Paul E. Gunnells- Weyl group multiple Dirichlet series

8/3/2019 Paul E. Gunnells- Weyl group multiple Dirichlet series

1/33

Weyl group multiple Dirichlet series

Paul E. Gunnells

UMass Amherst

August 2010

Paul E. Gunnells (UMass Amherst) Weyl group multiple Dirichlet series August 2010 1 / 33


2/33

Basic problem

Let be an irreducible root system of rank r.

Our goal: explain general construction of multiple Dirichlet series in rcomplex variables s = (s1, . . . , sr)

Z(s) =

c1,...,cr

a(c1, . . . , cr)cs11 . . . c

srr

satisfying a group of functional equations isomorphic to the Weylgroup W of .

The functional equations intermix all the variables, and are closelyrelated to the usual action of W on the space containing .



3/33

Example

Let = A2, W = 1, 2 | 2i = 1, 121 = 212. The desiredfunctional equations look like

1 : s1 2 s1, s2 s1 + s2 1, 2 : s1 s1 + s2 1, s2 2 s2s1=s2

s1=1

s2=1

s1+s2=2

s1+2s2=3

2s1+s2=3

e1

2

12

21121



4/33

Why?

Such series provide tools for certain problems in analytic numbertheory (moments, mean values, . . . ).

Conjecturally these series arise as FourierWhittaker coefficients ofEisenstein series on metaplectic groups

1 n

G(AF) G(AF) 1This has been proved in some cases (type A and type B (doublecovers)).

The series are built out of arithmetically interesting data, such as

Gauss sums, nth power residue symbols, Hilbert symbols, and(sometimes) L-functions.

The objects that arise in the construction have interestingrelationships with combinatorics, representation theory, and

statistical mechanics.Paul E. Gunnells (UMass Amherst) Weyl group multiple Dirichlet series August 2010 4 / 33


5/33

Maass and the half-integral weight Eisenstein series

Let E(z, s) be the half-integral weight Eisenstein series on 0(4):

E(z, s) =

\0(4)

j1/2(, z)1(z)s/2.

Maass showed that its dth Fourier coefficient is essentially

L(s, d),

where d is the quadratic character attached to Q(

d/Q).

Essentially means up to the Euler 2-factor, archimedian factors, andcertain correction factors that have to be inserted when d isntsquarefree.



6/33

Siegel, GoldfeldHoffstein

Siegel (1956), GoldfeldHoffstein (1985):

Z(s, w) =

0

(E(iy,s/2) const term) yw dyy

.

The result is a Dirichlet series roughly of the form

Z(s, w) d

L(s, d)dw

.

This behaves well in s since its built from the Dirichlet L-functions,and it turns out to have nice analytic properties in w as well.

GoldfeldHoffstein used this to get estimates for sums like

|d|


7/33

Siegel, GoldfeldHoffstein

Z(s, w) satisfies a functional equation in s, again because of theDirichlet L-functions. But it turns out that it satisfies extra functionalequations.

In fact, Z satisfies a group of 12 functional equations, and is an

example of a Weyl group multiple Dirichlet series of type A2. There isa subgroup of functional equations isomorphic to S3 = W(A2), and anextra one swapping s and w that corresponds to the outerautomorphism of the Dynkin diagram:

s w



8/33

Connection to A2

Why is this series related to root system A2 (besides the fact that thereare two variables and I drew the picture that way)?

Imagine expanding the L-functions in the rough definition:

Z(s, w) =d

L(s, d)dw

=d

dwc

dc

cs =

d,c

dc

csdw.

c ddc



9/33

The general shape

Heuristically, the multiple Dirichlet series looks like

Z(s) =

c1,...,cr

a(c1, . . . , cr)

cs11 . . . csrr

where a(c1, . . . , cr) is a product of nth power residue symbolscorresponding to the edges of the Dynkin diagram.

For instance D4, n = 2 leads to a series related to the third moment ofquadratic Dirichlet L-functions.

c1

c2

c3c4



10/33

Setup

F number field with 2nth roots of unity

S set of places of F containing archimedian, ramified, and such

that OS is a PID irreducible simply-laced root system of rank r

{1, . . . , r} the simple rootsm = (m1, . . . , mr) r-tuple of integers in OSs = (s1, . . . , sr) r-tuple of complex variables



11/33

Setup

FS =

vSFv

M() certain finite-dimensional space of complex-valued

functions on (FS )r (to deal with Hilbert symbols and units)

M()H(c;m) to be specified later . . . this is the most important partof the definition



12/33

The multiple Dirichlet series

Then the multiple Dirichlet series looks like

Z(s;m, ; , n) =c

H(c;m)(c) |ci|si ,

where c = (c1, . . . , cr) and each ci ranges over (OS {0})/OS.



13/33

The function H

The coefficients H have to be carefully defined to guarantee that Zsatisfies the desired group of functional equations. Generalconsiderations tell us how to define H in the following cases:

When c1 cr and c1 cr are relatively prime, one uses atwisted multiplicativity to construct H(cc;m) from H(c;m)

and H(c;m). One puts

H(cc;m) = (c, c)H(c;m)H(c;m),

where (c, c) is a root of unity built out of residue symbols and

root data:

(c, c) =r

i=1

cici

cici

ij

cicj

cicj

.



14/33

The function H

When (c1 cr, m1 mr) = 1, we can define H(c;mm) in termsof H(c;m) and certain power residue symbols:

H(c;mm) =r

j=1

mjcj

H(c;m)



15/33

The function H

So we reduce the definition of H to that of

H( k1 , . . . ,

kr ; l1 , . . . ,

lr),

where is a prime in OS.This leads naturally to the generating function

N = N(x1, . . . , xr)

= k1,...,kr0

H( k1 , . . . ,

kr ; l1 , . . . ,

lr)xk11

xkrr

(m is fixed). One can ask what properties this series has to satisfy sothat one can prove Z satisfies the right group of functional equations.



16/33

The function N

N = N(x1, . . . , xr) =

k1,...,kr0

H( k1 , . . . ,

kr)xk11 xkrr .

If one puts xi = qsi , where q = |OS/

|, then one can see that theglobal functional equations imply N must transform a certain wayunder a certain W-action.

This leads to a connection with characters of representations of g, the

simple complex Lie algebra attached to .In this relationship the monomials correspond to certain weight spaces.



17/33

Building N

The connection with characters leads to two approaches to defining N:

Crystal graphs. These are models for g representations that havevarious combinatorial incarnations (GelfandTsetlin patterns,tableaux, Proctor patterns, Littlemann path model, . . . ). One

tries to extract a statistic from the combinatorial model to definethe coefficients of N. (BrubakerBumpFriedberg,BeinekeBrubakerFrechette, ChintaPG)

Weyl character formula. This is an explicit expression for a givencharacter as a ratio of two polynomials. We take this approach

and define a deformation of Weyls formula that reflects themetaplectacity (metaplectaciousness?) of the setup. (ChintaPG,BucurDiaconu)



18/33


19/33

Deformation of the WCF

Our goal now is to define the W-action leading to H. For theapplication to multiple Dirichlet series, we normalize things slightlydifferently. Thus we work with the root lattice, introduce someq = |OS/

| powers, shift the character around, . . .

root lattice of

d : Z height function on the rootsA C[x11 , . . . , x1r ] complex group ring of (xi i)A

C(x1, . . . , xr) fraction field ofA

= +

lii a strictly dominant weight (recall that weredefining H(c;m) when m = ( l1 , . . . ,

lr))



20/33

The action on monomials

We let the Weyl group act on monomials through a change ofvariables map. This is essentially the same as the geometric action of

W on the root lattice (except for the q power).If f(x) = x, we put

f(wx) = qd(w1)xw

1.



21/33

Affine action ofW

Given any , we put

w = w( ) + ,

where the action on the right hand side is the usual action on the rootlattice. This just performs an affine reflection of R (the same asthe usual w reflection but shifted to have center ).

If i is a simple reflection, we put

i() = d(i ).This is just the multiple of i needed to go from to i .



22/33

Affine action ofW

1



23/33

Gauss sums

Choose some complex numbers (i), i = 1, . . . , n 1 such that(i)(n i) = 1/q. Put (0) = 1.Ultimately these numbers will be Gauss sums (the same onesappearing in the metaplectic cocycle), but actually any complexnumbers satisfying these relations will work.

Extend (i) to all integers by reducing i mod n.



24/33

Homogeneous decomposition

The action on a monomial f(x) = x depends on the congruence classof the monomial mod n.

To treat general rational functions, we decompose A into homogeneousparts

A =

/nA.

A consists of those rational function f/g where all monomials in g liein n and those in f map to modulo n.

e.g., 1 xyx2 y2 =

1

x2 y2 xy

x2 y2



25/33

Finally

Theorem (ChintaPG) Suppose f A. Let i be a simplereflection and let (k)n be the remainder upon division of k by n. Then

(f|i)(x) = (qxi)li+1(i())n 1 1/q1 qn1xni f(ix) (P)

(i()) (qxi)li+1n 1 (qxi)n

(1 qn1xni )f(ix) (Q)

extends to a W-action on C(x1, . . . , xr).



26/33

The W-action



27/33


28/33

The W-action

x ix



29/33

The W-action

x ix

P

Q


M ki h l i l Di i hl i


30/33

Making the multiple Dirichlet series

Theorem (ChintaPG)

Put (x) =

>0(1 qnxn) and D(x) =>0(1 qn1xn).Then

h(x) =wW

(1|w)(x)(wx)

is a rational function such that hD is a polynomial.Let N = hD, define H by

N =

k1,...,kr0

H( k1 , . . . ,

kr ; l1 , . . . ,

lr)xk11 xkrr ,

and specialize the (i) to the appropriate Gauss sums. Then theresulting multiple Dirichlet series Z(s;m, ; , n) has analyticcontinuation to Cr and satisfies a group of functional equationsisomorphic to W.


A l ( 2)


31/33

A2 examples (n = 2)

Here g1 = q(1) and the notation (a, b) means

= (a + 1)1 + (b + 1)2.

(0, 0): 1 + g1

x + g1

y

g1

qx2y

g1

qxy2

q2x2y2

(1, 0): 1 qx2 + g1y g1qx2y + g1q2x2y + q3x3y g1q3x2y3 q4x3y3(1, 1): 1 qx2 qy2 + q2x2y2 q3x2y2 + q4x4y2 + q4x2y4 q5x4y4(2, 1):1

qx2 +q2x2 +g1

q2x3

qy2 +q2x2y2

2q3x2y2 +q4x2y2

g

1q3x3y2 +

g1q4x3y2 +q4x4y2q5x4y2g1q5x5y2 +q4x2y4q5x2y4g1q5x3y4 +

g1q6x3y4 q5x4y4 + q6x4y4 + g1q6x5y4 g1q7x5y4 + q7x3y5 q8x5y5


O ti


32/33

Open questions

The WCF method works for all , whereas the crystal graphapproach has only been worked out for some (classical) . Canone do the latter for all uniformly? (KimLee, McNamara)

Prove that Z is a Whittaker coefficient of a metaplectic Eisenstein

series. (ChintaOffen)Prove that the crystal graph descriptions and the WCFdescriptions coincide. (ChintaOffen + McNamara)

Develop multiple Dirichlet series on affine Weyl groups and

crystallographic Coxeter groups (BucurDiaconu, Lee)What is the geometric interpretation of Weyl group multipleDirichlet series over function fields?


R f


33/33

References

Gautam Chinta and PG, Weyl group multiple Dirichlet seriesconstructed from quadratic twists, Invent. Math. 167 (2007), no.2,327353.

Gautam Chinta, Sol Friedberg, and PG, On the p-parts ofquadratic Weyl multiple Dirichlet series, J. Reine Angew. Math.623 (2008), 123.

Gautam Chinta and PG, Weyl group multiple Dirichlet series oftype A2, to appear in the Lang memorial volume.

, Constructing Weyl group multiple Dirichlet series, J.Amer. Math. Soc. 23 (2010), 189215.


Paul E. Gunnells- Weyl group multiple Dirichlet series

Documents