Multiple Dirichlet Series

Gautam Chinta

2013-02-18 Mon

Previous talks of Sol Friedberg

1. Fourier expansion of (metaplectic) GL2 Eisenstein series

2. The d th Fourier coefficient is a Dirichlet series withI functional equationsI analytic continuation

3. These properties came from the analytic properties of theEisenstein series

Previous talks of Sol Friedberg

1. Fourier expansion of (metaplectic) GL2 Eisenstein series

2. The d th Fourier coefficient is a Dirichlet series withI functional equationsI analytic continuation

3. These properties came from the analytic properties of theEisenstein series

Previous talks of Sol Friedberg

1. Fourier expansion of (metaplectic) GL2 Eisenstein series

2. The d th Fourier coefficient is a Dirichlet series withI functional equationsI analytic continuation

3. These properties came from the analytic properties of theEisenstein series

This talk

I Focus on n = 2: Fourier coeffs now involve quadraticDirichlet L-functions:

E (z , s) = ∗+∑d 6=0

L(s ′, χd)e2πixdWs(dy)

I Mellin transform produces a “double Dirichlet series”∑d 6=0

L(s, χd)

|d |w

I Properties can be established independent of theory ofEisenstein series (mostly)

I “Axiomatic development” of the theory of WMDS

This talk

I Focus on n = 2: Fourier coeffs now involve quadraticDirichlet L-functions:

E (z , s) = ∗+∑d 6=0

L(s ′, χd)e2πixdWs(dy)

I Mellin transform produces a “double Dirichlet series”∑d 6=0

L(s, χd)

|d |w

I Properties can be established independent of theory ofEisenstein series (mostly)

I “Axiomatic development” of the theory of WMDS

This talk

I Focus on n = 2: Fourier coeffs now involve quadraticDirichlet L-functions:

E (z , s) = ∗+∑d 6=0

L(s ′, χd)e2πixdWs(dy)

I Mellin transform produces a “double Dirichlet series”∑d 6=0

L(s, χd)

|d |w

I Properties can be established independent of theory ofEisenstein series (mostly)

I “Axiomatic development” of the theory of WMDS

This talk

I Focus on n = 2: Fourier coeffs now involve quadraticDirichlet L-functions:

E (z , s) = ∗+∑d 6=0

L(s ′, χd)e2πixdWs(dy)

I Mellin transform produces a “double Dirichlet series”∑d 6=0

L(s, χd)

|d |w

I Properties can be established independent of theory ofEisenstein series (mostly)

I “Axiomatic development” of the theory of WMDS

Transition to local parts

I Emphasis on construction of local parts. Two methods:I Functional Equations (Chinta/Gunnells)I Crystal basis description (Brubaker, Bump, Friedberg)

I Two methods are (more or less) equivalent when theyboth apply, but it’s not so easy to see

Transition to local parts

I Emphasis on construction of local parts. Two methods:I Functional Equations (Chinta/Gunnells)I Crystal basis description (Brubaker, Bump, Friedberg)

I Two methods are (more or less) equivalent when theyboth apply, but it’s not so easy to see

A2 quadratic double Dirichlet series

Goal: construct a two variable Dirichlet series which is roughlyof the form

Z (s,w) =∑d

L(s, χd)

dw

We will require that this series satisfy two functional equationsas

I (s,w) 7→ (1− s, s + w − 1/2)

I (s,w) 7→ (s + w − 1/2, 1− w)

Where do these functional equations come from?

A2 quadratic double Dirichlet series

Goal: construct a two variable Dirichlet series which is roughlyof the form

Z (s,w) =∑d

L(s, χd)

dw

We will require that this series satisfy two functional equationsas

I (s,w) 7→ (1− s, s + w − 1/2)

I (s,w) 7→ (s + w − 1/2, 1− w)

Where do these functional equations come from?

A2 quadratic double Dirichlet series

Goal: construct a two variable Dirichlet series which is roughlyof the form

Z (s,w) =∑d

L(s, χd)

dw

We will require that this series satisfy two functional equationsas

I (s,w) 7→ (1− s, s + w − 1/2)

I (s,w) 7→ (s + w − 1/2, 1− w)

Where do these functional equations come from?

Background on quadratic Dirichlet L-functionsDefine

L(s, χd0) =∞∑n=1

χd0(n)

ns=∏p

(1− χd0(p)

ps

)−1

I Euler product

I Analytic continuationI For d0 a fundamental discriminant, L(s, χd0) is entire

unless d0 = 1

I Functional equationI For d0 a fundamental discriminant,

L(s, χd0) = ∗|d |1/2−sL(1− s, χd0)

But we need to define for all d . How to do this withoutmessing up the earlier properties?

Background on quadratic Dirichlet L-functionsDefine

L(s, χd0) =∞∑n=1

χd0(n)

ns=∏p

(1− χd0(p)

ps

)−1

I Euler product

I Analytic continuationI For d0 a fundamental discriminant, L(s, χd0) is entire

unless d0 = 1

I Functional equationI For d0 a fundamental discriminant,

L(s, χd0) = ∗|d |1/2−sL(1− s, χd0)

But we need to define for all d . How to do this withoutmessing up the earlier properties?

Background on quadratic Dirichlet L-functionsDefine

L(s, χd0) =∞∑n=1

χd0(n)

ns=∏p

(1− χd0(p)

ps

)−1

I Euler product

I Analytic continuationI For d0 a fundamental discriminant, L(s, χd0) is entire

unless d0 = 1

I Functional equationI For d0 a fundamental discriminant,

L(s, χd0) = ∗|d |1/2−sL(1− s, χd0)

But we need to define for all d . How to do this withoutmessing up the earlier properties?

Weighting polynomials

Define L(s, d) = L(s, χd0) · P(s, d) for P a weightingpolynomial.

What properties do we want this weighting polynomial tohave?

I Euler product: P(s, d) =∏

pα||d Pp(p−s , d)

I functional equation: P(s, d) = (d/d0)1/2−sP(1− s, d).We want this so that L(s, d) = ∗d1/2−sL(1− s, d)

If this last condition is satisfied, then the double Dirichlet serieswill have a functional equation (s,w) 7→ (1− s, s + w − 1/2)

Weighting polynomials

Define L(s, d) = L(s, χd0) · P(s, d) for P a weightingpolynomial.

What properties do we want this weighting polynomial tohave?

I Euler product: P(s, d) =∏

pα||d Pp(p−s , d)

I functional equation: P(s, d) = (d/d0)1/2−sP(1− s, d).We want this so that L(s, d) = ∗d1/2−sL(1− s, d)

If this last condition is satisfied, then the double Dirichlet serieswill have a functional equation (s,w) 7→ (1− s, s + w − 1/2)

Weighting polynomials

Define L(s, d) = L(s, χd0) · P(s, d) for P a weightingpolynomial.

What properties do we want this weighting polynomial tohave?

I Euler product: P(s, d) =∏

pα||d Pp(p−s , d)

I functional equation: P(s, d) = (d/d0)1/2−sP(1− s, d).We want this so that L(s, d) = ∗d1/2−sL(1− s, d)

If this last condition is satisfied, then the double Dirichlet serieswill have a functional equation (s,w) 7→ (1− s, s + w − 1/2)

Passage to p-parts

Two types of p-parts for L(s, d) = L(s, χd0) · P(s, d):

I if (p, d0) = 1:

(1− χd0(p)p−s)−1 times Pp(p−s , d)

I if p|d0,

Pp(p−s , d)

We will next describe the construction of the p-part as inChinta-Gunnells

Passage to p-parts

Two types of p-parts for L(s, d) = L(s, χd0) · P(s, d):

I if (p, d0) = 1:

(1− χd0(p)p−s)−1 times Pp(p−s , d)

I if p|d0,

Pp(p−s , d)

We will next describe the construction of the p-part as inChinta-Gunnells

A generating series

Two variable generating function

F (x , y) =∑k≥0

(1− x)−1Pp(x , p2k)y 2k +∑k≥0

Pp(x , p2k+1)y 2k+1

We need F to satisfy

1. (1− x) [F (x , y) + F (x ,−y)] and 1y

[F (x , y)− F (x ,−y)]

are invariant under (x , y) 7→(

1px, xy√p).

2. Also need 2nd functional equation

3. Limiting behavior F (x , 0) = 11−x

=⇒ F (x , y) =1− xy

(1− x)(1− y)(1− px2y 2)

A generating series

Two variable generating function

F (x , y) =∑k≥0

(1− x)−1Pp(x , p2k)y 2k +∑k≥0

Pp(x , p2k+1)y 2k+1

We need F to satisfy

1. (1− x) [F (x , y) + F (x ,−y)] and 1y

[F (x , y)− F (x ,−y)]

are invariant under (x , y) 7→(

1px, xy√p).

2. Also need 2nd functional equation

3. Limiting behavior F (x , 0) = 11−x

=⇒ F (x , y) =1− xy

(1− x)(1− y)(1− px2y 2)

A generating series

Two variable generating function

F (x , y) =∑k≥0

(1− x)−1Pp(x , p2k)y 2k +∑k≥0

Pp(x , p2k+1)y 2k+1

We need F to satisfy

1. (1− x) [F (x , y) + F (x ,−y)] and 1y

[F (x , y)− F (x ,−y)]

are invariant under (x , y) 7→(

1px, xy√p).

2. Also need 2nd functional equation

3. Limiting behavior F (x , 0) = 11−x

=⇒ F (x , y) =1− xy

(1− x)(1− y)(1− px2y 2)

Definition of a group action

Let f be the monomial xaybzc . Define

(σ1f )(x , y , z) =

{f (y , x , z) if a − b evenf (y , x , z) x2−tx1

x1−tx2 if a − b odd

and define σ2f similarly.

Extend to polynomials by linearity and then to rationalfunctions. This will define an action of S3 on rationalfunctions.

Definition of a group action

Let f be the monomial xaybzc . Define

(σ1f )(x , y , z) =

{f (y , x , z) if a − b evenf (y , x , z) x2−tx1

x1−tx2 if a − b odd

and define σ2f similarly.Extend to polynomials by linearity and then to rationalfunctions. This will define an action of S3 on rationalfunctions.

Construction of p-parts

For a ≥ b ≥ c define the polynomials

Na,b,c(x , y , z) =

∑w∈W (−1)length(w)w(xa+2yb+1zc)

∆(x)· D(x)

where

D(x) =∏i<j

(x2i − t2x2j ) and ∆(x) =∏i<j

(x2i − x2j ).

These polynomials arise in the construction of the A2 multipleDirichlet series. They also arise as Whittaker functions on thedouble cover of GL3.

Construction of p-parts

For a ≥ b ≥ c define the polynomials

Na,b,c(x , y , z) =

∑w∈W (−1)length(w)w(xa+2yb+1zc)

∆(x)· D(x)

where

D(x) =∏i<j

(x2i − t2x2j ) and ∆(x) =∏i<j

(x2i − x2j ).

These polynomials arise in the construction of the A2 multipleDirichlet series. They also arise as Whittaker functions on thedouble cover of GL3.

Comparison with the Gelfand-Tsetlin formula

Will describe the “right-leaning” Gelfand-Tsetlin formula ofBBFH (I think).

Let I be the GT patterna00 a01 a02

a11 a12a22

.

Define the weight of I to be

(a00 + a01 + a02 − a11 − a12, a11 + a12 − a22, a22)

and “(right sum) — (up-and-right sum)” coordinates

r11 = (a11 − a01) + (a12 − a02), r12 = a12 − a02, r22 = a22 − a12.

Comparison with the Gelfand-Tsetlin formula

Will describe the “right-leaning” Gelfand-Tsetlin formula ofBBFH (I think). Let I be the GT pattern

a00 a01 a02a11 a12

a22

.

Define the weight of I to be

(a00 + a01 + a02 − a11 − a12, a11 + a12 − a22, a22)

and “(right sum) — (up-and-right sum)” coordinates

r11 = (a11 − a01) + (a12 − a02), r12 = a12 − a02, r22 = a22 − a12.

Comparison with the Gelfand-Tsetlin formula

Will describe the “right-leaning” Gelfand-Tsetlin formula ofBBFH (I think). Let I be the GT pattern

a00 a01 a02a11 a12

a22

.

Define the weight of I to be

(a00 + a01 + a02 − a11 − a12, a11 + a12 − a22, a22)

and “(right sum) — (up-and-right sum)” coordinates

r11 = (a11 − a01) + (a12 − a02), r12 = a12 − a02, r22 = a22 − a12.

GT-formula (cont.)

We say

I I is nonstrict if the rows of I are not strictly decreasing

I I is right-leaning at (i , j) if ai ,j = ai−1,jI I is left-leaning at (i , j) if ai ,j = ai−1,j−1

Define

G (I ) =

{0 if I is nonstrict

G (I ; 1, 1)G (I ; 1, 2)G (I ; 2, 2) otherwise

where

left-leaning right-leaning non-leaningrij even −t2 1 1− t2

rij odd t 1 0

GT-formula (cont.)

We say

I I is nonstrict if the rows of I are not strictly decreasing

I I is right-leaning at (i , j) if ai ,j = ai−1,jI I is left-leaning at (i , j) if ai ,j = ai−1,j−1

Define

G (I ) =

{0 if I is nonstrict

G (I ; 1, 1)G (I ; 1, 2)G (I ; 2, 2) otherwise

where

left-leaning right-leaning non-leaningrij even −t2 1 1− t2

rij odd t 1 0

GT-formula (cont.)

We say

I I is nonstrict if the rows of I are not strictly decreasing

I I is right-leaning at (i , j) if ai ,j = ai−1,jI I is left-leaning at (i , j) if ai ,j = ai−1,j−1

Define

G (I ) =

{0 if I is nonstrict

G (I ; 1, 1)G (I ; 1, 2)G (I ; 2, 2) otherwise

where

left-leaning right-leaning non-leaningrij even −t2 1 1− t2

rij odd t 1 0

GT-formula (cont.)

Finally we define

Ma,b,c(x , y , z) =∑I

G (I )xwt(I )

where the sum is over all GT -patterns with top rowa + 2, b + 1, c .

“Conjecture” For all a ≥ b ≥ c we have Na,b,c = Ma,b,c .

(Follows from work of McNamara, Chinta-Offen,Chinta-Gunnells. But more direct proof??)

GT-formula (cont.)

Finally we define

Ma,b,c(x , y , z) =∑I

G (I )xwt(I )

where the sum is over all GT -patterns with top rowa + 2, b + 1, c .

“Conjecture” For all a ≥ b ≥ c we have Na,b,c = Ma,b,c .

(Follows from work of McNamara, Chinta-Offen,Chinta-Gunnells. But more direct proof??)

GT-formula (cont.)

Finally we define

Ma,b,c(x , y , z) =∑I

G (I )xwt(I )

where the sum is over all GT -patterns with top rowa + 2, b + 1, c .

“Conjecture” For all a ≥ b ≥ c we have Na,b,c = Ma,b,c .

(Follows from work of McNamara, Chinta-Offen,Chinta-Gunnells. But more direct proof??)

Other problems

1. Construct/study G2 quadratic Dirichlet seriesI Weighted sum of quadratic Dirichlet L-functionsI What is it counting?

2. SU(3) Gauss sums (as in McNamara)I Can be used to construct multiple Dirichlet seriesI What are they?

3. Function fieldsI Multiple Dirichlet series are rational functions

Other problems

1. Construct/study G2 quadratic Dirichlet seriesI Weighted sum of quadratic Dirichlet L-functionsI What is it counting?

2. SU(3) Gauss sums (as in McNamara)I Can be used to construct multiple Dirichlet seriesI What are they?

3. Function fieldsI Multiple Dirichlet series are rational functions

Other problems

1. Construct/study G2 quadratic Dirichlet seriesI Weighted sum of quadratic Dirichlet L-functionsI What is it counting?

2. SU(3) Gauss sums (as in McNamara)I Can be used to construct multiple Dirichlet seriesI What are they?

3. Function fieldsI Multiple Dirichlet series are rational functions

Multiple Dirichlet series arising in other settings

Multiple Dirichlet series arise in many different settings, manyof which are not (obviously) related to WMDS, but..

I Are they related?

I Can techniques introduced/refined in the study of WMDSbe used in different settings

Multiple Dirichlet series arising in other settings

Multiple Dirichlet series arise in many different settings, manyof which are not (obviously) related to WMDS, but..

I Are they related?

I Can techniques introduced/refined in the study of WMDSbe used in different settings

Some other sources of multiple Dirichlet series

1. Point evaluations of higher rank Eisenstein seriesI Representation numbers of quadratic formsI Count integral points on flag varietiesI Predict or give evidence for automorphic liftings

2. Zeta functions of prehomogeneous vector spacesI Well-developed theory for reductive groupsI Rich arithmetic theory evident in spaces coming from

parabolic subgroups of reductive spaces

Some other sources of multiple Dirichlet series

1. Point evaluations of higher rank Eisenstein seriesI Representation numbers of quadratic formsI Count integral points on flag varietiesI Predict or give evidence for automorphic liftings

2. Zeta functions of prehomogeneous vector spacesI Well-developed theory for reductive groupsI Rich arithmetic theory evident in spaces coming from

parabolic subgroups of reductive spaces