Brubaker, Bump, Chinta, Friedberg and Gunnells- Crystals and Multiple Dirichlet Series

8/3/2019 Brubaker, Bump, Chinta, Friedberg and Gunnells- Crystals and Multiple Dirichlet Series

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Crystals and MultipleDirichlet Series

Brubaker, Bump, Chinta, Friedberg, Gunnells

Work In Progress

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Quantum groups and Crystal Bases

Quantum groups (Drinfeld, Jimbo) are deformations (ina suitable category) of Lie groups.

Crystal Bases were introduced by Kashiwara in connec-tion with quantum groups.

From Groups to Quantum Groups

Lie groups do not have deformations in their own category,so one works in the category of Hopf algebras.

If G is a Lie group, we have maps:

multiplication: m: GG

G

diagonal: : G GGsatisfying the associative laws and a compatibility property

GGG .m1 GG 1m m

GG . m G

G.

GG

1

GG .1 GGG

GG . GGGG m

G .

GG

rightmost vertical map:(x , y , z , w)

(m(x, z), m(y , w))

The universal enveloping algebra U(g) of g = Lie(G) is afunctor from groups to C-algebras. So U = U(g) becomes analgebra and a co-algebra, with associative maps

multiplication: m: UU

U

diagonal: : U

UUand a compatibility that makes it a Hopf algebra.

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Deformation

The deformation (Drinfeld, Jimbo) Uq(g) is a Hopf algebradepending on a parameter q C.

If q = 1 it is U.

q 1 is like classical limit 0 of quantum mechanics.

Representations

The irreducible modules of G or U are the same as theirreducible modules of Uq(g). If : G GL(V) is a repre-sentation, V is a module for U and also for Uq(g) for all q.

However Uq(g)-module structure on V1 V2 (two modules)varies with q.

The significance of the comultiplication is that it determines amultiplicative structure on the tensor category of representations.

If V is a G-module it is a U-module.

If V1, V2 are modules then V1

V2 is too. Comultiplication

: U UU implements this for the Hopf algebra U. For Uq(g) comult is required for this. (Group is gone.) The Hopf algebra U is cocommutative a property it

inherits from the diagonal map:

G G G

GG

U U U

U U

: (x, y) (y, x) : x y y x

The deformation Uq(g) is no longer cocommutative.

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Highest Weight Modules (Weyl)

Let G be a semisimple complex Lie group of rank r. Let T be amaximal torus. Then X(T)

@

Zr.

Elements of X(T)

@

Zr are called weights

R X(T) @ Rr has a fundamental domain C+ for theWeyl group W called the positive Weyl chamber.

If (, V) is a representation then restricting to T, themodule V decomposes into a direct sum of weighteigenspaces V() with multiplicity m() for weight .

There is a unique highest weight wrt partial order. Wehave C

+

and m() = 1. V gives a bijection between irreducible representa-tions and weights in C+.

Here are the weightsof an irreducible V

for G = SL3, showingmultiplicities and thepos. Weyl chamber.

m() = 1

m() = 2

C+

Legend

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Root Operators

Let 1, , r be the simple positive roots. For each there is a pairof root operators in the Lie algebra gU:

Ei corresponding to i,Fi corresponding to i.

For example if G = SL(3):

Ei=

0 1 00 0 0

0 0 0

, E2 =

0 0 00 0 1

0 0 0

, Fi =

0 0 01 0 0

0 0 0

, F2 =

0 0 00 0 0

0 1 0

.

The root operators shift the weight. ThusEi: V() V( + i)Fi: V() V(i)

+ 1

E1

+ 2

1

2

E2

The root operator E1maps the 2-dimensional

vector space V() intoV( + 1), and E2 mapsV() V( + 2).

For pictorial purposesassociate a color witheach root operator.

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Crystalization (Kashiwara)

It is not actually possible to take q = 0 in the Hopf algebra Uq(g).

In the limit q 0 the semisimple part ofg breaks down.

Still Ei and Fi can be continued to q = 0. Their effect beomes very simple. It is possible to choose a crystal basis vi such that In the

limit q = 0: each Ei or Fi takes a basis element toanother basis element or 0.

The action of the Ei and Fi on the crystal basis can beillustrated very simply by giving the crystal graph C.

Existence proofs were given by Kashiwara, Kashiwara andNakashima, Lusztig and Littelmann (path model).

Advertisement: SAGE 3.0 has Support for Crystals !

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Weyl Group Multiple Dirichlet Series

Given a root system and a field F containing the n-th roots ofunity, we may (always or often) construct a family of multipleDirichlet series

ci

H(c1, , cr; m1, , mr)i=1

r

Nci2si

having analytic continuation and functional equations in the si.

Here ci are ideals of S-integers oS (a PID for suitable S). The coefficients are twisted multiplicative. Let

bethe n-th power residue symbol. If gcd(

Ci,

Ci) = 1:

H(C1C1,

, CrCr , m1, , mr)

H(C1, , Cr, m1, , mr) H(C1,

, Cr , m1, , mr)

=

i=1

r CiCi

i2CiCi

i2i


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Three Approaches

There are about 3 approaches to defining the functions H. Asnoted above, we only need to define the p-part

H(pk1, , pkr;pl1, , plr).

The Crystal Description generalizes the Gelfand-Tsetlin description found for type Ar by Brubaker,Bump, Friedberg and Hoffstein. This definition may betranslated into sums over crystals. This works well some-times, and in other cases will require refinement.

The Chinta-Gunnells Description involves alternatingsums over Weyl groups analogous to the Weyl characterformula.

The Eisenstein-Whittaker Description describes the p-part in terms of metaplectic Whittaker coefficients. Inapproach, the global MDS is regarded as a Whittaker coef-ficient of an Eisenstein series.

It is not clear at the outset that these three approaches are equiv-

alent. However progress is being made towards unifying them.

For the Gelfand-Tsetlin (Crystal) description Brubaker,Bump, Friedberg proved equivalence of two versions of thedefinition, related by the Schtzenberger involution. Thisimplies analytic continuation of the Weyl Group MDS.

The method of Chinta and Gunnells now works in the mostgeneral case but only in certain cases are the coefficientsvery explicitly known. They can be computed by computerand checked to agree with other definitions.

The third method awoke in 2008 after a long nap. In at least one case all three methods are known to agree.

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The Nonmetaplectic Case

When n = 1 all three methods agree in at least two cases:

Type A: Tokuyama + Casselman-Shalika

Type C: Hamel and King and Beineke, Brubaker, FrechetteCartan type A or C when n=1 (nonmetaplectic)

For other Cartan types, the Crystal description needs moreinvestigation but the picture should remain valid.

The other links are valid for all Cartan types when n = 1. The underlying Lie group is the Langlands L-group. These Weyl Group MDS are Euler products.

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Tokuyamas formula (Type A)

Tokuyamas formula (1988) predates crystals. It was statedin the language of Gelfand-Tsetlin patterns.

The following data are equivalent (for type A):

Tableaux Gelfand-Tsetlin Patterns Vertices in Crystal Graphs

We can translate Tokuyamas formula into the crystal language.

It is clear that a Weyl character with highest weight can be expressed as a sum over the crystal graph C. Thisis not Tokuyamas formula.

Tokuyamas formula gives as a ratio with numerator asum over C+ where =

1

2

+

. (+ = pos. roots.)

Weyl Character Formula (WCF)

We return to a Lie group G with maximal torus T. The Weylcharacter formula takes place in the character ring C[] of T. If

= X(T) is a weight, we will denote its image in this char-

acter ring as e. Thus ee = e+ since we are writing the grouplaw in @ Zr additively.

Let C+ be weight in the positive Weyl chamber, and let be the character of its highest weight module. WCF asserts:

=

wW

( 1)l(w)w(e+)

wW ( 1)

l(w)w(

e

)

(W = Weyl group).

Tokuyamas formula gives a deformation of the numerator,which may be expressed as a sum over the crystal C+.

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Berenstein-Zelevinsky-Littelmann stringsFix a reduced decomposition of the long Weyl group element intosimple reflections.

w0 = si1si2 siN , si = si.

Now place a turtle at a vertex v of the crystal. The sequence

i1, i2, , iN

will be a code telling the turtle the order in which to follow theroot operators.

Let k1 be the largest integer such that Fi1k1(v) 0. In other words, the turtle can move along the i1-colored

edge a distance of k1 then no further.Then the process is repeated.

k2 is the largest integer such that Fi2k2Fi1k1v 0. The turtle winds up at the vertex with weight w0().

The Turtles Walk

v

w0()

BZL =(2, 3, 1)

The BZL string is (k1, k2, ). It uniquely determines v.

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Decorating the BZL string

Before we can state Tokuyamas theorem we must decorate theBZL string. We are in type Ar and use the decomposition

w0 = s1(s2s1)(s3s2s1) (srsr1 s2s1).

We follow Littelmann, arranging the BZL by inserting entries intoa triangular array from top to bottom and right to left.

Thus (2, 3, 1) becomes

1 23

.

0 0

0

0 0

1

1 1

0

1 2

0

1 2

1

1 3

0

1 3

1

1 3

2

1 4

0

1 4

1

1 4

2

1 4

3

0 1

0

0 1

1

0 1

2

0 2

0

0 2

1

0 2

2

0 2

3

0 3

0

0 3

1

0 3

2

0 3

3

0 3

4

We will decorate certain entries by boxing or circling them:?>=


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Boxing

The boxing rule is quite simple to understand.

If (k1, k2, k3, ) is the BZL, so that Fimkm Fi1k1(v) is thelocation of the turtle after m moves, then we box km if

eimfim1km1

fi1k1(v) = 0.

This means the turtle travels the entire length of the rootstring corresponding to the m-th move.

v

w0()

In this example, the turtle travels the entire length of thefirst segement (of length 2) and the last (of length 1) butnot the second. So we box 2 and 1 but not 3:

1 3

2

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Circling

Littelmann proved that if the the BZL pattern is arrangedin a triangle as above, the rows are nonincreasing. So

ai1 ai2 ai,r+1i 0

for the i-th row (ai1, ai2, , ai,r+1i).

If any of these inequalities is an equality, we circle the cor-responding entry. Thus if ai,j = ai,j+1 we circle ai, j or ifthe last entry vanishes we circle it.

3 52

2 43

1 34

0 25

3 51

2 42

1 33

0 24

3 41

2 32

1 23

0 14

3 50

2 41

1 32

0 23

3 40

2 31

1 22

0 13

3 30

2 21

1 12

0 03

2 40

1 31

0 22

2 30

1 21

0 12

2 20

1 11

0 02

1 30

0 211 2

0

0 11

1 10

0 01

0 20

0 10

0 00

The second row is (2, 2) with equality 2 = 2 so we circle the first 2:?>=


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Tokuyamas Theorem

We may now formulate Tokuyamas theorem in crystal terms. Ifa > 0 let g(a) = qa1, h(a) = (q 1)qa1. If v C+ define

G(v) =

aBZL(v)

qa if a is circled,g(a) if z is boxed,h(a) if neither,0 if both.

Let s be the Schur polynomial so that if g GLr+1(C) has eigen-values 1, , r+1 then

(g) = s(1, , r+1).

Tokuyamas theorem may be stated as follows:

qr2r1 r1s(1, q2, q23, , q

rr+1) =

vC+

G(v)ewt(v)

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The Gelfand-Tsetlin Description

Brubaker, Bump, Friedberg and Hoffstein gave a descrip-tion of Weyl group MDS for the n-fold cover of Type Ar interms of Gelfand-Tsetlin patterns.

If n = 1 then this boils down to Tokuyamas formula. Brubaker, Bump and Friedberg gave a proof of the func-

tional equations of the MDS by proving a difficult combina-torial statement.

The Chinta-Gunnells description also produces a Weylgroup MDS with functional equations. Conjecturally they

are equivalent but this has only been proved in certaincases.

Eisenstein series on the n-fold metaplectic cover of GLr+1give yet another Weyl group MDS. Very recently it wasrelated to Gelfand-Tsetlin description by pushingthrough a formerly intractable computation (BBF).

These are cases where Whittaker models are not unique. The reformulation of the Gelfand-Tsetlin description for

type A is just a paraphrase but it is a very suggestive onethat points the way to generalizations.

Crystals are well adapted to Kac-Moody frameworkand there is good hope that Weyl group multipleDirichlet series can be extended to this setting.

A recent preprint of Bucur and Diaconu in functionfield case shows that a Weyl group multiple Dirichlet seriesfor the affine Weyl group D4

(1)produces an object resem-

bling the infinite Weyl-denominator, supporting this hope.

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The p-part in Type A

Let the ground field contain the n-th roots of unity, and define:

g(m, c) = a m o d c(a,c)=1

a

c ma

c , = additive char.

ac

= powerresidue

symbol.

Fix p and for a > 0 let g(a) = g(pa1, pa), h(a) = g(pa, pa).

If n = 1 these reduce to g(a) = qa1, h(a) = (q 1)qa1as in Tokuyamas formula. Now q= N p.

The p-part of the Weyl group MDS is given by exactly the sameformula as in Tokuyamas recipe, multiplied by ew0(). It is

vC+

G(v)ewt(v+w0()), G(v) =

aBZL(v)

qa if a is circled,g(a) if z is boxed,h(a) if neither,0 if both.

The weight ewt(v+w0()) may be interpreted asi=1

r

Np2isi where

i is the number of steps of color i in the turtles walk.

The Weyl denominator as in Tokuyamas formulabecomes the normalizing factor.

+

(1

qnki()(2si1)1)1,

where =

ki() i (i = simple roots).

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A Riddle

The Br theory when n = 2 is particularly striking so we willdescribe it. This is related to Whittaker models on the meta-plectic double cover Sp(2r,Qp) of Sp(2r). But first, a riddle.

What is the L-group of Sp(2r,Qp) ?

The L-group of Sp(2r) is SO(2r + 1). Langlands did not define an L-group for metaplectic

groups, but there are reasons to say that if G = Sp(n)(2r) isthe n-th metaplectic cover, then

LG =

SO(2r + 1) if n is odd,Sp(2r) if n is even.

Evidence: Savin computed (affine) Iwahori Hecke algebras(IHA) of the n-fold covers of semisimple split groups, andthe IHA of the n cover of Sp(2r) is isomorphic to the IHAof Sp(2r) if n is odd or spin(2r + 1) if if n is even.

Evidence: Andrianov and Zhuravlev gave Rankin-Selbergconstructions that on Sp(n)(2r) produce a degree 2r + 1 L-function when n = 1 and a degree 2r L-function when n = 2.

A Paradox

Yet for Sp(2)(2r) it is orthogonal Br crystals we employand we will encounter representations of SO(2r + 1,C).

This is most unexpected for the above reasons. Both representations of SO(2r + 1, C) and Sp(2r, C) will

appear.

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Three Descriptions

Equivalence of the three descriptions is (nearly) proved. Inpreparation: Brubaker, Bump, Chinta and Gunnells. Priorwork: Bump, Friedberg, Hoffstein (Duke 1991).

In this case, Whittaker models are unique. Eisenstein-Whittaker description (BFH 1991) expresses the

spherical Whittaker function as sum over the Weyl group ofeither Br = SO(2r + 1) or Cr = Sp(2r).

The Weyl groups are the same so take your pick.

But the in the Weyl character formula tells us the com-

putation is related to the representation theory of Sp(2r),not SO(2r + 1).

Or does it? We will come back to this point.

The Chinta-Gunnells description is also a sum over the

same Weyl group, and is equivalent to BFH 1991 formula.

The equivalence of the Crystal description to Eisenstein-Whittaker description amenable to proof by induction.

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The Br Crystal

Here is a B2 crystal.(Br works the same way.)

w0 = s1s2s1s2(red, green, red, green)

BZL(v) =

?>==


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The p-Part for Type Br

The r axial entries (here a1 and a3) correspond to the shortroots.

long long short long longlong short long

short

If = (x) is the root correspond to the turtle walk of length x:

G(v) =

xL

g(x) if x is boxed but not circled,h(x) if x neither circled nor boxed,

q if x is circled but not boxed,0 if x is both circled and boxed.

Here g is a quadratic Gauss sum for the short roots but aRamanujan sum for the long roots.

Both can be evaluated explicitly. Assuming the groundfield contains the 4-th roots of unity,

g(m) =

qm1 for long roots , qm1 for short roots , m even,q(m

1

2)

for short roots , m odd,

h(m) =

(q 1)qm1 for long roots ,(q 1)qm1 for short roots , m even,0 for short roots , m odd,

Note that square roots of q appear: g(1)= q if short.

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Type B2Let be a dominant weight, and take the B2 crystal with highestweight vector + . Take the sum of

G()xa2+a4ya1+a3.

Then we obtain

(1x)(1+ q1/2y)(1+ q3/2xy)(1 q2xy2)times a polynomial P(x, y; q1/2) that is given by the followingtable (i are the fundamental weights).

P(x, y; q

)

0 11 1 + q x q3/2 x y + q3 x y2 + q4 x2 y22 (1 q1/2y)(1 q3/2xy)

1 + 2 (1 + qx)(1 q 1/2y)(1 q3/2xy)(1+ q3xy2)

21

x4y4q8 + x3y4q7 + x2y4q6x3y3q11/2+ x3y2q5x2y3q9/2 + 2x2y2q4

+ x y2q3

x2y q5/2 + x2q2

x y q3/2 + x q + 1

22

x2y4q6x2y3q9/2 + x2y2q4x y3q7/2+ x y2q3 + x y2q2+ y2q2x y q3/2 y q1/2 + 1

When we specialize q 1, these polynomials becomecharacters of spin(5). Paradoxical: This is in contrast with the popular beliefthat the L-group of Sp(2r) is Sp(2r).

Paradoxical: Also in contrast with this popular belief isthe fact that we used a type B (orthogonal) crystal.

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Type B3Here are the corresponding polynomials for B3:

P(x, y , z, q

)

0 11 = (1, 0, 0)

1 + qz + q2yz q5/2xyz+ q4x2yz + q5x2y2z + q6x2y2z2

2 = (1, 1, 0)

1 + q y q3/2x y + q3x2y+ q4x2y2 + q2y z q5/2x y z + q4x2y z q7/2x y2z + q4x2y2z + 2 q5x2y2z

q11/2x3y2z + q6x2y3z

q13/2x3y3z

+ q8x4y3z + q6x2y2z2 + q7x2y3z2

q15/2x3y3z2 + q9x4y3z2 + q10 x4y4z2

3 =

12

,12

,12

(1 q x)(1 q3/2xy)(1 q5/2xyz)

Again, when q

1, these become the characters of irre-ducible representations of spin(7).

The paradox is that these polynomials are defined in termsof the representation theory of Sp(2r) yet their crystalinterpretation involves type Br (orthogonal) crystals andthe expressions specialize to characters of Spin(2r + 1)when q

1.

We explain the paradox by proving that this always hap-pens for all r.

The explanation does not dispel the mystery.

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Paradox Explained, Mystery not DispelledAfter q1/2 1 specialization P becomes

Sp(2r)1 A

1k1+r2

k2+r1

rkr+r

i=1

r

(1 + i1)

=

A

1k1+r

1

22k2+r

3

2

rkr+

1

2

i=1r (1/2 + i

1/2)

Sp(2r)

.

The product inside is invariant under the Weyl group action andso we can pull it out, and using

i=1

r

(1/2

+ i1/2

)

SO(2r+1)

Sp(2r) = 1we get

SO(2r+1)(k1, , kr).

The BFH formula will need to be extended to GSp beforewe see the spin characters.

This shows that when q1/2 1 this symplectic alter-nating sum becomes orthogonal. This explanation of the paradox raises more questionsthan it answers.

Wanted: Quantum interpretation. Wanted: Connection with symmetric function theory

(Hall-Littlewood polynomials, etc.)

Documents

Brubaker, Bump, Chinta, Friedberg and Gunnells- Crystals and Multiple Dirichlet Series