1 Introduction

Langlands L-functions, which play a fundamental role in modern number theory,are functions of one complex variable, with Euler products and functional equationsassociated with automorphic forms on reductive algebraic groups. In prior work, theprincipal investigators have been studying Weyl group multiple Dirichlet series. LikeL-functions, these are Dirichlet series with multiplicative coefficients and functionalequations. But they are functions of several complex variables whose group of func-tional equations is a Weyl group. Moreover, the multiplicativity of the coefficientsis “twisted,” so they are not Euler products. That is, the prime power supportedcoefficients (henceforth “p-parts”) combine by means of roots of unity coming fromthe metaplectic cocycle, whose existence is related to the reciprocity laws of numbertheory.

Weyl group multiple Dirichlet series arise naturally as the Fourier-Whittakercoefficients of Eisenstein series on metaplectic covers of reductive groups. Their p-parts, which are key objects of study, are metaplectic Whittaker functions over alocal field. They may be viewed as generalizations of characters of finite-dimensionalirreducible representations of Lie groups in which the weights of the representationare modified by n-th order Gauss sums, n being the degree of the metaplectic cover.Indeed in the nonmetaplectic case (n = 1) the Casselman-Shalika formula identifiesthese p-parts with such characters.

The remarkable Casselman-Shalika formula is of key importance in automorphicforms, being central in both the Rankin-Selberg method and the Langlands-Shahidimethod. Recently the Casselman-Shalika formula has appeared in a crucial wayin the geometric Langlands theory and in connection with mirror symmetry. It isto be expected that the Whittaker functions of metaplectic automorphic forms willbe of similar key importance, but until recently, these coefficients have been verymysterious.

Our recent work has led us to uncover surprising relations between these p-partsand combinatorial representation theory. The p-parts may be realized as functionson Kashiwara crystals, which are important combinatorial structures on bases ofrepresentations of quantum groups. They can also be constructed by an averag-ing procedure in the spirit of the Weyl character formula. They can be related toMirkovic-Vilonen cycles in the affine Grassmannian and to periods of automorphicforms. They can even be described in terms of certain two-dimensional lattice mod-els originating in statistical mechanics. These recent developments and others give agreat deal of evidence that there should exist an unexpected and far reaching connec-tion between the p-parts of multiple Dirichlet series (or more generally of Whittaker

1

coefficients of automorphic forms) and quantum groups, and this is the focus of theproposed work.

2 Results of Prior NSF Support

The prior FRG grant “FRG Collaborative Research: Combinatorial representationtheory, multiple Dirichlet series and moments of L-functions” (July 1, 2007 to June30, 2010) supported 9 principal investigators and 1 postdoctoral researcher. Wereport on the prior results from the 4 principal investigators, Brubaker, Bump, Chintaand Friedberg, who are submitting this new proposal. This research includes manycommon papers with Gunnells, who is a fifth principal investigator on this proposal.We give a somewhat lengthy treatment of parts of this prior work since it will helpput the new work and the potential for establishing a connection to quantum groupsin context.

Our first body of results concerns Whittaker functions, Weyl group multipleDirichlet series, and their p-parts. In their 2007 paper [19], Brubaker, Bump, Fried-berg and Hoffstein studied the Whittaker coefficients of minimal parabolic Eisensteinseries on an n-fold cover of SL3. More precisely, let F be a totally complex numberfield containing the group µn of nth roots of unity, with ring of integers o. Definethe metaplectic Eisenstein series E(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 sothat the Kubota map κ : Γ(f)→ µn is a homomorphism on the principal congruencesubgroup Γ(f) in GL3(o). (See Section 1 of [19] for details.) Then given elementsm1,m2 ∈ o, they consider 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 char-acter of conductor o. They showed that these Whittaker coefficients were multipleDirichlet series in two complex variables whose p-parts could be described combina-torially: each p-part is a weighted sum over strict Gelfand-Tsetlin (or ‘GT’) patterns,where the weights are products of n-th order Gauss sums. Recall that GT patterns

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are triangular arrays of integers whose consecutive rows interleave; strict patternshave strictly decreasing rows. The set of all (rather than strict) GT patterns withfixed top row is in bijection with basis vectors for the highest weight representationwhose highest weight can be read from this top row. In our context this top row, orhighest weight representation, is determined by the p-adic valuation of m1 and m2

in (1).Based on this evidence, they made two conjectures in [19]. First, they defined a

family of multiple Dirichlet series in r complex variables whose p-parts had a similarcombinatorial description; these series were conjectured to have analytic continuationand a group of functional equations isomorphic to the Weyl group Sr. The definitionof this p-part they gave is equivalent to the crystal graph definition below in (3).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 andFriedberg have proven both these conjectures. In the book [16] (which makes useof [15]), we give a direct proof of the conjectured functional equations. In [17], weprove the connection to minimal parabolic Eisenstein series. Both these works arethe first steps in new projects below. We now briefly outline the methods of proof.

The proof of functional equations in [15], [16] uses an induction argument onthe rank r, where the case r = 1 is essentially work of Kubota [47], tailored toour situation in [10]. The inductive step is achieved by offering two definitionsof the p-part of a multiple Dirichlet series in terms of Gelfand-Tsetlin patterns andproving their equivalence. The two definitions, which come from two nested sequencesof parabolic subgroups in SLr+1, each inherit functional equations from one ranklower whose union generates the entire group. The proof of the equivalence of thesedefinitions is quite subtle and requires number-theoretic information.

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 graphfor the associated highest weight representation of Uq(slr+1), each associated to adifferent reduced decomposition of the long element of the Weyl group. Indeed manyof the aspects of the weighting function on patterns and methods of proof have morenatural definitions in terms of the crystal, as we will indicate below. A large portionof [16] is devoted to these connections. Second, the equality of the two descriptionscan be reformulated in the language of statistical mechanics as the commutativityof two transfer matrices for a 2-dimensional lattice model. At least when n = 1,the method of Baxter [3] can be used so that an alternate proof may be given interms of the Yang-Baxter equation. We discuss these two approaches and programsof research built around them in the project description.

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The second conjecture, identifying the Weyl group multiple Dirichlet series withmetaplectic Whittaker functions, is proved in [17]. The proof is based on inductionin stages—that is, realizing the metaplectic SLr+1 Eisenstein series as a maximalparabolic Eisenstein series whose inducing data is a metaplectic SLr Eisenstein series.The coset representatives may then be chosen in a particular way which leads to aformula that is compatible with a corresponding inductive property of the Gelfand-Tsetlin description.

Peter McNamara, a student of PI Brubaker, has provided further insight into therelationship between Whittaker functions and crystals in [52]. Working over a localfield, he demonstrates that the Whittaker integral attached to a spherical vector in anunramified principal series representation can be computed via an explicit Iwasawadecomposition which decomposes a unipotent radical of the Borel into “cells.” Thealgorithm for computing this decomposition is quite general, applying to any simplyconnected Chevalley group. McNamara provides an identification between these cellsand Lusztig’s canonical bases [49], as well as to a geometric realization of crystals– Mirkovic-Vilonen cycles in the affine Grassmannian (cf. [2]). The decompositionof the unipotent again depends on a reduced expression for the long element of theWeyl group, and McNamara has worked this out explicitly for w0 = s1s2s1 · · · ofAr, where he demonstrates that the p-adic Whittaker function exactly matches thep-part of the multiple Dirichlet series, whose crystal definition will be presented in(3) in the Project Description below. For arbitrary reduced expressions of the longword, it seems quite difficult to give an explicit formula for the cell decomposition.

Combining the work of McNamara with work of Chinta-Offen [34] gives anotherway of proving the connection between Whittaker functions and Weyl group multipleDirichlet series.

Add: work of Bump-Nakasuji, Chinta-Gunnells, CFG, Chinta-Offen,Chinta-Mohler. Some of this could be mentioned very briefly or notedwith the sentence “We defer a more detailed discussion of this work to thesection on proposed research below.” I think we don’t want the additionsto be more than 2 pages, so that the majority of the grant is the newprojects.

Another approach to building Weyl group multiple Dirichlet series, pursued byPIs Chinta and Gunnells, involves a deformation of the Weyl character formula.Rather than constructing the p-part directly, Chinta–Gunnells translate the de-sired global functional equations into a W -action on the field of rational functionsC(x1, . . . , xr). They then construct an invariant rational function fλ by averagingover W . The rational function fλ can be written as the ratio of two polynomialsNλ/D, where the denominator is independent of λ and is a deformation of the usual

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Weyl denominator. If we write Nλ =∑a(k1, . . . , kr)x

k11 · · ·xkrr , then the p-part is

determined by H(pk1 , . . . pkr ;λ) = a(k1, . . . , kr).This method has been developed by PIs Chinta and Gunnells in [25, 28, 30, 29]

and in [27] together with PI Friedberg. In [28] they treat simply-laced root systems,quadratic symbols (n = 2), and λ trivial. This is extended to nontrivial λ in [27],again for simply-laced root systems and n = 2, where combinatorics of the p-partare investigated and agreement is proved with [13] for the coefficients attached tothe vertices of the Newton polytope of Nλ. In [30] the authors study the case ofn ≥ 2 for A2 and trivial λ; the arguments there generalize to the simply-laced caseand trivial λ. Finally in [29] Chinta–Gunnells treat the case of general Weyl groups,general twists, and general n.

Thanks to work of Chinta–Offen [?] and McNamara [?], we now know that thep-parts constructed by averaging agree with those constructed using crystal graphswhen the root system is of type A.

The principal investigators wrote or co-authored the following papers during theperiod of prior support (2007–present):

[1, 4, 5, 14, 15, 16, 17, 18, 19, 20, 23, 27, 28, 29, 30, 31, 32, 26, 33, 34, 39]Add to list: Chinta-Mohler, other papers of Chinta and of Gunnells.

Please check to see that bibliography is up to date.

Training and dissemination efforts. The grant supported one postdoctoral re-searcher, Bucur, who has accepted a tenure-track position at UCSD. Three PIs areadvising doctoral students (Brubaker is advising Lennon, McNamara and Tabony,Bump is advising Ivanov, Chinta is advising Mohler), many of whom are workingon projects closely related to the proposal. Friedberg will begin advising studentsnow that Boston College has approved its new doctoral program, and Gunnells hasjust had a student finish (Boland). Bump also worked extensively with postdoctoralresearcher Nakasuji. The PIs helped organize and run a 5-day workshop on “MultipleDirichlet series and applications to automorphic forms” at the International Centrefor Mathematical Sciences, Edinburgh, United Kingdom in August 2008. This wasattended by over 60 researchers from 12 countries and many graduate students andincluded a detailed series of introductory lectures about the area. A proceedingsvolume is in preparation, to be published by Birkhauser. In June 2009 a secondweek-long workshop was held with a more intensive research focus but also a train-ing component, featuring detailed discussion of open problems and time for workgroups 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)and M. Nakasuji (Stanford).

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3 Project description

Whittaker functions and crystals. Both papers [16] and [17] give a translationof the definitions of the p-part of Whittaker coefficients for G = SLr+1 in terms ofcrystal graphs. We first review this definition and then describe the open problemsassociated to p-parts on other Chevalley groups.

Crystal graphs were invented by Kashiwara in connection with the representationsof quantized enveloping algebras. Rather than using the highest weight representa-tions of the complex Lie group LG or its Lie algebra Lg, we may use the universalenveloping algebra U(Lg) or its quantum group Uq(

Lg). The representations of LG,U(Lg) and Uq(

Lg) are all the same. The crystal is a combinatorial structure onthe representation space that appears when q→ 0. In particular, Littelmann [48]and Berenstein-Zelevinsky [8] give a combinatorial realization of the crystal graph interms of paths in the crystal which are traversed according to a factorization of thelong element of the Weyl group.

Let w0 = si1 · · · sit be a reduced decomposition of the long element into sim-ple reflections. Given an element b of the crystal basis, we apply the Kashiwaralowering operator fαi1 repeatedly to b. Let a1 be the maximal integer such thatb′ := fa1

αi1(b) 6= 0. Repeating this process with b′ we obtain a sequence of integers

(a1, . . . , at) representing the basis element b and such that fatαit · · · fa2αi2fa1αi1

(b) is thelowest weight vector in the crystal graph. The collection of all such sequences, rang-ing over all possible highest weights, are the integral points of a cone in Rt, whosebounding hyperplanes depend on the choice of reduced decomposition. The set ofsequences for a fixed choice of highest weight vector form a polytope cut from thiscone by additional bounding hyperplanes (see for example Proposition 1.5 of [48]).

For example, if we consider the root system A2 with w0 = s1s2s1, then the coneconsists of all sequences (a1, a2, a3) with ai ≥ 0 and a2 ≥ a3. We place these 3integers in an array reflecting these inequalities and so that the path lengths for eachroot operator are all in the same column, 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 pathfrom one element of the crystal basis to the lowest weight vector are illustrated inFigure 1, which depicts a crystal graph of highest weight λ + ρ = 2ε1 + 2ε2. Anentry ci,j of the BZL pattern will be called “minimal” if ci,j = ci,j+1 (i.e. the coneinequality is an equality), and “maximal” if the hyperplane inequality for ci,j is anequality.

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

f2

f1

v

vlow

Figure 1: The BZL pattern (a1, a2, a3) = (2, 3, 1) depicted as a path in the crystalgraph Cλ+ρ with λ+ ρ = 2ε1 + 2ε2. The entry a2 is maximal, while a1, a3 are neithermaximal nor minimal.

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

7

as in (2), we associate the function

G(P ) =∏

1≤i≤j≤r

γ(ci,j) where γ(c) =

g(pc−1, pc) if c is maximal

g(pc, pc) if c is neither max’l nor min’l

pc if c is minimal

0 if c is both max’l and min’l

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

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

of w0 = s1s2s1 . . . srsr−1 . . . s1 agrees with the p-part of the associated metaplecticWhittaker coefficient on SLr+1. While there are many combinatorial descriptionsof bases for highest weight representations, what is striking here is the very naturalway in which the BZL patterns – paths in crystals – may be used to represent aWhittaker coefficient.

Note that the definitions we have made in (3) and (5) for SLr+1 and a particu-lar reduced decomposition of w0 in terms of paths in a crystal make sense for anyChevalley group and any reduced long word. Unfortunately, this definition does notmatch the associated Whittaker coefficient in general. For example, for SL4, thereare 16 reduced expressions. Some progress has been made for other classical groups(cf. [4] and [31]).

NEED TO KEEP GOING WITH THIS PROJECT DESCRIPTION.JUST GETTING STARTED... projects include expressions on crystalsholding for all classical groups, all long words. Two main approaches –Paul’s repeated testing to make conjectures, new method of coset reps forG/P, P - maximal parabolic to actually compute exponential sum fromWhittaker integration.

Whittaker functions and quantum groups. In order to understand the mo-tivation behind this project, we begin with the non-metaplectic case. In this case,

8

the Casselman-Shalika formula describes the values of a spherical p-adic Whittakerfunction in a surprising and beautiful way. The values are simply the characters ofthe finite-dimensional irreducible representations of the dual group evaluated at theLanglands parameter conjugacy class. This formula has many applications in auto-morphic forms and number theory, being basic to both the Rankin-Selberg methodand the Langlands-Shahidi method of constructing L-functions. It has also recentlyattracted the attention of physicists, as in the work of Gerasimov, Lebedev andOblezin in connection with Givental’s work on mirror symmetry.

Let G be a reductive algebraic group and let λ be a dominant weight of the rootsystem Φ of the Langlands dual group LG. This indexes an element tλ of G. Forexample if G = GLr+1 then λ = (λ1, · · · , λr+1) ∈ Zr+1 (identified with the weightlattice) with λ1 > λ2 > · · · and we may take tλ to be a lift to G of a diagonal matrixwith eigenvalues pλi .

The spherical representations of G or G are parametrized by conjugacy classesin LG. Let z ∈ LG parametrize such a representation, and let W : G −→ C bethe p-adic Whittaker function. If n = 1, the Casselman-Shalika formula says thatthe spherical Whittaker function W (tλ) is, up to a normalization factor, χλ(z), thecharacter of an the irreducible module of LG with highest weight vector λ.

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 (3), 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 denominatorby its deformation ∏

α∈Φ+

(1− q−1znα) (7)

with q = |p|, the cardinality of the residue field, we have a striking identity for thecharacter χλ. It is actually not new, but was discovered in a purely combinatorialcontext (in terms of GT patterns) by Tokuyama [62]. The crystal definition suggestsa deeper connection with the theory of quantum groups, and we now explain howthe work of Brubaker, Bump and Friedberg in [18] uses Tokuyama’s identity as astarting point for these connections. We then propose further work to complete thisprogram.

Quantum groups were introduced by Drinfeld and Jimbo in order to explainexactly solved models in statistical mechanics and integrable systems in quantummechanics. We have recently discovered that one well-studied model, the six-vertex

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model which corresponds to two-dimensional ice, can be used as a basis for investi-gating Whittaker functions, at least on SLr+1 and Sp2r. Moreover, such a descriptionextends to metaplectic covers.

We consider a rectangular grid, with vertices placed at each crossing. Every edgeof the grid is to be assigned one of two “spins,” + or −. In our six-vertex model, theadjacent edges to any vertex can come in only 6 possible configurations. Dependingon the adjacent spins, each vertex is assigned a Boltzmann weight . The Boltzmannweight of the entire configuration is the product of the weights at each vertex. Thepartition function is then the sum of the Boltzmann weights over all states.

To “solve” a statistical model is to determine its partition function. The six-vertex model was originally solved in 1967 by Lieb and Sutherland on a latticewith periodic boundary conditions, where the Boltzmann weights satisfy a certainsymmetry. There is an immense literature concerning solvable lattice models. Wehighlight the visionary contribution of Baxter [3], who gave a treatment in terms ofwhat is now known as the Yang-Baxter equation (‘YBE’) that is at the heart of thefollowing discussion.

We now explain how to obtain the value of the Whittaker function at tλ as apartition function on a lattice or “ice” with certain boundary conditions. The sixpossible spin configurations around a vertex, together with their associated Boltz-mann weights in both the non-metaplectic (i.e. degree of the cover n = 1) andmetaplectic cases, are given in Table 1 below. We call this recipe “Gamma ice”(as there will be other models and weights to come). The index i records the rownumber of the lattice, and will be explained momentarily. To each such i, zi and tiare arbitrary complex numbers. The metaplectic weights (n ≥ 1), will be explainedlater.

Gamma ice

i i i i i i

Boltzmannweight (n = 1)

1 zi(ti + 1) 1 ti zi zi

weight for n ≥ 1 1 zih(a) 1 zig(a) zi zi

Table 1: Boltzmann weights for Gamma ice

These weights will be applied to ice with the following boundary conditions,determined by the index of the Whittaker coefficient λ+ ρ = (λ1 + ρ1, . . . , λr + ρr, 0)

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where ρ = (r, r − 1, · · · , 0). (See [18] for further information.) The ice will haver + 1 rows and λ1 + ρ1 + 1 columns. We require that the edge spins along the leftand bottom edges be +; along the right edges, they are to be −. Along the toprow, we label the columns from right to left, and place a − on the top edge in thecolumns labeled λj + ρj. A typical example of such boundary conditions is given inthe 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 columnslabeled 5, 2, and 0. The set of all possible fillings of this ice with boundary, using thesix types of vertices, give the total collection of admissible states. One such fillinghas been depicted above.

In evaluating the ice, we take the Boltzmann weights from Table 1 using the valueszi, ti in the i-th row. For each highest weight λ+ ρ, the resulting partition functionZ, the sum of Boltzmann weights (themselves a product of weights at each vertex)over the set of all admissible Gamma ice configurations SΓ

λ, has been computed.

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 [62] stated this theorem in the language of strict Gelfand-Tsetlin patterns.Our set of ice with boundary SΓ

λ is in bijection with strict GT patterns having highestweight λ + ρ. Hamel and King [41] proved a symplectic version of Tokuyama’stheorem, and provided a translation of these formulas into the language of ice.

While they were interested in such deformations for their combinatorial structure,we now explain how this represents a p-adic Whittaker coefficient on G = GLr+1. If

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we take ti = −q−1 for all i, and z = (z1, · · · , zr+1) the eigenvalues of the conjugacyclass in LG, this represents δ−1/2(tλ)W (tλ), where we take the Whittaker functionin the normalization coming from its standard representation as an integral on thep-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 where we haveused the shorthand g(a) = q−ag(pa−1, pa) and h(a) = q−ag(pa, pa) where g(m, d) is ann-th order p-adic Gauss sum, the analogue of (4) with residue symbol replaced by thelocal Hilbert symbol. (We assume that n does not divide the residue characteristic.)In the Boltzmann weights, the argument a in h(a) or g(a) is the number of + edgesin the row to the right of the vertex.

In Brubaker, Bump and Friedberg [18], a new proof of Theorem 1 is given us-ing the Yang-Baxter equation in the spirit of Baxter [3]. We briefly explain thisconnection and then outline a host of project possibilities resulting from the link tostatistical mechanics, including most importantly the creation of a quasitriangularHopf algebra (quantum group) that may be the key to understanding the relevanceof crystals in Whittaker coefficients of automorphic forms.

Baxter [3] gave a method for solving lattice models based on what he called thestar-triangle relation. In the context of our work, this may be explained as follows.Define another type of ice, called “Gamma-Gamma ice” (because it can be used tobraid two strands of Gamma ice), whose weights are given in 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

Boltzmannwt. (n = 1)

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

Table 2: Boltzmann weights for Gamma-Gamma ice

Now consider pair of mini-ensembles, each with just three vertices and six bound-ary spins α, β, σ, τ, ρ, θ that are to be prescribed.

The star-triangle relation states that these two mini-ensembles have the samepartition function. That is, if we sum over the possible spins that can be assigned tothe edges γ, ν, µ the left-hand side gives the same evaluation as summing the states

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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.

of the right-hand side, summing over δ, φ, ψ. This may be confirmed for the weightswe 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 entryin Table 2 as defining a map on basis vectors, our endomorphism can be expressedas a matrix consisting 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 fromTable 1. If R is an endomorphism of V ⊗ V let R12, R13 and R23 be endomorphismsof V ⊗V ⊗V in which Rij acts on the i-th and j-th components, and as the identityon the remaining one. Then the star-triangle relation may be written R12M13N23 =N23M13R12. In our case, it is satisfied for R = RΓΓ(i, j), M = MΓ(i) and N = MΓ(j).

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

µ,ν,γ

Rραµγ(j, k)Rθγ

νβ(i, k)Rνµστ (i, j) =

∑δ,φ,ψ

Rψδτβ(j, k)Rφα

νδ (i, k)Rθρφψ(i, j).

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This may be diagrammed as follows:

k

j

i

i

j

k

β

τ

σ

ν γ

µ

θ

ρ

α

k

j

i

i

j

k

β

τ

σ

δ

ψ

φ

θ

ρ

α

It can be thought of as a braid relation in which three strands of gamma ice arewoven, to give vertices in Gamma-Gamma ice at each crossing.

In future work, we intend to investigate the underlying Hopf algebra structurefor this R matrix. Indeed, Drinfeld [35], Faddeev, Reshetikhin and Takhtajan [36]and Majid [51] have given results which allow one to construct a quasitriangularHopf algebra (quantum group) using generators and relations determined by theR-matrix. From this parametrized YBE, the resulting Hopf algebra HΓΓ will haveone two-dimensional module V (z, t) for every pair of nonzero complex numbers zand t, and RΓΓ(i, j) is the endomorphism of V (zi, ti) ⊗ V (zj, tj) whose compositionwith the interchange x ⊗ y → y ⊗ x is an intertwining map V (zi, ti) ⊗ V (zj, tj) →V (zj, tj)⊗ V (zi, ti).

A host of open problems appear. A first question is whether this Hopf algebracan be related to known quantum groups. A second urgent question is that whilethe vertex-model definition of the p-parts extends to the metaplectic case n > 1,we do not yet have the Yang-Baxter equation in that context. It is also interestingto ask whether, since we are able to take the second set ti of spectral parametersto be unequal in the n = 1 case, whether this is also true when n > 1. Anotherquestion is whether this connection with quantum groups explains the relevance ofcrystal bases. Finally, when n = 1, the action of the Weyl group on the spectralparameters is reflected in the Yang-Baxter equation, and it seems likely that this isconnected with the work of Chinta-Gunnells and Chinta-Offen, where such a Weylgroup action also occurs. Also, knot invariants, such as the Jones polynomial andVassiliev invariants, have been shown to result from the categories of representationsof quasitriangular Hopf algebras, and there could be applications to knot theory.And in the special case where the second set of spectral parameters ti are all equalto a fixed value t, we know from work of Nichita and Parashar [54] that it is possibleto introduce another parameter σ and obtain a seven vertex model. The implicationsof this are unknown.

We computed a partition function for a lattice composed of Gamma ice, arriv-ing at a p-adic Whittaker function. But we could consider more general boundary

14

conditions, which might result in interesting partition functions. Moreover, we couldinvestigate the partition function for a lattice composed of Gamma-Gamma ice aswell. The existence of a YBE for Gamma-Gamma ice already implies a certain sym-metry of the resulting partition function. Using rather different weighting systems,Fomin and Kirillov [?] constructed symmetric functions - Schubert and Grothendieckpolynomials arising naturally in the cohomology and K-theory of the flag variety -based on the YBE. It would be interesting to have a general theory for producingsuch geometrically motivated bases of symmetric functions based on solutions of theYBE which incorporated both sets of results.

The structure we have described is really only the beginning of the story. Aswe noted in the Prior Work section, the proof of functional equations for multipleDirichlet series in [16] was reduced to the equivalence of two definitions of the p-partof the multiple Dirichlet series. These definitions, in the crystal language, are relatedby the Schutzenberger involution on the crystal Bλ+ρ. The Schutzenberger involutioncan be built up in smaller steps, each the reflection in a root string. The number ofsuch steps equals the length of the long element of the Weyl group. Each of thesesteps is accomplished by an application of an involution for “short Gelfand-Tsetlinpatterns” and this statement (see “Statement B” in [16]) translates beautifully intothe vertex operator setting.

Thus there is, in addition to the Gamma ice, another type of ice called Delta icecorresponding to the second definition of the p-part. In Baxter’s statistical-mechanicslanguage, Statement B is equivalent to the commutativity of two transfer operatorscorresponding to Gamma and Delta ice. This equivalence remains valid even in themetaplectic case. At least if n = 1 it can be proved using a star-triangle relation.This requires the invention of suitable R-matrices, Delta-Delta ice, Gamma-Deltaice and Delta-Gamma ice (see [18] for their Boltzmann weights). Putting Gamma-Gamma, Gamma-Delta, Delta-Gamma and Delta-Delta ice together one obtains agrand Hopf algebra HΓΓ ⊗H∆∆. As a Hopf algebra this is just the coproduct of theHopf algebras HΓΓ and H∆∆ but its quasitriangular structure is quite a bit richersince it also has an R-matrix in End(V ⊗W ) where V is a module of HΓΓ (a directsum of the irreducible two-dimensional modules V (z, t)) and W is a module of H∆∆.At least if the ti are equal, one has also the data needed to define a Yang-Baxtersystem as introduced by Hlavaty [44] based on earlier work of Freidel and Maillet[38]. This data allows the construction of a quasitriangular Hopf algebra that is aDrinfeld double.

Whittaker functions for non-Borel Eisenstein Series. If G is a split connectedreductive algebraic group over F , P = MN is a maximal parabolic subgroup withLevi decomposition, and π is an automorphic representation of M(AF ), the adelic

15

points of M , then one may form an Eisenstein series of G(AF ) induced from π.Its Whittaker function may be expressed in terms of the Langlands L-functionsL(s, π, ri), where r = ⊕mi=1ri is the adjoint action of LM on the complex Lie algebraLn. This is the beginning of Langlands-Shahidi theory (see Shahidi [59, 60] and thereferences therein).

As a vast generalization of the Borel Eisenstein series case, we propose to computethe coefficients of more general metaplectic Eisenstein series. We have already workedout one case, inducing from GLr to GLr+1, in [17], and are in the process of writingup another [20]. From these computations, we once again expect the theory ofcrystals to intervene. However, twisted multiplicativity must be modified, in a waythat reflects the choice of P . Even giving Tokuyama-type identities in this situationshould be very interesting, as it could give new identities blending the coefficients ofautomorphic forms with the combinatorics of crystals. And when one induces lowerrank Eisenstein series, this should give a way to study the crystal description whenthe long word is factored into other good decompositions, and obtain a theory thatis valid for all such decompositions.

A metaplectic Casselman-Shalika formula. Casselman and Shalika [24] explic-itly compute the spherical Whittaker function of a p-adic reductive group, gener-alizing Shintani’s formula for GLr [61]. Chinta and Offen [34] have extended thisformula to include the n-fold metaplectic cover of GLr and we intend to furthergeneralize this result to n-fold cover of an arbitrary reductive groups. For n > 1, acentral difficulty is the failure of uniqueness of Whittaker functionals (and thereforeof spherical Whittaker functions of a fixed Hecke eigenvalue). In [?], Y. Hironakacomputed explicitly the spherical functions on the space of non-singular Hermitianmatrices with respect to an unramified quadratic extension of p-adic fields. This isa case where multiplicity one fails. Hironaka’s approach to the Casselman-Shalikamethod in case of multiplicities (see §1 of [loc. cit.]) is our guideline for this project.

Roughly speaking, a spherical function can be expressed as the value of a certainlinear form applied to translates of the unramified vector in an unramified principalseries representation. The idea behind the Casselman-Shalika method is to reduce thecomputation for the value of the linear form on a translate of the element invariantunder the maximal compact subgroup to the computation of simpler expressions forelements invariant under a smaller open compact - the Iwahori subgroup. There arethree main steps in carrying out the method.

The first has to do solely with the group and not with the particular linearform we consider. It is an expansion of the unramified element of a principal seriesrepresentation in terms of a ‘well chosen’ basis of the Iwahori invariant subspace -the Casselman-Shalika basis. In [58], Sakellaridis provides a formula for spherical

16

functions in the general setting of spherical varieties for a split reductive group (this,however, does not contain our case as long as n > 1). His characterization of theCasselman-Shalika basis simplifies the computation.

The second step is to obtain Weyl group functional equations between the spher-ical functions. The unramified principal series representations are parameterized bya variable, say s, in some complex variety on which a related Weyl group acts. Cru-cial to the computation of the spherical functions is to relate explicitly between thespherical functions associated to s and those associated to ws for any Weyl elementw. When n = 1, the space of Whittaker functionals of an unramified principal seriesrepresentation is one dimensional. There is then a one dimensional space of sphericalWhittaker functions for a given parameter s (i.e. for a fixed Hecke eigenvalue) and thefunctional equations are therefore scalar valued. For metaplectic groups multiplicityone fails, e.g. on the n-fold metaplectic cover of GLr, the space of Whittaker func-tionals for an unramified principal series representation is of dimension nr

gcd(n,2rc+r−1).

This complicates the computation of the functional equations. Once a basis of Whit-taker functionals for any parameter s has been fixed, there is a matrix associatedto any Weyl element w that expresses the basis for ws in terms of the basis for s.For GLr such a functional equation was established by Kazhdan and Patterson [46].Generalizing this to an arbitrary reductive group will require a careful reworking ofthe results of Kazhdan and Patterson. Aiding us in this is the fact that we knowwhat the functional equations should be: a local version of the Eisenstein conjecturein [13] implies that the functional equations of the Whittaker functions should berelated 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 Iwahoriinvariant functions in the Casselman-Shalika basis. This, in our case, is not muchmore complicated than in the Shintani, Casselman-Shalika case.

Once this is done, we expect the resulting formula for the spherical Whittakerfunction to coincide with the p-part of the Weyl group multiple Dirichlet series in[29], thereby providing a link between these multiple Dirichlet series and metaplecticWhittaker functions. Until now, such a connection has been rigorously establishedonly for G = GLr.

Orthogonal periods and Multiple Dirichlet Series. Brubaker, Bump andFriedberg [17] establish a relation between the Weyl group multiple Dirichlet seriesand Fourier coefficients of Eisenstein series on metaplectic covers of GLr. Langlandsfunctoriality conjectures expect, in particular, the existence of transfers of automor-phic forms between metaplectic groups and reductive groups. It is therefore likelythat Weyl group multiple Dirichlet series appear naturally also in the theory of au-

17

tomorphic forms on algebraic reductive groups.Let G be a reductive group defined over a number field F and let H be a closed

subgroup. Set G = G(F ) and H = H(F ). Whenever convergent, the H-periodintegral of an 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 contextof a symmetric space, i.e. when H is the group of fixed points of an involution on Gthere is another group G′ associated to G/H and a functorial transfer of automorphicforms from G′ to G that is expected to capture non vanishing of the period integral.The value PH(φ) when not zero often carries interesting arithmetic information. Incases where the period factorizes it is expected to be related to automorphic L-functions. When there is no factorization the arithmetic meaning of the period isnot yet understood.

Consider now the case where G = GLr over F and H is an orthogonal subgroup.Using the formalism of the relative trace formula and evidence from the r = 2 case,Jacquet conjectured that in this setting orthogonal periods of a form on G shouldbe related to a a period of a form on G′, the metaplectic double cover of G [45]. ForG′ local multiplicity one of Whittaker functionals fails. This leads us to expect thatthe period integral PH(φ) of a cusp form is not factorizable. To date, the arithmeticinterpretation of the period at hand is a mystery, and precise conjectures are yetto be made. Often, studying the period integral of an Eisenstein series is moreapproachable then that of a cusp form and may help to predict expectations for thecuspidal case (this was the case for G = GL2 and H an anisotropic torus, where theclassical formula of Hecke for the period of an Eisenstein series in terms of the zetafunction of an imaginary quadratic field significantly predates the analogous formulaof Waldspurger for the (absolute value squared) of the period of a cusp form).

Following a suggestion of Bump and Venkatesh, Chinta and Offen [33] computethe period integral PH(E(ϕ, λ)) in the special case that r = 3, H is the orthogonalgroup associated to the identity matrix and E(ϕ, λ) is the unramified Eisensteinseries induced from the Borel subgroup. The formula we obtain expresses the periodintegral as a finite sum of products of the quadratic A2 double Dirichlet series, whichare essentially (see [12]) Whittaker coefficients of the metaplectic Eisenstein series.This fits perfectly into Jacquet’s formalism, and it is our hope that the formulain this very special case can shed a light on the arithmetic information carried byorthogonal periods of cuspforms. We propose to develop the study of periods andtheir relation to multiple Dirichlet series, and to investigate the implication of the

18

connection to quantum groups in this context as well.

The Affine case. There are reasons to believe that this theory will extend to otherKac-Moody root systems. Affine root systems are an important case. Such rootsystems have infinite Weyl groups, so the multiple Dirichlet series, originally definedin a product of right half places of Cr (r=number of simple roots), are expected tocontinue to a proper subset of Cr rather than to the full space; see Bump, Friedbergand Hoffstein [22]. There is a character formula which admits q-deformations, andthere is reason to believe that there are also metaplectic deformations.

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

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

of Bucur and Diaconu [21]. (Their result requires working over the rational func-tion field; it builds on work of Chinta and Gunnells [29].) If one could establishthe analytic properties of such series in full generality, one would have a powerfultool for studying moments of L-functions. On the other hand, the p-parts of suchmultiple Dirichlet series will have connections with representation theory (generaliz-ing those outlined above) quite independent of applications to number theory, sincethese would be metaplectic deformations of the Kac-Weyl character formula.

The Formula of Gindikin and Karpelevich. Let G be a reductive p-adic group,which we assume for simplicity to be simply laced. The formula of Gindikin andKarpelvich 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 thequantized enveloping algebra of N−, and if one applies the recipe (??) to this crystal,Bump and Nakasuji [23] proved that∑

v∈B(∞)

G(v) =∏

α ∈ Φ+

1− q−1zα

1− zα.

19

This is striking, since it shows that an integral over the group N− can be replaced bya sum over a basis of its quantized enveloping algebra. These results have extensionsto the n-fold metaplectic cover.

The proof of the Casselman-Shalika formula makes use of a particular basis ofthe 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)

when b ∈ B(F ), the Borel subgroup, and k ∈ J , the Iwahori subgroup. Here χ isa fixed character of a split maximal torus in T (F ) ⊂ B(F ) and δ is the modularquasicharacter of B(F ). This Casselman-basis , indexed by the Weyl group W , isdual to the standard intertwining operators. It is difficult to compute, but oneparticular basis vector may be computed, and as Casselman noted, this is enoughfor the applications.

However, Bump and Nakasuji have a partial generalization of the Gindikin-Karpelevich formula which provides a partial evaluation of the Casselman basis.It is shown that the difficulty of evaluating the Casselman basis is related to thegeometry of Schubert cells and Kazhdan-Lusztig polynomials.

This is work in progress, and the results stated below are conjectural, but someor all are expected to be proved. We assume that G is simply-laced. If u ∈ W (theWeyl 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 . Accordingto the Deodhar conjecture, which was proved by Carrell-Peterson, Polo and Dyer, ifu 6 v the cardinality of S(u, v) = {α ∈ Φ+|u 6 v · rα < v} is > l(v) − l(u), where lis the length function on the Weyl group and rα is the reflection in the hyperplaceperpendicular to the positive root α. Moreover, it is known that |S(u, v)| = l(v)−l(u)if and only if the Kazhdan-Lusztig polynomial Pu,v(q) = 1. This is equivalent to theSchubert cell S(u) not being contained in the singularities of S(v). Assuming this,Bump and Nakasuji conjecture and hope 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 Karpelvich. At this writing, this is provedif S(u, v) consists of a single root, and verified by computer calculations (using the

20

affine Hecke algebra) for GLr up to about r = 6. If Pu,v 6= 1, then this fails, andthe integral does not have a simple expression, though even in this case there areinteresting patterns to be found. Let mu,v be (8), and regard M = (mu,v)u,v∈W to bea matrix indexed by the Weyl group (upper triangular in the Bruhat order). ThenBump and Nakasuji conjecture that the 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 theCasselman basis. A combinatorial conjecture related to the Deodhar conjecturewould show that this conjecture follows from the first conjecture (8).

Justification for group effort

The principal investigators have worked together for many years, sharing ideasand work as soon as it is carried out. We share a common vision of the area (in-deed, we are responsible for many of the developments in it) but bring differentperspectives (algebraic, geometrical, analytic, computational) to it, in a way thatcomplements each other. Carrying out research in this collaborative way has led usto a great number of breakthroughs including establishing unexpected connections.So we believe it is best to continue doing so.

Timeline/Justification for the duration

First year: We will extend the results on MDS to other crystals of root systemsbeyond type A, and define MDS for non-minimal parabolic subgroups. We will focuson the local field setting to make connections to quantum groups via [?] and extendthe work of [33] to deal with orthogonal periods of minimal parabolic Eisensteinseries on GLr. Second and third years: We will further extend the results to generalKac-Moody Weyl groups and consider the implications (automorphic and local) ofa quantized “metaplectic” Langlands program. Justification: This proposal seeks todevelop a new kind of connection between automorphic forms and quantum groups.Exploring this will allow us to apply techniques from disparate areas but this willrequire three full years to develop. The training modes described below also willrequire this much time.

Dissemination plan

The papers we produce will be posted on our websites and submitted to refereedjournals. All of the PIs give many external lectures and we will present our advances

21

as such opportunities arise. In addition, we will systematically present our results atthe workshops described below and at other conferences to which we are invited.

Note: Material from here on does not count in the 20 page limit.Modes of Collaboration and Training

We plan to hold two workshops during the summers 2011 and 2012 (since theworkshops may be in June we have put them in the budgets for the first and secondyears of the grant, respectively). The first workshop will have an initial focus ongraduate students, and feature 3 days of talks aimed specifically at them. Shortcourses will be given describing the emerging new techniques of the field, whichincreasingly draws upon methods from different areas (automorphic forms, quantumgroups, statistical mechanics). This will be followed by two days of research talksthat highlight not only recent progress but also open problems in the field. Themodels for this workshop are the highly successful 2005 workshop in Bretton Woodsand 2008 workshop in Edinburgh. But the field is expanding so rapidly and drawingon new methods so extensively that we believe that an intensive teaching componentis necessary to welcome new workers to the area.

The second workshop will be more targeted towards active workers (both es-tablished ones and newer recruits) with extensive problem sessions. Models for thesecond workshop will be the 2006 and 2009 Stanford workshops, where lectures werescheduled to allow substantial time for groups to interact and make progress. In eachcase this led to significant accomplishments in a short time.

These two workshops will be complemented by the Workshop on Whittaker Func-tions, Crystal Bases, and Quantum Groups being held at the Banff InternationalResearch Station, June 6-11, 2010, organized by the PIs. In addition, two of the PIsare co-organizers of the BC-MIT Number Theory Seminar. This seminar has beenused to increase the exposure of MIT graduate students to the latest advances innumber theory, and with the new Ph.D. program at Boston College it will do thesame at BC.

Management Plan

Among the PIs, the group effort will be coordinated and decisions made in a man-ner consistent with the way we have conducted our collaboration for many years. Weare all in more or less constant email and/or telephone contact. Progress is exchangedimmediately and obstacles discussed openly. Each PI will be responsible for super-vising post docs and graduate students at the local instititution. Administrativetasks associated with the workshops will be shared by the PIs.

22

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