabout the area. A proceedings volume is in preparation, to be published by Birkhauser.In June 2009 a second week-long workshop was held with a more intensive research focusbut also a training component, featuring detailed discussion of open problems and time forwork groups to meet and interact. These efforts facilitated the entry of additional mathe-maticians into the research area, including O. Offen (Technion), K.-H. Lee (U Conn) andM. Nakasuji (Stanford).

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3 Project description

Whittaker functions and crystals. The papers [16,17] give a translation of the defini-tions of the p-part of Whittaker coefficients for G = SLr+1 in terms of crystal graphs. Wefirst review this definition and then describe the open problems associated to p-parts onother Chevalley groups.

Crystal graphs were invented by Kashiwara in connection with the representations ofquantized enveloping algebras. Rather than using the highest weight representations of thecomplex Lie group LG or its Lie algebra Lg, we may use the universal enveloping algebraU(Lg) or its quantum group Uq(Lg). The representations of LG, U(Lg), and Uq(Lg) for qgeneric are all the same. The crystal is a combinatorial structure on the representationspace that appears when q→0. In particular, Littelmann [50] and Berenstein-Zelevinsky [8]give a combinatorial realization of the crystal graph in terms of paths in the crystal whichare traversed according to a factorization of the long element of the Weyl group.

Let w0 = si1 · · · sit be a reduced decomposition of the long element into simple reflec-tions. Given an element b of the crystal basis, we apply the Kashiwara lowering operatorfαi1 repeatedly to b. Let a1 be the maximal integer such that b′ := fa1

αi1(b) 6= 0. Repeating

this process with b′ we obtain a sequence of integers (a1, . . . , at) representing the basiselement b and such that fatαit · · · f

a2αi2fa1αi1

(b) is the lowest weight vector in the crystal graph.The collection of all such sequences, as one ranges over all possible highest weights, formthe integral points of a cone in Rt, whose bounding hyperplanes depend on the choice ofreduced decomposition. The set of sequences for a fixed choice of highest weight vectorforms a polytope cut from this cone by additional bounding hyperplanes (see for exampleProposition 1.5 of [50]).

For example, if we consider the root system A2 and write w0 = s1s2s1, then the coneconsists of all sequences (a1, a2, a3) with ai ≥ 0 and a2 ≥ a3. We place these 3 integers inan array reflecting these inequalities and so that the path lengths for each root operatorare aligned in columns, and renumber them in matrix notation:[

a2 a3

a1

]=:[c1,1 c1,2

c2,2

]. (2)

We will refer to such arrays as “BZL patterns.” These sequences and a sample path fromone element of the crystal basis to the lowest weight vector are illustrated in Figure 1,which depicts a crystal graph of highest weight λ+ρ = 2ε1 + 2ε2. An entry ci,j of the BZLpattern will be called “minimal” if ci,j = ci,j+1 (i.e. the cone inequality is an equality), and“maximal” if the hyperplane inequality for ci,j is an equality.

We now describe how to attach a monomial to each BZL pattern to form the p-partof a multiple Dirichlet series for Whittaker functions on SLr+1 for any rank r. For thep-part of the (m1,m2, . . . ,mr)th Whittaker coefficient (the analogue of the (m1,m2)thcoefficient in the rank two case in (1)), we use the crystal graph Cλ+ρ with weight λ =λ(p) = ordp(m1)ε1 + · · ·+ ordp(mr)εr, where ε1, . . . , εr are the fundamental weights. Then

6

000

001

002

110

111

220

230

231

240

241

242

120

121

122

130

131

132

133

010

011

012

013

020

021

022

023

024

f1

f1f2

f2

f2f1

v

vlow

Figure 1: The BZL pattern (a1, a2, a3) = (2, 3, 1) depicted as a path in the crystal graphCλ+ρ with λ + ρ = 2ε1 + 2ε2. The entry a2 is maximal, while a1, a3 are neither maximalnor minimal.

to each BZL pattern P in Cλ+ρ with matrix notation as in (2), we associate the function

G(P ) =∏

1≤i≤j≤rγ(ci,j), where γ(c) =

g(pc−1, pc) if c is maximal,pc if c is minimal,g(pc, pc) if c is neither maximal nor minimal,0 if c is both maximal and minimal,

(3)and where g(m, d) is the usual Gauss sum formed with nth power residue symbol, namely

g(m, d) =∑

a (mod d)

(ad

)ne(mad

). (4)

Then the p-part of the Whittaker coefficient is given by∑v∈Bλ+ρ

G(v)|p|−2 wt1(v)s1−···−2 wtr(v)sr with wtj(v) =r∑i=1

ci,r+1−j , (5)

where Bλ+ρ is the crystal with highest weight vector λ+ ρ.The main theorem of [17] is that this definition with reduced decomposition w0 =

s1s2s1 . . . srsr−1 . . . s1 agrees with the p-part of the associated metaplectic Whittaker co-efficient on SLr+1. While there are many combinatorial descriptions of bases for highest

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weight representations, what is striking here is the very natural way in which the BZLpatterns—paths in crystals—may be used to represent a Whittaker coefficient.

The definitions we have made in (3) and (5) for SLr+1 and a particular reduced decom-position of w0 in terms of paths in a crystal make sense for any Chevalley group and anylong word. Unfortunately, the resulting generating function does not match the p-part ofan associated Whittaker coefficient in general. For example, for SL4, there are 16 reducedexpressions of w0, including for example s1s3s2s1s3s2. Given a highest weight λ + ρ, wemay apply the analogous crystal definition for this expression, and we find that the re-sulting generating function on the crystal matches that for the decomposition s1s2s1s3s2s1

given in (5) roughly 90% of the time, with errors occurring in the interior of the polytope.One can attempt to tweak the basic recipe in (3) for any given group and long word tomatch either a known p-part of a Whittaker function or to match a conjectural descriptionof the Whittaker coefficient as given in [29]. Using these methods, we have recently givendefinitions of generating functions which conjecturally match the p-parts of a Whittakercoefficients for other classical groups [4, 31].

To prove that these definitions actually do match a p-adic Whittaker coefficient forarbitrary choice of highest weight vector, one can employ the methods of [17], as extendedin [20, 54]. That is, we may compute the Whittaker coefficient (the analogue of (1)) bysuccessively inducing from maximal parabolic subgroups. This produces a recursion whichincludes intricate character sums. The generating functions also behave nicely with respectto parabolic induction in stages, leading to a very similar recursion. It then remains toshow the exponential sum from the integration matches the inductively defined piece ofthe generating function. A closer investigation of these exponential sums, in conjunctionwith conjectured combinatorial expressions in terms of BZL patterns, should reveal theunderlying geometric properties of the polytope which can be used to give a unified defi-nition for the p-part of a Whittaker coefficient for arbitrary choice of group and long worddecomposition.

Whittaker functions and quantum groups. In order to understand the motivationbehind this project, we begin with the non-metaplectic case (n = 1). In this case, theCasselman-Shalika formula describes the values of a spherical p-adic Whittaker functionin a surprising and beautiful way. The values are simply the characters of the finite-dimensional irreducible representations of the dual group evaluated at the Langlands pa-rameter conjugacy class. This formula has many applications in automorphic forms andnumber theory, being basic to both the Rankin-Selberg method and the Langlands-Shahidimethod of constructing L-functions. It has also recently attracted the attention of physi-cists, as in the work of Gerasimov, Lebedev and Oblezin in connection with Givental’swork on mirror symmetry.

Let G be a reductive algebraic group, let G be the n-fold metaplectic cover, and let λbe a dominant weight of the root system Φ of the Langlands dual group LG. The weightλ indexes an element tλ of G. For example if G = GLr+1 then λ = (λ1, · · · , λr+1) ∈ Zr+1

8

(identified with the weight lattice) with λ1 > λ2 > · · · , and we may take tλ to be a lift toG of a diagonal matrix with eigenvalues pλi .

The spherical representations of G or G are parametrized by conjugacy classes in LG.Let z ∈ LG parametrize such a representation, and let W : G −→ C be the p-adicWhittaker function. If n = 1, the Casselman-Shalika formula says that the sphericalWhittaker function W (tλ) is, up to a normalization factor, the character χλ(z) of an theirreducible module of LG with highest weight vector λ. That is,

W (tλ) = (normalization)× χλ(z), χλ(z) =∑

w∈W sgn(w)zw(λ+ρ)+ρ∏α∈Φ+(1− zα)

, (6)

where W is the Weyl group and ρ is half the sum of the positive roots.In (5), we expressed the p-adic Whittaker function of SLr+1 as a generating function on

a crystal. Combining with (6), where we replace the Weyl denominator by its deformation∏α∈Φ+

(1− q−1znα) (7)

with q = |p|, the cardinality of the residue field, we have a striking identity for the characterχλ. It is actually not new, but was discovered in a purely combinatorial context (in termsof GT patterns) by Tokuyama [64]. The crystal definition suggests a deeper connectionwith the theory of quantum groups, and we now explain how the work of Brubaker, Bumpand Friedberg in [18] uses Tokuyama’s identity as a starting point for these connections.We then propose further work to complete this program.

Quantum groups were introduced by Drinfeld and Jimbo in order to explain exactlysolved models in statistical mechanics and integrable systems in quantum mechanics. Wehave recently discovered that one well-studied model, the six-vertex model , which corre-sponds to two-dimensional ice, can be used as a basis for investigating Whittaker functions,at least on SLr+1 and Sp2r. Moreover, such a description extends to metaplectic covers.

We consider a rectangular grid with vertices placed at each crossing. Every edge of thegrid is to be assigned one of two “spins,” + or −. In our six-vertex model, the adjacentedges to any vertex can come in only 6 possible configurations. Depending on the adjacentspins, each vertex is assigned a Boltzmann weight . The Boltzmann weight of the entireconfiguration is the product of the weights at each vertex. The partition function is thenthe sum of the Boltzmann weights over all states.

To “solve” a statistical model is to determine its partition function. The six-vertexmodel was originally solved in 1967 by Lieb and Sutherland on a lattice with periodicboundary conditions, where the Boltzmann weights satisfy a certain symmetry. Thereis an immense literature concerning solvable lattice models. We highlight the visionarycontribution of Baxter [3], who gave a treatment in terms of what is now known as theYang-Baxter equation (‘YBE’). This equation lies at the heart of the following discussion.

We now explain how to obtain the value of the Whittaker function at tλ as a partitionfunction on a lattice or “ice” with certain boundary conditions. The six possible spin

9

configurations around a vertex, together with their associated Boltzmann weights in boththe non-metaplectic (i.e. degree of the cover n = 1) and metaplectic cases, are givenin Table 1 below. We call this recipe Gamma ice (as there will be other models andweights to come). The index i records the row number of the lattice, and will be explainedmomentarily. The parameters zi and ti are arbitrary complex numbers. The metaplecticweights (n ≥ 1), which specialize to the Boltzmann weights when n = 1, will be explainedlater.

Gamma ice

i i i i i i

Boltzmannweight (n = 1)

1 zi(ti + 1) 1 ti zi zi

metaplecticweight (n ≥ 1)

1 ziq−ah(a) 1 ziq

−ag(a) zi zi

Table 1: Boltzmann weights for Gamma ice

These weights will be applied to ice with the following boundary conditions, determinedby the index of the Whittaker coefficient λ + ρ = (λ1 + ρ1, . . . , λr + ρr, 0) where ρ =(r, r − 1, · · · , 0). (See [18] for further information.) The ice will have r + 1 rows andλ1 + ρ1 + 1 columns. We require that the edge spins along the left and bottom edges be+; along the right edges, they are to be −. Along the top row, we label the columns fromright to left, and place a − on the top edge in the columns labeled λj + ρj . A typicalexample of such boundary conditions is given in the following figure.

5 4 3 2 1 0

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

Thus in the example, λ = (3, 1, 0), so λ+ρ = (5, 2, 0) and we put − in the columns labeled5, 2, and 0. One such state has been depicted above.

In evaluating the ice, we take the Boltzmann weights from Table 1 using the valueszi, ti in the ith row. For each highest weight λ+ ρ, the resulting partition function Z, thesum of Boltzmann weights (themselves a product of weights at each vertex) over the set ofall admissible Gamma ice configurations SΓ

λ, has been computed.

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Theorem 1 (Tokuyama, Hamel-King)

Z(SΓλ) =

∏i<j

(tizj + zi)sλ(z1, · · · , zr+1)

where sλ is the Schur polynomial of highest weight λ.

Tokuyama [64] stated this theorem in the language of strict Gelfand-Tsetlin patterns. Ourset of ice with boundary SΓ

λ is in bijection with strict GT patterns having highest weightλ + ρ. Hamel and King [42] proved a symplectic version of Tokuyama’s theorem, andprovided a translation of these formulas into the language of ice.

While they were interested in such deformations for their combinatorial structure, wenow explain how this represents a p-adic Whittaker coefficient on G = GLr+1. If we taketi = −q−1 for all i, and let z = (z1, · · · , zr+1) be the eigenvalues of the conjugacy class inLG, then the partition function represents δ−1/2(tλ)W (tλ), where we take the Whittakerfunction in the normalization coming from its standard representation as an integral onthe p-adic group. The factor

∏i<j(zi − q−1zj) is just zρ times (7) and δ−1/2(tλ) is a

normalization.If n ≥ 1 we may represent the p-adic Whittaker function as a similar sum over states.

For this, we use the weights listed in the bottom row of Table 1 with g(a) = q−ag(pa−1, pa)and h(a) = q−ag(pa, pa) where g(m, d) is an nth order p-adic Gauss sum, the analogue of(4) with residue symbol replaced by the local Hilbert symbol. (We assume that n does notdivide the residue characteristic.) In the Boltzmann weights, the argument a in h(a) org(a) is the number of + edges in the row to the right of the vertex.

In Brubaker, Bump and Friedberg [18], a new proof of Theorem 1 is given using theYang-Baxter equation in the spirit of Baxter [3]. Baxter [3] gave a method for solvinglattice models based on what he called the star-triangle relation. In the context of ourwork, this may be explained as follows. Define another type of ice, called Gamma-Gammaice (because it can be used to braid two strands of Gamma ice), whose weights are givenin Table 2.

Gamma-Gamma ice

j i

i j

j i

i j

j i

i j

j i

i j

j i

i j

j i

i j

Boltzmannweight (n = 1)

tjzi + zj (ti + 1)zi zi − zj tizj − tjzi (tj + 1)zj tizj + zi

Table 2: Boltzmann weights for Gamma-Gamma ice

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j

i

τ

σ

ν

µ

β

γ

α

θ

ρ

i

j=

i

j

τ

σ

β

δ

α

ψ

φ

θ

ρ

j

i

Figure 2: The star-triangle identity.

Now consider pair of mini-ensembles, each with just three vertices and six boundaryspins α, β, σ, τ, ρ, θ that are to be prescribed. The star-triangle relation states that thesetwo mini-ensembles have the same partition function. That is, if we sum over the possiblespins that can be assigned to the edges γ, ν, µ the left-hand side gives the same evaluationas summing the states of the right-hand side, summing over δ, φ, ψ. This may be confirmedfor the weights we have given, and is the basis of the proof of (1) in [18].

This can all be stated algebraically in terms of endomorphisms of V ⊗ V , where Vis a two-dimensional complex vector space with basis {v+, v−}. Regarding each entry inTable 2 as defining a map on basis vectors, our endomorphism can be expressed as a matrixconsisting of Boltzmann weights of Gamma-Gamma ice:

RΓΓ(i, j) =

tjzi + zj 0 0 0

0 tizj − tjzi (tj + 1)zj 00 (ti + 1)zi zi − zj 00 0 0 tizj + zi

.

We can arrange a similar matrix MΓ(i) for Gamma ice using the values from Table 1.If R is an endomorphism of V ⊗V let R12, R13 and R23 be endomorphisms of V ⊗V ⊗V inwhich Rij acts on the ith and jth components, and as the identity on the remaining one.Then the star-triangle relation may be written R12M13N23 = N23M13R12. In our case, itis satisfied for R = RΓΓ(i, j), M = MΓ(i) and N = MΓ(j).

Most significantly RΓΓ itself satisfies the parametrized Yang-Baxter equation, a specialcase of the star-triangle identity:∑

µ,ν,γ

Rραµγ(j, k)Rθγνβ(i, k)Rνµστ (i, j) =∑δ,φ,ψ

Rψδτβ(j, k)Rφανδ (i, k)Rθρφψ(i, j).

It can be thought of as a braid relation in which three strands of gamma ice are woven, togive vertices in Gamma-Gamma ice at each crossing.

In future work, we intend to investigate the underlying Hopf algebra structure for thisR-matrix. Indeed, Drinfeld [36], Faddeev, Reshetikhin and Takhtajan [37] and Majid [53]

12

have given results which allow one to construct a quasitriangular Hopf algebra (quantumgroup) using generators and relations determined by the R-matrix. From this parametrizedYBE, the resulting Hopf algebra HΓΓ will have one two-dimensional module V (z, t) forevery pair of nonzero complex numbers z and t, and RΓΓ(i, j) is the endomorphism ofV (zi, ti)⊗V (zj , tj) whose composition with the interchange x⊗y → y⊗x is an intertwiningmap V (zi, ti)⊗ V (zj , tj)→ V (zj , tj)⊗ V (zi, ti).

A host of open problems appears. A first question is whether this Hopf algebra canbe related to known quantum groups. A second urgent question is that while the vertex-model definition of the p-parts extends to the metaplectic case n > 1, we do not yet havethe Yang-Baxter equation in that context. It is also interesting to ask whether, since weare able to take the second set ti of spectral parameters to be unequal in the n = 1 case,whether this is also true when n > 1. Another question is whether this connection withquantum groups explains the relevance of crystal bases. Finally, when n = 1, the action ofthe Weyl group on the spectral parameters is reflected in the Yang-Baxter equation, andit seems likely that this is connected with the work of Chinta-Gunnells and Chinta-Offen,where such a Weyl group action also occurs.

We computed a partition function for a lattice composed of Gamma ice, arriving at ap-adic Whittaker function. But we could consider more general boundary conditions, whichmight result in interesting partition functions. Moreover, we could investigate the partitionfunction for a lattice composed of Gamma-Gamma ice as well. The existence of a YBE forGamma-Gamma ice already implies a certain symmetry of the resulting partition function.Using rather different weighting systems, Fomin and Kirillov [38] constructed symmetricfunctions—Schubert and Grothendieck polynomials arising naturally in the cohomologyand K-theory of the flag variety—based on the YBE. It would be interesting to have ageneral theory for producing such geometrically motivated bases of symmetric functionsbased on solutions of the YBE which incorporated both sets of results.

The structure we have described is really only the beginning of the story. As we noted inthe Prior Work section, the proof of functional equations for multiple Dirichlet series in [16]was reduced to the equivalence of two definitions of the p-part of the multiple Dirichletseries. These definitions, in the crystal language, are related by the Schutzenberger involu-tion on the crystal Bλ+ρ. The Schutzenberger involution can be built up in smaller steps,each the reflection in a root string. The number of such steps equals the length of thelong element of the Weyl group. Each of these steps is accomplished by an application ofan involution for “short Gelfand-Tsetlin patterns” and this statement (see “Statement B”in [16]) translates beautifully into the vertex operator setting.

Thus, in addition to the Gamma ice, there is another type of ice, called Delta ice, corre-sponding to the second definition of the p-part. In Baxter’s statistical-mechanics language,Statement B is equivalent to the commutativity of two transfer operators correspondingto Gamma and Delta ice. This equivalence remains valid even in the metaplectic case. Atleast if n = 1 it can be proved using a star-triangle relation. This requires the inventionof suitable R-matrices, Delta-Delta ice, Gamma-Delta ice and Delta-Gamma ice (see [18]

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for their Boltzmann weights). Putting Gamma-Gamma, Gamma-Delta, Delta-Gamma andDelta-Delta ice together one obtains a grand Hopf algebra HΓΓ ⊗ H∆∆. As a Hopf alge-bra this is just the coproduct of the Hopf algebras HΓΓ and H∆∆, but its quasitriangularstructure is quite a bit richer since it also has an R-matrix in End(V ⊗W ), where V is amodule of HΓΓ (a direct sum of the irreducible two-dimensional modules V (z, t)) and Wis a module of H∆∆.

The special case where the ti are equal is most relevant to the p-parts of multipleDirichlet series. In this case, using ideas in Hlavaty [45], one can construct a different Hopfalgebra besides the one described above that contains HΓΓ. There is reason to think thatit is possible that this is the more important of the two, or that the two could possibly becombined. Also in this special case where the second set of spectral parameters ti are allequal to a fixed value t, we know from work of Nichita and Parashar [56] that it is possibleto introduce another parameter σ and obtain a seven vertex model. The implications ofthis are unknown.

Whittaker functions for non-Borel Eisenstein Series. If G is a split connectedreductive algebraic group over F , P = MN is a maximal parabolic subgroup with Levidecomposition, and π is an automorphic representation of M(AF ), the adelic points of M ,then one may form an Eisenstein series of G(AF ) induced from π. Its Whittaker functionmay be expressed in terms of the Langlands L-functions L(s, π, ri), where r = ⊕mi=1ri is theadjoint action of LM on the complex Lie algebra Ln. This is the beginning of Langlands-Shahidi theory (see Shahidi [61,62] and the references therein).

As a vast generalization of the Borel Eisenstein series case, we propose to computethe coefficients of more general parabolic metaplectic Eisenstein series. We have alreadyworked out one case, inducing from GLr to GLr+1, in [17], and are in the process of writingup the case of GL(2) × GL(2) to GL(4) in [20]. This latter case is illuminating as it isthe first case in which one needs a non-trivial Plucker relation to parametrize the cosetsin P\G. The key advance here is a new way of constructing coset representatives for P\Gwhich allows for a simple expression of the Kubota symbol defined from the metaplecticcocycle.

Based on these computations, we once again expect the theory of crystals to intervene.However, twisted multiplicativity must be modified in a way that reflects the choice of P .Even giving Tokuyama-type identities for the non-metaplectic case in this situation shouldbe very interesting, as it could give new identities blending the coefficients of automorphicforms with the combinatorics of crystals. And when one induces lower rank Eisensteinseries, this should give a way to study the crystal description when the long word isfactored into other good decompositions, and obtain a theory that is valid for all suchdecompositions.

A metaplectic Casselman-Shalika formula. Casselman and Shalika [24] explicitlycompute the spherical Whittaker function of a p-adic reductive group, generalizing Shin-tani’s formula for GLr [63]. Chinta and Offen [34] have extended this formula to include

14

the n-fold metaplectic cover of GLr, and we intend to further generalize this result ton-fold cover of an arbitrary reductive groups. For n > 1, a central difficulty is the failureof uniqueness of Whittaker functionals (and therefore of spherical Whittaker functions ofa fixed Hecke eigenvalue). In [44] Y. Hironaka computed explicitly the spherical functionson the space of non-singular Hermitian matrices with respect to an unramified quadraticextension of p-adic fields. This is a case where multiplicity one fails. Hironaka’s approachto the Casselman-Shalika method in case of multiplicities is our guideline for this project.

Roughly speaking, a spherical function can be expressed as the value of a certain linearform applied to translates of the unramified vector in an unramified principal series rep-resentation. The idea behind the Casselman-Shalika method is to reduce the computationfor the value of the linear form on a translate of the element invariant under the maximalcompact subgroup to the computation of simpler expressions for elements invariant undera smaller open compact—the Iwahori subgroup. There are three main steps in carryingout the method.

The first has to do solely with the group and not with the particular linear form weconsider. It is an expansion of the unramified element of a principal series representationin terms of a ‘well chosen’ basis of the Iwahori invariant subspace—the Casselman basis.In [60], Sakellaridis provides a formula for spherical functions in the general setting ofspherical varieties for a split reductive group (this, however, does not contain our case aslong as n > 1). His characterization of the Casselman basis simplifies the computation.

The second step is to obtain Weyl group functional equations between the sphericalfunctions. The unramified principal series representations are parameterized by a variable,say s, in some complex variety on which a related Weyl group acts. Crucial to the compu-tation of the spherical functions is to relate explicitly the spherical functions associated tos and those associated to ws for any Weyl element w. When n = 1, the space of Whittakerfunctionals of an unramified principal series representation is one dimensional. There isthen a one dimensional space of spherical Whittaker functions for a given parameter s(i.e. for a fixed Hecke eigenvalue) and the functional equations are therefore scalar val-ued. For metaplectic groups multiplicity one fails. This complicates the computation ofthe functional equations. Once a basis of Whittaker functionals for any parameter s hasbeen fixed, there is a matrix associated to any Weyl element w that expresses the basisfor ws in terms of the basis for s. For GLr such a functional equation was established byKazhdan and Patterson [47]. Generalizing this to an arbitrary reductive group will requirea careful reworking of the results of Kazhdan and Patterson. Aiding us in this is the factthat we know what the functional equations should be: a local version of the Eisensteinconjecture in [13] implies that the functional equations of the Whittaker functions shouldbe related to the functional equations of the p-part of the Weyl group multiple Dirichletseries constructed by Chinta and Gunnells [29].

The third step is the evaluation of the linear forms on translates of the Iwahori invariantfunctions in the Casselman basis. This, in our case, is not much more complicated than inthe Shintani, Casselman-Shalika case.

15

Once this is done, we expect the resulting formula for the spherical Whittaker functionto coincide with the p-part of the Weyl group multiple Dirichlet series in [29], thereby pro-viding a link between these multiple Dirichlet series and metaplectic Whittaker functions.Until now, such a connection has been rigorously established only for G = GLr.

Orthogonal periods and Multiple Dirichlet Series. Brubaker, Bump and Fried-berg [17] establish a relation between the Weyl group multiple Dirichlet series and Fouriercoefficients of Eisenstein series on metaplectic covers of GLr. Langlands functoriality con-jectures expect, in particular, the existence of transfers of automorphic forms betweenmetaplectic groups and reductive groups. It is therefore likely that Weyl group multi-ple Dirichlet series appear naturally also in the theory of automorphic forms on algebraicreductive groups.

Let G be a reductive group defined over a number field F and let H be a closedsubgroup. Set G = G(F ) and H = H(F ). Whenever convergent, the H-period integral ofan automorphic form φ on G\GAF is defined by

PH(φ) =∫H\HAF

φ(h) dh.

To study period integrals Jacquet developed his relative trace formula. In the context ofa symmetric space, i.e. when H is the group of fixed points of an involution on G thereis another group G′ associated to G/H and a functorial transfer of automorphic formsfrom G′ to G that is expected to capture non vanishing of the period integral. The valuePH(φ) when not zero often carries interesting arithmetic information. In cases where theperiod factorizes it is expected to be related to automorphic L-functions. When there isno factorization the arithmetic meaning of the period is not yet understood.

Consider now the case where G = GLr over F and H is an orthogonal subgroup. Usingthe formalism of the relative trace formula and evidence from the r = 2 case, Jacquetconjectured that in this setting orthogonal periods of a form on G should be related to aa period of a form on G′, the metaplectic double cover of G [46]. For G′ local multiplicityone of Whittaker functionals fails. This leads us to expect that the period integral PH(φ)of a cusp form is not factorizable. To date, the arithmetic interpretation of the period athand is a mystery, and precise conjectures are yet to be made. Often, studying the periodintegral of an Eisenstein series is more approachable then that of a cusp form and mayhelp to predict expectations for the cuspidal case (this was the case for G = GL2 and Han anisotropic torus, where the classical formula of Hecke for the period of an Eisensteinseries in terms of the zeta function of an imaginary quadratic field significantly predatesthe analogous formula of Waldspurger for the (absolute value squared) of the period of acusp form).

Following a suggestion of Bump and Venkatesh, Chinta and Offen [33] compute theperiod integral PH(E(ϕ, λ)) in the special case that r = 3, H is the orthogonal groupassociated to the identity matrix and E(ϕ, λ) is the unramified Eisenstein series induced

16

from the Borel subgroup. The formula we obtain expresses the period integral as a finitesum of products of the quadratic A2 double Dirichlet series, which are essentially (see [12])Whittaker coefficients of the metaplectic Eisenstein series. This fits perfectly into Jacquet’sformalism, and it is our hope that the formula in this very special case can shed a lighton the arithmetic information carried by orthogonal periods of cuspforms. We proposeto develop the study of periods and their relation to multiple Dirichlet series, and toinvestigate the implication of the connection to quantum groups in this context as well.

The Affine case. There are reasons to believe that this theory will extend to other Kac-Moody root systems. Affine root systems are an important case. Such root systems haveinfinite Weyl groups, so the multiple Dirichlet series, originally defined in a product of righthalf places of Cr (where r is the number of simple roots), are expected to continue to aproper subset of Cr rather than to the full space; see Bump, Friedberg and Hoffstein [22].There is a character formula that admits q-deformations, and there is reason to believethat there are also metaplectic deformations.

A first case of multiple Dirichlet series having infinite group of functional equations(the affine Weyl group D

(1)4 in Kac’s classification)at may be found in the work of Bucur

and Diaconu [21]. (Their result requires working over the rational function field; it buildson work of Chinta and Gunnells [29].) If one could establish the analytic properties of suchseries in full generality, one would have a powerful tool for studying moments of L-functions.On the other hand, the p-parts of such multiple Dirichlet series will have connections withrepresentation theory (generalizing those outlined above) quite independent of applicationsto number theory, since these would be metaplectic deformations of the Kac-Weyl characterformula.

Generalizations of the Formula of Gindikin and Karpelevich. Let G be a reductivep-adic group, which we assume for simplicity to be simply laced. The formula of Gindikinand Karpelevich says that if φ◦ is the standard spherical vector in V (χ)K , where K is themaximal compact subgroup, then∫

N∩wN−w−1

φ◦(w−1n) dn =∏

α ∈ Φ+

w(α) ∈ Φ−

1− q−1zα

1− zα,

where z ∈ LG is related to χ by the Satake isomorphism and N,N− are the unipotentradicals of B and its opposite B−. For the long element w = w0, this means∫

N−

φ◦(n) dn =∏

α ∈ Φ+

1− q−1zα

1− zα,

On the other hand, Kashiwara defined a particular crystal called B(∞) which is the quan-tized enveloping algebra of N−. By applying the recipe (5) to this crystal, Bump and

17

Nakasuji [23] proved that ∑v∈B(∞)

G(v) =∏

α ∈ Φ+

1− q−1zα

1− zα.

This is striking, since it shows that an integral over the group N− can be replaced by asum over a basis of its quantized enveloping algebra. These results have extensions to then-fold metaplectic cover.

As mentioned above, the proof of the Casselman-Shalika formula makes use of a par-ticular basis of the space V (χ)J of Iwahori fixed vectors in a spherical representation V (χ)of G(F ), where G is a reductive algebraic group over the nonarchimedean local field F .V (χ)J is the space of functions φ that satisfy

φ(bgk) = (δ1/2χ)(b)φ(g)

where b ∈ B(F ), the Borel subgroup, and k ∈ J , the Iwahori subgroup. Here χ is a fixedcharacter of a split maximal torus in T (F ) ⊂ B(F ) and δ is the modular quasicharacterof B(F ). This Casselman basis, indexed by the Weyl group W , is dual to the standardintertwining operators. It is difficult to compute, but one particular basis vector may becomputed, and as Casselman [24] noted, this is enough for the applications.

However, Bump and Nakasuji have a partial generalization of the Gindikin-Karpelevichformula which provides a partial evaluation of the Casselman basis. It is shown that thedifficulty of evaluating the Casselman basis is related to the geometry of Schubert cells andKazhdan-Lusztig polynomials.

We assume that G is simply-laced. If u ∈W (the Weyl group) define

ψu(bv−1k) ={δ1/2χ(b) if v > u,0 otherwise,

for b ∈ B(F ), k ∈ J and v ∈ W . Here > is the Bruhat order on W . According to theDeodhar conjecture, which was proved by Carrell-Peterson, Polo and Dyer, if u 6 v thecardinality of S(u, v) = {α ∈ Φ+|u 6 v · rα < v} is > l(v) − l(u), where l is the lengthfunction on the Weyl group and rα is the reflection in the hyperplace perpendicular tothe positive root α. Moreover, it is known that |S(u, v)| = l(v) − l(u) if and only if theKazhdan-Lusztig polynomial Pu,v(q) = 1. This is equivalent to the Schubert cell S(u) notbeing contained in the singularities of S(v). Assuming this, Bump and Nakasuji conjectureand propose to prove that∫

N∩vN−v−1

ψu(n) dn =∏

α∈S(u,v)

1− q−1zα

1− zα. (8)

If u = 1, this is the formula of Gindikin and Karpelevich. In the general case, it is muchdeeper, however. At this writing, this is also proved if S(u, v) consists of a single root,

18

and verified by computer calculations (using the affine Hecke algebra) for GLr up to aboutr = 6. If Pu,v 6= 1, then this fails, and the integral does not have a simple expression,though even in this case there are interesting patterns to be found. Let mu,v denote theleft-hand side of (8), and regard M = (mu,v)u,v∈W to be a matrix indexed by the Weylgroup (upper triangular in the Bruhat order). Then Bump and Nakasuji conjecture thatthe coefficient of u, v in M−1 is

(−1)|S′(u,v)|

∏α∈S′(u,v)

1− q−1zα

1− zα,

where S′(u, v) = {α|u < urα 6 v} . Complete knowlege of M−1 would evaluate the Cassel-man basis. A combinatorial conjecture related to the Deodhar conjecture would show thatthis conjecture follows from the first conjecture, the equality in (8).

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Justification for group effort

The principal investigators have worked together for many years, sharing ideas andwork as soon as it is carried out. We share a common vision of the area (indeed, we areresponsible for many of the developments in it) but bring different perspectives (algebraic,geometrical, analytic, computational) to it, in ways that complement each other. Carryingout research in this collaborative way has led us to a great number of breakthroughsincluding establishing unexpected connections.

Timeline/Justification for the duration

First year: We will attempt to obtain Yang-Baxter equations generalizing the workof [18] to metaplectic covers. We will extend the results on crystal graph descriptions toother root systems systems beyond type A, and other reduced decompositions of the longWeyl group element. We will try to understand whether the new connections to quantumgroup theory explain the relevance of crystal bases to Weyl group multiple Dirichlet series.We will extend the work of [33] to deal with orthogonal periods of minimal parabolicEisenstein series on GLr.

Second and third years: We will continue the above mentioned investigations. Beyondthat, we will investigate MDS for non-minimal parabolic subgroups and affine Kac-MoodyWeyl groups.

Justification: The proposal calls for the development of a new kind of connection be-tween metaplectic Whittaker functions, which are special functions arising from the theoryof automorphic forms, and quantum groups. Whittaker functions are of great importancein number theory, and metaplectic Whittaker functions are still mysterious. The proposedwork includes both the development of a new theory and of the solution to specific prob-lems. Exploring this will require us to apply techniques from disparate areas and this willrequire three full years to develop. The training modes described below also will requirethis much time.

Dissemination plan

The papers we produce will be posted on our websites and submitted to refereed jour-nals. All of the PIs give many external lectures and we will present our advances as suchopportunities arise. In addition, we will systematically present our results at the workshopsdescribed below and at other conferences to which we are invited.

20

Modes of Collaboration and Training

We plan to hold two workshops during the summers 2011 and 2012 (since the workshopsmay be in June we have put them in the budgets for the first and second years of the grant,respectively). The first workshop will be focussed on graduate students, with several daysof lectures devoted to developing the broad background needed to work in this rapidlyexpanding field, which increasingly draws upon methods from different areas (automorphicforms, quantum groups, statistical mechanics). This will be followed by several additionaldays to disseminate emerging new techniques to the widest possible audience; the latertalks will highlight not only recent progress but also open problems in the field. Theworkshops will be advertised, and support for graduate students is included in the budget;we will make a particular effort to reach out to previously underrepresented groups. Wehave a history of inclusive practices in previous workshops and post-doctoral mentoring,and three of the researchers whose work is discussed in this grant—Beineke, Frechette, andNakasuji—are members of an underrepresented group who have been introduced to thefield in this way.

We propose to build upon two existing models for this conference, the 2005 workshopin Bretton Woods and 2008 workshop in Edinburgh in which the Principal Investigatorsplayed lead organizing roles. Previous workshops featured a series of lectures to introducethe audience to the new approaches. However, the field is expanding so rapidly and drawingon new methods so extensively, we believe that an intensive teaching component is necessaryto welcome new workers to the area.

The second workshop will be targeted towards researchers who are engaged in or aboutto be engaged in this specific area of research. It will give an opportunity for workersin the field to share their advances and to map out future research projects. Studentswho have completed their research and recent Ph.D.s will be encouraged to present theirresults at the workshop. Models for the second workshop will be the 2006 and 2009Stanford workshops, where lectures were scheduled to allow substantial time for groups tointeract and make progress. In each case, conference attendees at the earlier internationalexpository conference became active researchers who gave talks at the smaller workshops.This influx of new talent and ideas, and the open schedule devoted to research, led tosignificant accomplishments in a short time at both Stanford workshops.

Though we do not include a budget for post-docs, we expect a number of our graduatestudents to be highly engaged post-docs within a few years, and we are committed totheir on-going mentoring. They will be involved in the workshops and invited to visit sitesinvolved in the project to give lectures.

These two workshops will be complemented by the Workshop on Whittaker Func-tions, Crystal Bases, and Quantum Groups being held at the Banff International ResearchStation, June 6-11, 2010, organized by the PIs. The participant list for this workshop un-derscores the growing community of researchers who have begun fruitful discussions withthe PIs and whose research problems are intimately connected to Whittaker functions and

21

quantum groups. Specialists in algebraic combinatorics, algebraic groups, representationtheory, infinite dimensional algebras, special function theory, geometric crystals, quan-tum groups and automorphic forms will all be participating. We hope to seize upon thismomentum with the future workshops proposed in this grant.

In addition, two of the PIs are co-organizers of the BC-MIT Number Theory Seminar.This seminar has been used to increase the exposure of MIT graduate students to thelatest advances in number theory, and with the new Ph.D. program at Boston College itwill do the same at BC. Friedberg will begin to support graduate students in years twoand three of the proposed grant. To foster dialogue between the groups of students at BCand MIT, Brubaker and Friedberg will offer graduate learning seminars held alternatelyat both schools and open to the entire Boston area math community. Students, post-docs,and the PIs would give the lecture on topics beyond the scope of graduate offerings at bothschools.

Management Plan

Among the PIs, the group effort will be coordinated and decisions made in a mannerconsistent with the way we have conducted our collaboration for many years. We are all inmore or less constant email and/or telephone contact. Progress is exchanged immediatelyand obstacles discussed openly. Each PI will be responsible for supervising post docsand graduate students at the local instititution. Administrative tasks associated with theworkshops will be shared by the PIs.

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References

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[3] R. J. Baxter, Exactly solved models in statistical mechanics. Academic Press Inc.[Harcourt Brace Jovanovich Publishers], London, 1982.

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[28] G. Chinta and P. Gunnells, “Weyl group multiple Dirichlet series constructed fromquadratic characters,” Invent. Math. 167, 327–353 (2007).

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