Andr´e’s reflection principle and functional equations in number theory

Andr´ e’s reflection principle and functional equations in number theory

RedaCHHAIBI

IMT, France

23 June 2016, MADACA

Sommaire

1 Introduction: the rank 1 story

2 Higher rank (Non-Archimedean)

3 The probabilistic proof of the Shintani-Casselman-Shalika formula

4 References

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 2 / 21

Introduction

Rank 1 example: Let us explain the relationship between Andr´e’s reflection principle:

Figure:Andr´e’s reflection principle (src:Wikipedia)

The functional equation in number theory

∀s∈C,Λ(s) = Λ(1−s) where

Λ(s) =π^−s2Γ ^s₂ ζ(s) P∞ 1

(Variant of) Andr´ e’s reflection principle

Letz>1 andW^(z)be SRW such that:

∀t, P

W_t+1^(z)−W_t^(z)= 1

= 1−P

W_t+1^(z)−W_t^(z)=−1

= z

z+z⁻¹ >1 2 andkilledupon exitingN.

Lemma

P^x

W^(z)survives

= 1−z^−2(x+1)

Corollary (Probabilistic representation ofSL₂-characters) s_(x)

z,z⁻¹

:=z^x+1−z^−(x+1)

z−z⁻¹ = z^x+1 z−z⁻¹Px

W^(z)survives where sλis the Schur function associated to the shapeλ.

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 4 / 21

(Variant of) Andr´ e’s reflection principle - Geometric case

Letµ >0 andW^(µ)be the sub-Markovian process onRwith generator:

L^(µ)=1

2∂x²+µ∂x−2e^−2x

i.e Brownian motion with driftµandkilling measureby a potentialV(x) = 2e^−2x. Proposition (Probabilistic representation of theK-Bessel/Whittaker function) If

Kµ(y) = Z ∞

0

dt

t t^µe^−y(^t+t−1) then

Kµ

e^−2x

= Γ(µ)e^µxP^x

W^(µ)survives

Remark: Kµ e^−2x

plays the role of character associated to aSL2-geometric crystals. In all cases, symmetric and analytic extension of survival probabilities.

Functional equation I: The higher peak of Eisenstein series

H={=z>0};g·z= ^az+b_cz+d usual action ofSL2 onH.

Non-holomorphic Eisentein seriedefined for<(s)>2 andz=x+iy∈H: E(z;s) =1

2 X

(m,n)6=0

y^s (mz+n)^s

E^∗(z;s) =π^−sΓ(s)ζ(2s)E(z;s) (Normalization) Properties:

Automorphic: ∀g ∈SL2(Z), E^∗(g·z;s) =E^∗(z;s) .

1-periodicity inx Expansion inFourier-Whittakercoefficients:

E^∗(z=x+iy;s) =X

n∈Z

An(y;s)e^i2πxn.

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 6 / 21

Functional equation II: Computing coefficients

E^∗(z=x+iy;s) =X

n∈Z

An(y;s)e^i2πxn .

An(y;s) =

( ∗ ifn= 0 y¹²σ_s−1

2(n)K_s−1

2(π|n|y) ifn6= 0 where

Ks(y) = R∞ 0

dt

t t^se^−y(^t+t−1) (Bessel K-function) σs(n) = P

d|n n d²

s

(Normalized divisor function)

Symmetries ofA1-type:

∀s∈C, Ks(y) =K−s(y), σs(n) =σ−s(n)

(Functional equation) E^∗(s;z)−E^∗(1−s;z) = 0

Functional equation II: Computing coefficients

E^∗(z=x+iy;s) =X

n∈Z

An(y;s)e^i2πxn .

An(y;s) =

( ∗ ifn= 0 y¹²σ_s−1

2(n)K_s−1

2(π|n|y) ifn6= 0 where

Ks(y) = R∞ 0

dt

t t^se^−y(^t+t−1) (Bessel K-function) σs(n) = P

d|n n d²

s

(Normalized divisor function)

Symmetries ofA1-type:

∀s∈C, Ks(y) =K−s(y), σs(n) =σ−s(n)

(Functional equation) E^∗(s;z)−E^∗(1−s;z) = 0

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 7 / 21

Functional equation II: Computing coefficients

E^∗(z=x+iy;s) =X

n∈Z

An(y;s)e^i2πxn .

An(y;s) =

( ∗ ifn= 0 y¹²σ_s−1

2(n)K_s−1

2(π|n|y) ifn6= 0 where

Ks(y) = R∞ 0

dt

t t^se^−y(^t+t−1) (Bessel K-function) σs(n) = P

d|n n d²

s

(Normalized divisor function)

Symmetries ofA1-type:

∀s∈C, Ks(y) =K−s(y), σs(n) =σ−s(n)

(Functional equation) E^∗(s;z)−E^∗(1−s;z) = 0

Relationship

An important remark: σsis weakly multiplicative σs(n) =Y

p|n

σs

p^νp(n)

σs

p^k

=sk p^s,p^−s

As such, the functional equations:

E^∗(s;z) =E^∗(1−s;z); Λ(s) = Λ(1−s) (Constant term*) are implied by theA1-symmetry in:

Ks(·) The Archimedean Whittaker function sk p^s,p^−s

The non-Archimedean Whittaker function at the primep Taking the problem backwards: What if these functions were expressed as survival probabilities from the start? Then

Functional equations + Analytic continuation⇔Andr´e’s reflection principle

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 8 / 21

Relationship

An important remark: σsis weakly multiplicative σs(n) =Y

p|n

σs

p^νp(n)

σs

p^k

=sk p^s,p^−s

As such, the functional equations:

E^∗(s;z) =E^∗(1−s;z); Λ(s) = Λ(1−s) (Constant term*) are implied by theA1-symmetry in:

Ks(·) The Archimedean Whittaker function sk p^s,p^−s

The non-Archimedean Whittaker function at the primep Taking the problem backwards: What if these functions were expressed as survival probabilities from the start? Then

Functional equations + Analytic continuation⇔Andr´e’s reflection principle

Sommaire

1 Introduction: the rank 1 story

2 Higher rank (Non-Archimedean)

3 The probabilistic proof of the Shintani-Casselman-Shalika formula

4 References

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 9 / 21

Notations

We consider the case ofGLnfor concreteness and not other (split) Chevalley-Steinberg groups.

K=F^q((T)): field of Laurent series in the formal variableT with coefficients in the finite fieldFq. Ifx∈ K, then val(x) is the index of the first non-zero monomial. If k= val(x):

x =akT^k+ak+1T^k+1+. . .

withak6= 0. The ring of integers is made of elements of non-negative valuation and denotedO=F^q[[T]].

G =GLn(K): group ofK-points.

K =GLn(O): maximal compact (open) subgroup.

A=

T^−µ=diag(T^µ1,T^µ2, . . . ,T^µn)|µ∈Zⁿ ≈Zⁿ A+=

T^−λ|λ1≥λ2· · · ≥λn ≈Nⁿ

N(K) is the unipotent subgroup of lower triangular matrices.

Facts:

G =NAK (Iwasawa or “Gram-Schmidt”) G=KA+K (Cartan or “Polar”)

Number theory facts

(Langlands, Shahidi) Eisenstein series generalize fromH=SL2(R)/SO2(R) to other Lie groups

Functions onG/K ,general symmetric spaces.

We restrict ourselves to a special kind of Eisenstein serie associated to the function with paramz∈Rⁿ:

Φz(g) :=e^hz,µi wheng∈NT^µK

(Jacquet’s thesis, 1967) Fourier-Whittaker coefficients of such generalized

Einsenstein series are given by theWhittaker functionWz. LetϕN:N→(C^∗,×) be a character trivial on integer points andw0is the permutation matrix (n. . .321).

Wz(g) := Z

N

Φz(w0ng)ϕN(n)⁻¹dn is convergent forz∈C.

Important message: As in rank 1, the functional equations for Eisenstein series is implied by theW =Sn-symmetry after a normalizationWfz

∀σ∈Sn,∀g ∈G,∀z∈Rⁿ,Wgσz(g) =Wfz(g)

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 11 / 21

Number theory facts

(Langlands, Shahidi) Eisenstein series generalize fromH=SL2(R)/SO2(R) to other Lie groups

Functions onG/K ,general symmetric spaces.

We restrict ourselves to a special kind of Eisenstein serie associated to the function with paramz∈Rⁿ:

Φz(g) :=e^hz,µi wheng∈NT^µK

(Jacquet’s thesis, 1967) Fourier-Whittaker coefficients of such generalized

Einsenstein series are given by theWhittaker functionWz. LetϕN:N→(C^∗,×) be a character trivial on integer points andw0is the permutation matrix (n. . .321).

Wz(g) := Z

N

Φz(w0ng)ϕN(n)⁻¹dn is convergent forz∈C.

Important message: As in rank 1, the functional equations for Eisenstein series is implied by theW =Sn-symmetry after a normalizationWfz

∀σ∈Sn,∀g ∈G,∀z∈Rⁿ,Wgσz(g) =Wfz(g)

Number theory facts

(Langlands, Shahidi) Eisenstein series generalize fromH=SL2(R)/SO2(R) to other Lie groups

Functions onG/K ,general symmetric spaces.

We restrict ourselves to a special kind of Eisenstein serie associated to the function with paramz∈Rⁿ:

Φz(g) :=e^hz,µi wheng∈NT^µK

(Jacquet’s thesis, 1967) Fourier-Whittaker coefficients of such generalized

Einsenstein series are given by theWhittaker functionWz. LetϕN:N→(C^∗,×) be a character trivial on integer points andw0is the permutation matrix (n. . .321).

Wz(g) :=

Z

N

Φz(w0ng)ϕN(n)⁻¹dn is convergent forz∈C.

Important message: As in rank 1, the functional equations for Eisenstein series is implied by theW =Sn-symmetry after a normalizationWfz

∀σ∈Sn,∀g ∈G,∀z∈Rⁿ,Wgσz(g) =Wfz(g)

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 11 / 21

Number theory facts

(Langlands, Shahidi) Eisenstein series generalize fromH=SL2(R)/SO2(R) to other Lie groups

Functions onG/K ,general symmetric spaces.

We restrict ourselves to a special kind of Eisenstein serie associated to the function with paramz∈Rⁿ:

Φz(g) :=e^hz,µi wheng∈NT^µK

(Jacquet’s thesis, 1967) Fourier-Whittaker coefficients of such generalized

Einsenstein series are given by theWhittaker functionWz. LetϕN:N→(C^∗,×) be a character trivial on integer points andw0is the permutation matrix (n. . .321).

Wz(g) :=

Z

N

Φz(w0ng)ϕN(n)⁻¹dn is convergent forz∈C.

Important message: As in rank 1, the functional equations for Eisenstein series is implied by theW =Sn-symmetry after a normalizationWfz

∀σ∈Sn,∀g ∈G,∀z∈Rⁿ,Wgσz(g) =Wfz(g)

The Shintani-Casselman-Shalika formula

Recall:

Wz(g) = Z

N

Φz(w0ng)ϕN(n)⁻¹dn The Whittaker function is essentially a function onA≈Zⁿ:

∀(n,T^−λ,k)∈N×A×K, Wz

nT^−λk

=ϕN(n)Wz(T^−λ)

Theorem (Shintani 76 forGL_n, Casselman-Shalika 81 for G general)

Wz

T^−λ

= (

e^12PiλiQ

i<j 1−q⁻¹e^zi−zj

sλ(z) if λdominant,

0 otherwise.

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 12 / 21

The Shintani-Casselman-Shalika formula

Recall:

Wz(g) = Z

N

Φz(w0ng)ϕN(n)⁻¹dn The Whittaker function is essentially a function onA≈Zⁿ:

∀(n,T^−λ,k)∈N×A×K, Wz

nT^−λk

=ϕN(n)Wz(T^−λ)

Theorem (Shintani 76 forGL_n, Casselman-Shalika 81 for G general)

Wz

T^−λ

= (

e^12PiλiQ

i<j 1−q⁻¹e^zi−zj

sλ(z) if λdominant,

0 otherwise.

Sommaire

1 Introduction: the rank 1 story

2 Higher rank (Non-Archimedean)

3 The probabilistic proof of the Shintani-Casselman-Shalika formula

4 References

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 13 / 21

Step 1: A crucial remark

There is a seemingly innocent remark in papers of Shintani and Lafforgue:

Remark (Shintani, L. Lafforgue)

IfHis a K bi-invariant measure on G , then:

Wz?H=S(H)(z)Wz , for a certainS(H)(z)∈C .

“The Whittaker function is a convolution eigenfunction” Harmonicity for a probabilist!

Proposition

There exists a left-invariant random walk Bt

W^(z)

;t≥0

on NA such that (Harmonicity) A simple transform ofWz is harmonic for

Bt

W^(z)

;t≥0 . Bt

W^(z)

=Nt

W^(z)

T^−Wt(z) and W^(z)is the lattice walk onNⁿ with:

P

W_t+1^(z)−Wt^(z)=ei

= zi

P

jzj

The N-part of the increments are Haar distributed on N(O). (Exit law) Nt

W^(z)_t→∞

−→ N∞

W^(z)

Step 1: A crucial remark

There is a seemingly innocent remark in papers of Shintani and Lafforgue:

Remark (Shintani, L. Lafforgue)

IfHis a K bi-invariant measure on G , then:

Wz?H=S(H)(z)Wz , for a certainS(H)(z)∈C .

“The Whittaker function is a convolution eigenfunction” Harmonicity for a probabilist!

Proposition

There exists a left-invariant random walk Bt

W^(z)

;t≥0

on NA such that (Harmonicity) A simple transform of Wz is harmonic for

Bt

W^(z)

;t≥0 . Bt

W^(z)

=Nt

W^(z)

T^−Wt(z) and W^(z)is the lattice walk onNⁿ with:

P

W_t+1^(z)−Wt^(z)=ei

= zi

P

jzj

The N-part of the increments are Haar distributed on N(O).

(Exit law) Nt

W^(z)_t→∞

−→ N∞

W^(z)

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 14 / 21

Step 1: GL

example

Consider W¹,W²

a simple random walk onN²by steps (1,0) and (0,1).

W_t^(z)=W_t¹−W_t², SRW Form the random walk onGL2(K):

Bt+1(W) =Bt(W) T^{−(Wt+11 −Wt1)} 0 Ut T^{−(Wt+12 −Wt2)}

!

where (Ut;t≥0) are i.i.d Haar distributed onO. A simple computation gives that:

Bt(W) = T^−Wt1 0

T^−Wt1Pt

s=0T^{−(Ws2−Ws1)}Us T^−Wt2

! ,

hence

N∞

W^(z)

= 1 0

P∞

s=0T^Ws(z)Us 1

! .

Step 2: Trivial key lemma

Imperfect relation to tropical calculus:

val (x+y)≥min (val(x),val(y))

The perfect marriage of probability and non-Archimedean fields appears in:

Lemma (“The trivial key lemma”)

If U and U⁰are independent and Haar distributed onO, then for any two integers a and b inZ:

T^aU+T^bU^{0 L}=T^min(a,b)U

In a sense, non-Archimedean Haar random variables are “tropical calculus friendly”. By this lemma:

N∞

W^(z)

= 1 0

P∞

s=0T^Ws(z)Us 1

!

=L 1 0

T^{min0≤sWs(z)}U 1

!

andU Haar distributed onO.

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 16 / 21

Step 3: Poisson formula

Wz(T^−λ) = e^12PiλiY

i<j

1−q⁻¹e^zi−zj e^hz,λiE

ϕN

T^−λN∞(W^(z))T^λ

∼ e^hz,λiE

ϕN

T^−λN∞(W^(z))T^λ

Def ofϕ_N

= e^hz,λiE

"n−1

Y

i=1

ψ

T^{λi−λi+1+min0≤sWsi−Wsi+1}Ui

#

AverageU_i

= e^hz,λiE

"_n−1 Y

i=1

1{λ_i−λ_i+1+min_0≤sW_sⁱ−W_sⁱ⁺¹≥0}

#

= e^hz,λiPλ

W^(z)remains inC

Shintani-Casselman-Shalika formula in short

Theorem

Non-Archimedean Whittaker functions are proportional to Schur functions.

Sketch of probabilistic proof.

Non-Archimedean Whittaker functions are harmonic for certain random walks

Bt

W^(z)

,t≥0

onGLn(K), driven by a lattice walkW^(z)onZⁿ (with drift).

Poisson formula (not the summation one!): The Whittaker function is an expectation of the random walks’ exit law.

The stationary distribution is made of many sums which can be simplified thanks to the “Trivial key lemma”, giving infinimas.

Integrate out the Haar random variables. Get indicator functions. Whittaker function is proportional to the probability of a lattice walkW staying inside the Weyl chamber.

The reflection principle gives a determinant. This determinant is the Schur function.

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 18 / 21

Sommaire

1 Introduction: the rank 1 story

2 Higher rank (Non-Archimedean)

3 The probabilistic proof of the Shintani-Casselman-Shalika formula

4 References

References

Casselman, Shalika. The unramified principal series forp-adic groups II. The Whittaker function. 1980.

Chhaibi. A probabilistic approach to the Shintani-Casselman-Shalika formula. 2014.

Lafforgue, “Comment construire des noyaux de fonctorialit´e”. 2009.

Langlands. Euler products. 1971.

Jacquet. Fonctions de Whittaker associ´ees aux groupes de Chevalley. 1967.

Shintani, On an explicit formula for class-1 Whittaker functions. 1976.

RedaChhaibi(IMT, France) Reflection principle and number theory 23 June 2016, MADACA 20 / 21

Acknowledgments

Thank you for your attention!