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Functional equation of an enhanced zeta distribution — the case of positive symmetric cone

joint work in progress with Bent Ørsted & Akihito Wachi Kyo Nishiyama 西山 享

Aoyama Gakuin University 青山学院大学 WORKSHOP ”Dirac operators and representation theory”

Zagreb University, June 18-22, 2018

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 1 / 23

Plan

Plan of talk

1 Zeta distributions/integrals

Classical examples of zeta integrals Prehomogeneous vector spaces Fundamental Theorem

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 2 / 23

Plan

Plan of talk

1 Zeta distributions/integrals

Classical examples of zeta integrals Prehomogeneous vector spaces Fundamental Theorem

2 Enhanced zeta integral and its meromorphic continuation

b-functions gamma factors and meromorphic continuation

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 2 / 23

Plan

Plan of talk

1 Zeta distributions/integrals

Classical examples of zeta integrals Prehomogeneous vector spaces Fundamental Theorem

2 Enhanced zeta integral and its meromorphic continuation

b-functions gamma factors and meromorphic continuation

3 Fourier transform and a functional equation

Fourier transform of kernel function Functional equation

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 2 / 23

Plan

Plan of talk

1 Zeta distributions/integrals

Classical examples of zeta integrals Prehomogeneous vector spaces Fundamental Theorem

2 Enhanced zeta integral and its meromorphic continuation

b-functions gamma factors and meromorphic continuation

3 Fourier transform and a functional equation

Fourier transform of kernel function Functional equation

4 Idea of Proof, Further problems

Idea of Proof Zeta integrals associated with orbits

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 2 / 23

Plan

Plan of talk

1 Zeta distributions/integrals

Classical examples of zeta integrals Prehomogeneous vector spaces Fundamental Theorem

2 Enhanced zeta integral and its meromorphic continuation

b-functions gamma factors and meromorphic continuation

3 Fourier transform and a functional equation

Fourier transform of kernel function Functional equation

4 Idea of Proof, Further problems

Idea of Proof Zeta integrals associated with orbits

5 Motivations

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 2 / 23

Zeta distributions/integrals

Tate’s zeta integral

Z Tpϕ, sq “ ż

R

ϕpzq|z|sdz ps P C, ϕ P S pRqq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 3 / 23

Zeta distributions/integrals

Tate’s zeta integral

Z Tpϕ, sq “ ż

R

ϕpzq|z|sdz ps P C, ϕ P S pRqq Convergence: Re s ą ´1

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 3 / 23

Zeta distributions/integrals

Tate’s zeta integral

Z Tpϕ, sq “ ż

R

ϕpzq|z|sdz ps P C, ϕ P S pRqq Convergence: Re s ą ´1 Meromorphic continuation to C

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 3 / 23

Zeta distributions/integrals

Tate’s zeta integral

Z Tpϕ, sq “ ż

R

ϕpzq|z|sdz ps P C, ϕ P S pRqq Convergence: Re s ą ´1 Meromorphic continuation to C Functional equation: Z Tpϕ, sq “ π´s´ 1

2 Γpps ` 1q{2q

Γp´s{2q Z Tpp ϕ, ´s ´ 1q

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 3 / 23

Zeta distributions/integrals

Tate’s zeta integral

Z Tpϕ, sq “ ż

R

ϕpzq|z|sdz ps P C, ϕ P S pRqq Convergence: Re s ą ´1 Meromorphic continuation to C Functional equation: Z Tpϕ, sq “ π´s´ 1

2 Γpps ` 1q{2q

Γp´s{2q Z Tpp ϕ, ´s ´ 1q ¨ ¨ ¨

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 3 / 23

Zeta distributions/integrals

Tate’s zeta integral

Z Tpϕ, sq “ ż

R

ϕpzq|z|sdz ps P C, ϕ P S pRqq Convergence: Re s ą ´1 Meromorphic continuation to C Functional equation: Z Tpϕ, sq “ π´s´ 1

2 Γpps ` 1q{2q

Γp´s{2q Z Tpp ϕ, ´s ´ 1q ¨ ¨ ¨ just copied it from Leticia’s paper (JFA 2004) [Bar04]

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 3 / 23

Zeta distributions/integrals

Tate’s zeta integral

Z Tpϕ, sq “ ż

R

ϕpzq|z|sdz ps P C, ϕ P S pRqq Convergence: Re s ą ´1 Meromorphic continuation to C Functional equation: Z Tpϕ, sq “ π´s´ 1

2 Γpps ` 1q{2q

Γp´s{2q Z Tpp ϕ, ´s ´ 1q ¨ ¨ ¨ just copied it from Leticia’s paper (JFA 2004) [Bar04] Related to Riemann zeta ζpsq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 3 / 23

Zeta distributions/integrals

Godement-Jacquet zeta integral

A generalization to pGLnpRq ˆ GLnpRq, MnpRqq: Z GJpϕ, sq “ ż

MnpRq

ϕpzq| det z|sdz Again we have

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 4 / 23

Zeta distributions/integrals

Godement-Jacquet zeta integral

A generalization to pGLnpRq ˆ GLnpRq, MnpRqq: Z GJpϕ, sq “ ż

MnpRq

ϕpzq| det z|sdz Again we have ¨ ¨ ¨

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 4 / 23

Zeta distributions/integrals

Godement-Jacquet zeta integral

A generalization to pGLnpRq ˆ GLnpRq, MnpRqq: Z GJpϕ, sq “ ż

MnpRq

ϕpzq| det z|sdz Again we have ¨ ¨ ¨ copied it from Leticia’s paper (JFA 2004) [Bar04] Convergence: Re s ą ´1 with meromorphic continuation to C Functional equation: π

1 2 nps`nq

Γnp s`n

2 q Z GJpϕ, sq “

π´ n

2 s

Γnp´s{2qZ GJpp ϕ, ´s ´ nq Related to zeta function ζpL, sq for a lattice L ¨ ¨ ¨ important in NT

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 4 / 23

Zeta distributions/integrals

Godement-Jacquet zeta integral

A generalization to pGLnpRq ˆ GLnpRq, MnpRqq: Z GJpϕ, sq “ ż

MnpRq

ϕpzq| det z|sdz Again we have ¨ ¨ ¨ copied it from Leticia’s paper (JFA 2004) [Bar04] Convergence: Re s ą ´1 with meromorphic continuation to C Functional equation: π

1 2 nps`nq

Γnp s`n

2 q Z GJpϕ, sq “

π´ n

2 s

Γnp´s{2qZ GJpp ϕ, ´s ´ nq Related to zeta function ζpL, sq for a lattice L ¨ ¨ ¨ important in NT D further generalizat’n to pGLnpRq, SymnpRqq (Shintani, Satake-Faraut, ...)

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 4 / 23

Zeta distributions/integrals

Godement-Jacquet zeta integral

A generalization to pGLnpRq ˆ GLnpRq, MnpRqq: Z GJpϕ, sq “ ż

MnpRq

ϕpzq| det z|sdz Again we have ¨ ¨ ¨ copied it from Leticia’s paper (JFA 2004) [Bar04] Convergence: Re s ą ´1 with meromorphic continuation to C Functional equation: π

1 2 nps`nq

Γnp s`n

2 q Z GJpϕ, sq “

π´ n

2 s

Γnp´s{2qZ GJpp ϕ, ´s ´ nq Related to zeta function ζpL, sq for a lattice L ¨ ¨ ¨ important in NT D further generalizat’n to pGLnpRq, SymnpRqq (Shintani, Satake-Faraut, ...) Question : What is a right frame work?

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 4 / 23

Zeta distributions/integrals

Prehomogeneous Vector Space

Frame work brought by Shintani and Mikio Sato („ 70’s, [SS74]):

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 5 / 23

Zeta distributions/integrals

Prehomogeneous Vector Space

Frame work brought by Shintani and Mikio Sato („ 70’s, [SS74]): Setting (roughly):

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 5 / 23

Zeta distributions/integrals

Prehomogeneous Vector Space

Frame work brought by Shintani and Mikio Sato („ 70’s, [SS74]): Setting (roughly):

1 pG, V q : PV {C, i.e., D an open orbit

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 5 / 23

Zeta distributions/integrals

Prehomogeneous Vector Space

Frame work brought by Shintani and Mikio Sato („ 70’s, [SS74]): Setting (roughly):

1 pG, V q : PV {C, i.e., D an open orbit 2 Ppzq P CrV s : fundamental relative invariant with character χP

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 5 / 23

Zeta distributions/integrals

Prehomogeneous Vector Space

Frame work brought by Shintani and Mikio Sato („ 70’s, [SS74]): Setting (roughly):

1 pG, V q : PV {C, i.e., D an open orbit 2 Ppzq P CrV s : fundamental relative invariant with character χP 3 VRztP “ 0u “ Ťℓ i“1 Oi : decomposition to open orbits

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 5 / 23

Zeta distributions/integrals

Prehomogeneous Vector Space

Frame work brought by Shintani and Mikio Sato („ 70’s, [SS74]): Setting (roughly):

1 pG, V q : PV {C, i.e., D an open orbit 2 Ppzq P CrV s : fundamental relative invariant with character χP 3 VRztP “ 0u “ Ťℓ i“1 Oi : decomposition to open orbits

Definition 2.1 (local zeta integral) Z pG,V q

i

pϕ, sq “ ż

Oi

ϕpzq|Ppzq|sdz

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 5 / 23

Zeta distributions/integrals

Fundamental Theorem of PV

Theorem 2.2 (Sato-Shintani [SS74]) Assume G is reductive and D Ppzq only one fundamental rel inv

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 6 / 23

Zeta distributions/integrals

Fundamental Theorem of PV

Theorem 2.2 (Sato-Shintani [SS74]) Assume G is reductive and D Ppzq only one fundamental rel inv

1 Zipϕ, sq converges in Re s ą 0 and continued meromorphically to C

ø b-function and Bernstein-Sato identity

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 6 / 23

Zeta distributions/integrals

Fundamental Theorem of PV

Theorem 2.2 (Sato-Shintani [SS74]) Assume G is reductive and D Ppzq only one fundamental rel inv

1 Zipϕ, sq converges in Re s ą 0 and continued meromorphically to C

ø b-function and Bernstein-Sato identity

2 local functional equation: n “ dim V , d “ deg P

Zipp ϕ, s ´ n

d q “ γps ´ n d q ℓ

ř

i“1

uijpsqZjpϕ, ´sq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 6 / 23

Zeta distributions/integrals

Fundamental Theorem of PV

Theorem 2.2 (Sato-Shintani [SS74]) Assume G is reductive and D Ppzq only one fundamental rel inv

1 Zipϕ, sq converges in Re s ą 0 and continued meromorphically to C

ø b-function and Bernstein-Sato identity

2 local functional equation: n “ dim V , d “ deg P

Zipp ϕ, s ´ n

d q “ γps ´ n d q ℓ

ř

i“1

uijpsqZjpϕ, ´sq Remark 2.3 Gamma factor γpsq can be described explicitly in terms of b-function uijpsq is a product of exponential function and a polynomial in e˘πis

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 6 / 23

Zeta distributions/integrals

Fundamental Theorem of PV

Theorem 2.2 (Sato-Shintani [SS74]) Assume G is reductive and D Ppzq only one fundamental rel inv

1 Zipϕ, sq converges in Re s ą 0 and continued meromorphically to C

ø b-function and Bernstein-Sato identity

2 local functional equation: n “ dim V , d “ deg P

Zipp ϕ, s ´ n

d q “ γps ´ n d q ℓ

ř

i“1

uijpsqZjpϕ, ´sq Remark 2.3 Gamma factor γpsq can be described explicitly in terms of b-function uijpsq is a product of exponential function and a polynomial in e˘πis The Fundamental Theorem is generalized to the case of several complex variables by Fumihiro Sato [Sat82a] [Sat83] [Sat82b]

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 6 / 23

Enhanced zeta integral and its meromorphic continuation

Enhanced space

Aim To investigate the zeta integral for PV with two fundamental relative invariants Generalization of existing results on MnpRq, SymnpRq, . . . etc

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 7 / 23

Enhanced zeta integral and its meromorphic continuation

Enhanced space

Aim To investigate the zeta integral for PV with two fundamental relative invariants Generalization of existing results on MnpRq, SymnpRq, . . . etc Setting: V “ SymnpRq : Euclidean Jordan algebra (cf. Faraut-Kor´

anyi [FK94])

action by L “ Str pV q “ GLnpRq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 7 / 23

Enhanced zeta integral and its meromorphic continuation

Enhanced space

Aim To investigate the zeta integral for PV with two fundamental relative invariants Generalization of existing results on MnpRq, SymnpRq, . . . etc Setting: V “ SymnpRq : Euclidean Jordan algebra (cf. Faraut-Kor´

anyi [FK94])

action by L “ Str pV q “ GLnpRq E “ Mn,dpRq “ ‘dRn : representation of V d-copies of natural rep, action by Str pV q “ GLnpRq & H “ GLdpRq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 7 / 23

Enhanced zeta integral and its meromorphic continuation

Enhanced space

Aim To investigate the zeta integral for PV with two fundamental relative invariants Generalization of existing results on MnpRq, SymnpRq, . . . etc Setting: V “ SymnpRq : Euclidean Jordan algebra (cf. Faraut-Kor´

anyi [FK94])

action by L “ Str pV q “ GLnpRq E “ Mn,dpRq “ ‘dRn : representation of V d-copies of natural rep, action by Str pV q “ GLnpRq & H “ GLdpRq W “ V ‘ E “ SymnpRq ‘ Mn,dpRq : enhanced space action of G “ L ˆ H “ GLnpRq ˆ GLdpRq via pg, hq ¨ pz, yq “ pgz tg, gy thq where pg, hq P L ˆ H “ G pz, yq P V ‘ E “ W

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 7 / 23

Enhanced zeta integral and its meromorphic continuation

Two relative invariants P1, P2

Extend the base field R to C GC ñ WC “ SymnpCqˆMn,dpCq: PV

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 8 / 23

Enhanced zeta integral and its meromorphic continuation

Two relative invariants P1, P2

Extend the base field R to C GC ñ WC “ SymnpCqˆMn,dpCq: PV Lemma 3.1 Assume d ď n. Then there are two fundamental relative invts of pGC, WCq: for pz, yq P WC“ Symn ˆMn,d, pg, hq P GC P1pz, yq “ det z with char χP1pg, hq “ pdet gq2 P2pz, yq “ p´1qd det ´ z y

ty 0

¯ with char χP2pg, hq “ pdet gq2pdet hq2 @ relative invariants are of the form Pm1

1 Pm2 2 pm1, m2 P Zě0q

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 8 / 23

Enhanced zeta integral and its meromorphic continuation

Two relative invariants P1, P2

Extend the base field R to C GC ñ WC “ SymnpCqˆMn,dpCq: PV Lemma 3.1 Assume d ď n. Then there are two fundamental relative invts of pGC, WCq: for pz, yq P WC“ Symn ˆMn,d, pg, hq P GC P1pz, yq “ det z with char χP1pg, hq “ pdet gq2 P2pz, yq “ p´1qd det ´ z y

ty 0

¯ with char χP2pg, hq “ pdet gq2pdet hq2 @ relative invariants are of the form Pm1

1 Pm2 2 pm1, m2 P Zě0q

Note: z P SymnpCq is regular ù ñ P2pz, yq “ det z ¨ det ` tyz´1y ˘

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 8 / 23

Enhanced zeta integral and its meromorphic continuation

Two relative invariants P1, P2

Extend the base field R to C GC ñ WC “ SymnpCqˆMn,dpCq: PV Lemma 3.1 Assume d ď n. Then there are two fundamental relative invts of pGC, WCq: for pz, yq P WC“ Symn ˆMn,d, pg, hq P GC P1pz, yq “ det z with char χP1pg, hq “ pdet gq2 P2pz, yq “ p´1qd det ´ z y

ty 0

¯ with char χP2pg, hq “ pdet gq2pdet hq2 @ relative invariants are of the form Pm1

1 Pm2 2 pm1, m2 P Zě0q

Note: z P SymnpCq is regular ù ñ P2pz, yq “ det z ¨ det ` tyz´1y ˘ If d ą n ù ñ P2 ” 0 &

• nly one rel inv P1 survives

6 We always assume d ď n below

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 8 / 23

Enhanced zeta integral and its meromorphic continuation

b-functions

inner product in WC: xpz, yq, pw, xqy “ Tr zw ` Tr tyx

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 9 / 23

Enhanced zeta integral and its meromorphic continuation

b-functions

inner product in WC: xpz, yq, pw, xqy “ Tr zw ` Tr tyx P˚

i pBz,yq : const coeff diff op defined by

P˚

i pBz,yqexpz,yq,pw,xqy “ Pipw, xqexpz,yq,pw,xqy

pz, w P Symn, y, x P Mn,dq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 9 / 23

Enhanced zeta integral and its meromorphic continuation

b-functions

inner product in WC: xpz, yq, pw, xqy “ Tr zw ` Tr tyx P˚

i pBz,yq : const coeff diff op defined by

P˚

i pBz,yqexpz,yq,pw,xqy “ Pipw, xqexpz,yq,pw,xqy

pz, w P Symn, y, x P Mn,dq Proposition 3.2 (Bernstein-Sato identity) b-functions for cplx parameters s “ ps1, s2q : b1,0psq “

d

ś

j“1

ps1 ` d`1

2

´ j´1

2 q n´d

ś

k“1

ps1 ` s2 ` n`1

2

´ k´1

2 q,

b0,1psq “

d

ś

j“1

ps2 ` d`1

2

´ j´1

2 qps2 ` n 2 ´ j´1 2 q n´d

ś

k“1

ps1 ` s2 ` n`1

2

´ k´1

2 q

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 9 / 23

Enhanced zeta integral and its meromorphic continuation

b-functions

inner product in WC: xpz, yq, pw, xqy “ Tr zw ` Tr tyx P˚

i pBz,yq : const coeff diff op defined by

P˚

i pBz,yqexpz,yq,pw,xqy “ Pipw, xqexpz,yq,pw,xqy

pz, w P Symn, y, x P Mn,dq Proposition 3.2 (Bernstein-Sato identity) b-functions for cplx parameters s “ ps1, s2q : b1,0psq “

d

ś

j“1

ps1 ` d`1

2

´ j´1

2 q n´d

ś

k“1

ps1 ` s2 ` n`1

2

´ k´1

2 q,

b0,1psq “

d

ś

j“1

ps2 ` d`1

2

´ j´1

2 qps2 ` n 2 ´ j´1 2 q n´d

ś

k“1

ps1 ` s2 ` n`1

2

´ k´1

2 q

ù ñ Bernstein-Sato identity: P˚

1 pBz,yq

´ P1pz, yqs1`1P2pz, yqs2 ¯ “ b1,0ps1, s2qP1pz, yqs1P2pz, yqs2, P˚

2 pBz,yq

´ P1pz, yqs1P2pz, yqs2`1¯ “ b0,1ps1, s2qP1pz, yqs1P2pz, yqs2.

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 9 / 23

Enhanced zeta integral and its meromorphic continuation

Enhanced positive cone

Return back to REAL WORLD R

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 10 / 23

Enhanced zeta integral and its meromorphic continuation

Enhanced positive cone

Return back to REAL WORLD R Unique open orbit {C

breaks up

Ý Ý Ý Ý Ý Ñ several open orbits {R

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 10 / 23

Enhanced zeta integral and its meromorphic continuation

Enhanced positive cone

Return back to REAL WORLD R Unique open orbit {C

breaks up

Ý Ý Ý Ý Ý Ñ several open orbits {R Among open orbits, get interested in enhanced positive cone: r Ω “ Ω ˆ M˝

n,dpRq

Ω “ Sym`

n pRq,

M˝

n,dpRq: full rank matrices

and . . .

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 10 / 23

Enhanced zeta integral and its meromorphic continuation

Enhanced positive cone

Return back to REAL WORLD R Unique open orbit {C

breaks up

Ý Ý Ý Ý Ý Ñ several open orbits {R Among open orbits, get interested in enhanced positive cone: r Ω “ Ω ˆ M˝

n,dpRq

Ω “ Sym`

n pRq,

M˝

n,dpRq: full rank matrices

and . . . enhanced zeta distribution: Zr

Ωpϕ, sq “

ż

r Ω

ϕpz, yqP1pz, yqs1P2pz, yqs2dzdy “ ż

Sym`

n pRq

pdet zqs1 dz ż

Mn,dpRq

ˇ ˇ ˇdet ´ z y

ty 0

¯ˇ ˇ ˇ

s2 dy

s “ ps1, s2q P C2, ϕpz, yq P S pW q, dz, dy : Lebesgue measures

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 10 / 23

Enhanced zeta integral and its meromorphic continuation

Meromorphic continuation

Zr

Ωpϕ, sq converges if Re s1, Re s2 " 0

ù C2 : meromorphic cont

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 11 / 23

Enhanced zeta integral and its meromorphic continuation

Meromorphic continuation

Zr

Ωpϕ, sq converges if Re s1, Re s2 " 0

ù C2 : meromorphic cont First we need a gamma factor: Γkpαq “ Γpαq Γpα ´ 1

2q ¨ ¨ ¨ Γpα ´ k´1 2 q “ śk j“1Γpα ´ j´1 2 q

Γr

Ωpsq “ Γdps1 ` d`1 2 q Γdps2 ` d`1 2 q Γdps2 ` n 2q Γn´dps1 ` s2 ` n`1 2 q

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 11 / 23

Enhanced zeta integral and its meromorphic continuation

Meromorphic continuation

Zr

Ωpϕ, sq converges if Re s1, Re s2 " 0

ù C2 : meromorphic cont First we need a gamma factor: Γkpαq “ Γpαq Γpα ´ 1

2q ¨ ¨ ¨ Γpα ´ k´1 2 q “ śk j“1Γpα ´ j´1 2 q

Γr

Ωpsq “ Γdps1 ` d`1 2 q Γdps2 ` d`1 2 q Γdps2 ` n 2q Γn´dps1 ` s2 ` n`1 2 q

Theorem 3.3 (Meromorphic Continuation) The zeta integral normalized by the gamma factor 1 Γr

Ωpsq Zr Ωpϕ, sq

is extended to an entire function in s “ ps1, s2q P C2, @ϕ P S pW q

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 11 / 23

Enhanced zeta integral and its meromorphic continuation

Meromorphic continuation

Zr

Ωpϕ, sq converges if Re s1, Re s2 " 0

ù C2 : meromorphic cont First we need a gamma factor: Γkpαq “ Γpαq Γpα ´ 1

2q ¨ ¨ ¨ Γpα ´ k´1 2 q “ śk j“1Γpα ´ j´1 2 q

Γr

Ωpsq “ Γdps1 ` d`1 2 q Γdps2 ` d`1 2 q Γdps2 ` n 2q Γn´dps1 ` s2 ` n`1 2 q

Theorem 3.3 (Meromorphic Continuation) The zeta integral normalized by the gamma factor 1 Γr

Ωpsq Zr Ωpϕ, sq

is extended to an entire function in s “ ps1, s2q P C2, @ϕ P S pW q ù Zr

Ωpϕ, sq extends to a meromorphic fun

with possible poles specified by Γr

Ωpsq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 11 / 23

Enhanced zeta integral and its meromorphic continuation

Meromorphic continuation

Zr

Ωpϕ, sq converges if Re s1, Re s2 " 0

ù C2 : meromorphic cont First we need a gamma factor: Γkpαq “ Γpαq Γpα ´ 1

2q ¨ ¨ ¨ Γpα ´ k´1 2 q “ śk j“1Γpα ´ j´1 2 q

Γr

Ωpsq “ Γdps1 ` d`1 2 q Γdps2 ` d`1 2 q Γdps2 ` n 2q Γn´dps1 ` s2 ` n`1 2 q

Theorem 3.3 (Meromorphic Continuation) The zeta integral normalized by the gamma factor 1 Γr

Ωpsq Zr Ωpϕ, sq

is extended to an entire function in s “ ps1, s2q P C2, @ϕ P S pW q ù Zr

Ωpϕ, sq extends to a meromorphic fun

with possible poles specified by Γr

Ωpsq

Remark 3.4 The case for d “ 1 is already studied by Suzuki [Suz79]

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 11 / 23

Fourier transform and a functional equation

Fourier transform

Write r z “ pz, yq & recall inner product on W “ SymnpRq ‘ Mn,dpRq: xr z, r wy “ Tr zw ` Tr tyx for r z “ pz, yq, r w “ pw, xq P W

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 12 / 23

Fourier transform and a functional equation

Fourier transform

Write r z “ pz, yq & recall inner product on W “ SymnpRq ‘ Mn,dpRq: xr z, r wy “ Tr zw ` Tr tyx for r z “ pz, yq, r w “ pw, xq P W Euclidean Fourier transform Fϕ “ p ϕ is defined as usual: p ϕpr wq “ ż

W

ϕpr zqe´2πixr

z, r wydr

z, q ψpr zq “ 2´ npn´1q

2

ż

W

ψpr wqe2πix r

w,r zyd r

w

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 12 / 23

Fourier transform and a functional equation

Fourier transform

Write r z “ pz, yq & recall inner product on W “ SymnpRq ‘ Mn,dpRq: xr z, r wy “ Tr zw ` Tr tyx for r z “ pz, yq, r w “ pw, xq P W Euclidean Fourier transform Fϕ “ p ϕ is defined as usual: p ϕpr wq “ ż

W

ϕpr zqe´2πixr

z, r wydr

z, q ψpr zq “ 2´ npn´1q

2

ż

W

ψpr wqe2πix r

w,r zyd r

w Want to consider the FT of the distribution: K `

s pr

zq “ $ ’ & ’ % pdet zqs1 ˇ ˇdet ´ z y

ty 0

¯ˇ ˇs2 r z P r Ω

• therwise.
• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 12 / 23

Fourier transform and a functional equation

Hyperfunction Ξs

Need one more notation: define hyperfunction Ξspr wq “ P1p`0 ´ 2πiw, xqs1P2p`0 ´ 2πiw, xqs2 “ lim

vÓ0 detpv ´ 2πiwqs1

´ p´1qd det ´ v ´ 2πiw x

tx

¯¯s2, where v P Ω “ Sym`

n pRq moves to 0 in the positive cone.

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 13 / 23

Fourier transform and a functional equation

Hyperfunction Ξs

Need one more notation: define hyperfunction Ξspr wq “ P1p`0 ´ 2πiw, xqs1P2p`0 ´ 2πiw, xqs2 “ lim

vÓ0 detpv ´ 2πiwqs1

´ p´1qd det ´ v ´ 2πiw x

tx

¯¯s2, where v P Ω “ Sym`

n pRq moves to 0 in the positive cone.

Ξspr wq should be interpreted as ż

W

Ξspr wqϕpr wqd r w “ lim

vÓ0

ż

W

P1pv ´ 2πiw, xqs1P2pv ´ 2πiw, xqs2ϕpr wqd r w

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 13 / 23

Fourier transform and a functional equation

Hyperfunction Ξs

Need one more notation: define hyperfunction Ξspr wq “ P1p`0 ´ 2πiw, xqs1P2p`0 ´ 2πiw, xqs2 “ lim

vÓ0 detpv ´ 2πiwqs1

´ p´1qd det ´ v ´ 2πiw x

tx

¯¯s2, where v P Ω “ Sym`

n pRq moves to 0 in the positive cone.

Ξspr wq should be interpreted as ż

W

Ξspr wqϕpr wqd r w “ lim

vÓ0

ż

W

P1pv ´ 2πiw, xqs1P2pv ´ 2πiw, xqs2ϕpr wqd r w If Re s1 ě 0 and Re s2 ě 0, we can take v Ó 0 inside the integral, and get Ξspr wq “ p´2πiqns1`pn´dqs2P1pr wqs1P2pr wqs2 with an appropriate choice of the branch of exponents.

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 13 / 23

Fourier transform and a functional equation

Hyperfunction Ξs

Need one more notation: define hyperfunction Ξspr wq “ P1p`0 ´ 2πiw, xqs1P2p`0 ´ 2πiw, xqs2 “ lim

vÓ0 detpv ´ 2πiwqs1

´ p´1qd det ´ v ´ 2πiw x

tx

¯¯s2, where v P Ω “ Sym`

n pRq moves to 0 in the positive cone.

Ξspr wq should be interpreted as ż

W

Ξspr wqϕpr wqd r w “ lim

vÓ0

ż

W

P1pv ´ 2πiw, xqs1P2pv ´ 2πiw, xqs2ϕpr wqd r w If Re s1 ě 0 and Re s2 ě 0, we can take v Ó 0 inside the integral, and get Ξspr wq “ p´2πiqns1`pn´dqs2P1pr wqs1P2pr wqs2 with an appropriate choice of the branch of exponents. In particular, we get : Ξp0,0qpr wq “ 1 (constant function)

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 13 / 23

Fourier transform and a functional equation

Fourier transform of K `

s

Theorem 4.1 (Fourier transform of rel inv) Fourier transform of K `

s

is given by

1 Γdps1` d`1 2 q Γdps2` n 2 q Γn´dps1`s2` n`1 2 q

y K `

s “ cpsq Γdp´s2q Ξ´ps1` d`1

2 q,´ps2` n 2 q

where cpsq “ p2πq

npn´1q 4

π´2dps2` n

4 q,

Γkpαq “ śk

j“1Γpα ´ j´1 2 q

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 14 / 23

Fourier transform and a functional equation

Fourier transform of K `

s

Theorem 4.1 (Fourier transform of rel inv) Fourier transform of K `

s

is given by

1 Γdps1` d`1 2 q Γdps2` n 2 q Γn´dps1`s2` n`1 2 q

y K `

s “ cpsq Γdp´s2q Ξ´ps1` d`1

2 q,´ps2` n 2 q

where cpsq “ p2πq

npn´1q 4

π´2dps2` n

4 q,

Γkpαq “ śk

j“1Γpα ´ j´1 2 q

In particular, K `

s

has a pole at s “ ´ 1

2pd ` 1, nq and there the first

residue is a constant multiple of the delta distribution: δ “ 1 cpsq Γdps2 ` d`1

2 q Γdp´s2q

Γr

Ωpsq

¨ K `

s

ˇ ˇ ˇ

s“´ 1

2 pd`1,nq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 14 / 23

Fourier transform and a functional equation

Functional equation

Corollary 4.2 If ϕ P S pW q is supported in the closure of the enhanced positive cone r Ω, we get a functional equation: 1 Γr

Ωpsq Zr Ωpp

ϕ; s1, s2q “ cpsqp´2πiq´pns1`pn´dqs2` npn`1q

2 q

Γdps2 ` d`1

2 q Γdp´s2q

Zr

Ωpϕ; ´ps1 ` d`1 2 q, ´ps2 ` n 2qq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 15 / 23

Fourier transform and a functional equation

Functional equation

Corollary 4.2 If ϕ P S pW q is supported in the closure of the enhanced positive cone r Ω, we get a functional equation: 1 Γr

Ωpsq Zr Ωpp

ϕ; s1, s2q “ cpsqp´2πiq´pns1`pn´dqs2` npn`1q

2 q

Γdps2 ` d`1

2 q Γdp´s2q

Zr

Ωpϕ; ´ps1 ` d`1 2 q, ´ps2 ` n 2qq

“ cpsqp´πq´d p´2πiqns1`pn´dqs2` npn`1q

2

śd

j“1 sinps2 ` d´j 2 qπ ˆ

Zr

Ωpϕ; ´ps1 ` d`1 2 q, ´ps2 ` n 2qq

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 15 / 23

Idea of Proof

Idea of Proof

¨ ¨ ¨ is straightforward

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 16 / 23

Idea of Proof

Idea of Proof

¨ ¨ ¨ is straightforward However, we need two facts:

1 Fourier transform of quadratic form (generalization of Epstein zeta

integral) proved by J.L. Clerc [Cle02] (see below) Theorem 5.1 (Clerc [Cle02, Th 2]) For a representation E “ Mn,dpRq of Vd “ SymdpRq, Fourier transform of the power of the quadratic form Qpyqs “ pdet tyyqs is given by ż

E

p ψpyqpdet tyyqsdy “ π´2dps` n

4 q Γdps ` n 2q

Γdp´sq ż

E

ψpxqpdet txxq´s´ n

2 dx

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 16 / 23

Idea of Proof

Idea of Proof

¨ ¨ ¨ is straightforward However, we need two facts:

1 Fourier transform of quadratic form (generalization of Epstein zeta

integral) proved by J.L. Clerc [Cle02] (see below)

2 Gindikin’s Gamma function (cf. Faraut-Kor´

anyi [FK94])

ş

Ω e´ Tr zpdet zqα∆dpzqβdz “ p2πq npn´1q 4

Γdpα ` β ` n`1

2 qΓn´dpα ` n´d`1 2

q

Theorem 5.1 (Clerc [Cle02, Th 2]) For a representation E “ Mn,dpRq of Vd “ SymdpRq, Fourier transform of the power of the quadratic form Qpyqs “ pdet tyyqs is given by ż

E

p ψpyqpdet tyyqsdy “ π´2dps` n

4 q Γdps ` n 2q

Γdp´sq ż

E

ψpxqpdet txxq´s´ n

2 dx

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 16 / 23

Idea of Proof

Idea of Proof

¨ ¨ ¨ is straightforward However, we need two facts:

1 Fourier transform of quadratic form (generalization of Epstein zeta

integral) proved by J.L. Clerc [Cle02] (see below)

2 Gindikin’s Gamma function (cf. Faraut-Kor´

anyi [FK94])

ş

Ω e´ Tr zpdet zqα∆dpzqβdz “ p2πq npn´1q 4

Γdpα ` β ` n`1

2 qΓn´dpα ` n´d`1 2

q

Theorem 5.1 (Clerc [Cle02, Th 2]) For a representation E “ Mn,dpRq of Vd “ SymdpRq, Fourier transform of the power of the quadratic form Qpyqs “ pdet tyyqs is given by ż

E

p ψpyqpdet tyyqsdy “ π´2dps` n

4 q Γdps ` n 2q

Γdp´sq ż

E

ψpxqpdet txxq´s´ n

2 dx

The calculation is a fun, but it is too much involved and we omit details

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 16 / 23

Zeta integrals associated with orbits

Further problems

So far, we could manage the enhanced positive cone

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 17 / 23

Zeta integrals associated with orbits

Further problems

So far, we could manage the enhanced positive cone Case of bilinear forms V “ SymnpRq ù pn ` 1q open orbits Ωpp, qq determined by signature zeta distributions: Zpp,qqpϕ, sq “ ż

Ωpp,qq

ϕpzq| det z|sdz Gamma factors and functional equations

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 17 / 23

Zeta integrals associated with orbits

Further problems

So far, we could manage the enhanced positive cone Case of bilinear forms V “ SymnpRq ù pn ` 1q open orbits Ωpp, qq determined by signature zeta distributions: Zpp,qqpϕ, sq “ ż

Ωpp,qq

ϕpzq| det z|sdz Gamma factors and functional equations ¨ ¨ ¨ complete results obtained by Satake-Faraut [SF84]

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 17 / 23

Zeta integrals associated with orbits

Further problems

So far, we could manage the enhanced positive cone Case of bilinear forms V “ SymnpRq ù pn ` 1q open orbits Ωpp, qq determined by signature zeta distributions: Zpp,qqpϕ, sq “ ż

Ωpp,qq

ϕpzq| det z|sdz Gamma factors and functional equations ¨ ¨ ¨ complete results obtained by Satake-Faraut [SF84] In our case, there appear more open orbits:

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 17 / 23

Zeta integrals associated with orbits

Further problems

So far, we could manage the enhanced positive cone Case of bilinear forms V “ SymnpRq ù pn ` 1q open orbits Ωpp, qq determined by signature zeta distributions: Zpp,qqpϕ, sq “ ż

Ωpp,qq

ϕpzq| det z|sdz Gamma factors and functional equations ¨ ¨ ¨ complete results obtained by Satake-Faraut [SF84] In our case, there appear more open orbits: Open orbit O Ø pz, yq : z P Ωpp, qq and signature of z´1|Im y

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 17 / 23

Zeta integrals associated with orbits

Further problems

So far, we could manage the enhanced positive cone Case of bilinear forms V “ SymnpRq ù pn ` 1q open orbits Ωpp, qq determined by signature zeta distributions: Zpp,qqpϕ, sq “ ż

Ωpp,qq

ϕpzq| det z|sdz Gamma factors and functional equations ¨ ¨ ¨ complete results obtained by Satake-Faraut [SF84] In our case, there appear more open orbits: Open orbit O Ø pz, yq : z P Ωpp, qq and signature of z´1|Im y Problem 6.1 Determine functional equation for arbitrary open orbits

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 17 / 23

Zeta integrals associated with orbits

Further and Further Problems

Another big issues are:

1 locate all poles (and zeros) (We only determined possible poles)

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 18 / 23

Zeta integrals associated with orbits

Further and Further Problems

Another big issues are:

1 locate all poles (and zeros) (We only determined possible poles) 2 compute residues

These are future subjects

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 18 / 23

Motivations

Jumbles

Q Why we are interested in the enhanced zeta integrals?

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 19 / 23

Motivations

Jumbles

Q Why we are interested in the enhanced zeta integrals? Here are some na¨ ıve reasons/expectations ¨ ¨ ¨ opinions/comments are welcome

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 19 / 23

Motivations

Jumbles

Q Why we are interested in the enhanced zeta integrals? Here are some na¨ ıve reasons/expectations ¨ ¨ ¨ opinions/comments are welcome

1 It is a rare example of explicit Functional Eqs in the case of several

variables (with several fundamental relative invts) However, D generaltheorey for several cplx variables by Fumihiro Sato [Sat82a] [Sat83] [Sat82b]

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 19 / 23

Motivations

Jumbles

Q Why we are interested in the enhanced zeta integrals? Here are some na¨ ıve reasons/expectations ¨ ¨ ¨ opinions/comments are welcome

1 It is a rare example of explicit Functional Eqs in the case of several

variables (with several fundamental relative invts) However, D generaltheorey for several cplx variables by Fumihiro Sato [Sat82a] [Sat83] [Sat82b]

2 Kernel K ` s

was used to construct intertwiners between degenerate principal series (N-Ørsted [NOr18])

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 19 / 23

Motivations

Jumbles

Q Why we are interested in the enhanced zeta integrals? Here are some na¨ ıve reasons/expectations ¨ ¨ ¨ opinions/comments are welcome

1 It is a rare example of explicit Functional Eqs in the case of several

variables (with several fundamental relative invts) However, D generaltheorey for several cplx variables by Fumihiro Sato [Sat82a] [Sat83] [Sat82b]

2 Kernel K ` s

was used to construct intertwiners between degenerate principal series (N-Ørsted [NOr18]) ù some information of images and kernels? small submodules?

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 19 / 23

Motivations

Jumbles

Q Why we are interested in the enhanced zeta integrals? Here are some na¨ ıve reasons/expectations ¨ ¨ ¨ opinions/comments are welcome

1 It is a rare example of explicit Functional Eqs in the case of several

variables (with several fundamental relative invts) However, D generaltheorey for several cplx variables by Fumihiro Sato [Sat82a] [Sat83] [Sat82b]

2 Kernel K ` s

was used to construct intertwiners between degenerate principal series (N-Ørsted [NOr18]) ù some information of images and kernels? small submodules? ù analytic cont of intertwiners and their residues

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 19 / 23

Motivations

Jumbles continued!

3 W “ V ‘ E is isomorphic to a boundary orbit of the positive cone

Sym`

n`dpRq related to unitary highest weight module (Wallach set)

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 20 / 23

Motivations

Jumbles continued!

3 W “ V ‘ E is isomorphic to a boundary orbit of the positive cone

Sym`

n`dpRq related to unitary highest weight module (Wallach set)

Our results related to a further continuation of submodules restricted to a certain subgroup?

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 20 / 23

Motivations

Jumbles continued!

3 W “ V ‘ E is isomorphic to a boundary orbit of the positive cone

Sym`

n`dpRq related to unitary highest weight module (Wallach set)

Our results related to a further continuation of submodules restricted to a certain subgroup?

4 The kernel K ` s

can be interpreted as the cplx power of a matrix coefficient of finite dimensional representation

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 20 / 23

Motivations

Jumbles continued!

3 W “ V ‘ E is isomorphic to a boundary orbit of the positive cone

Sym`

n`dpRq related to unitary highest weight module (Wallach set)

Our results related to a further continuation of submodules restricted to a certain subgroup?

4 The kernel K ` s

can be interpreted as the cplx power of a matrix coefficient of finite dimensional representation ù similar theory for Knapp-Stein kernels? (Barchini-Sepanski-Zierau, Ben-Sa¨ ıd-Clerc-Koufany, M¨

• llers-Oshima-Ørsted, ¨ ¨ ¨ )
• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 20 / 23

Motivations

Jumbles continued!

3 W “ V ‘ E is isomorphic to a boundary orbit of the positive cone

Sym`

n`dpRq related to unitary highest weight module (Wallach set)

Our results related to a further continuation of submodules restricted to a certain subgroup?

4 The kernel K ` s

can be interpreted as the cplx power of a matrix coefficient of finite dimensional representation ù similar theory for Knapp-Stein kernels? (Barchini-Sepanski-Zierau, Ben-Sa¨ ıd-Clerc-Koufany, M¨

• llers-Oshima-Ørsted, ¨ ¨ ¨ )

ù constructing invariant differential operators as residues?

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 20 / 23

Motivations

Thank you for your attention &

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 21 / 23

Motivations

Thank you for your attention & Congratulation!! to Prof Kashiwara Determination of b-functions [SKKO81] Algorithm for calculating Fourier transform of zeta integrals [KM75]

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 21 / 23

References

References I

[Bar04]

• L. Barchini, Zeta distributions and boundary values of Poisson transforms, J. Funct.
• Anal. 216 (2004), no. 1, 47–70. MR 2091356

[Cle02] Jean-Louis Clerc, Zeta distributions associated to a representation of a Jordan algebra, Math. Z. 239 (2002), no. 2, 263–276. MR 1888224 [FK94] Jacques Faraut and Adam Kor´ anyi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994, Oxford Science Publications. MR 1446489 [KM75] Masaki Kashiwara and Tetsuji Miwa, Microlocal calculus and Fourier transforms of relative invariants of prehomogeneous vector spaces, Sˆ urikaisekikenkyˆ usho K´

• kyˆ

uroku (1975), no. 238, 60–147, Theory of prehomogeneous vector spaces and its applications (Short Courses, Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1974). MR 0650981 [NOr18] Kyo Nishiyama and Bent Ø rsted, Real double flag varieties for the symplectic group, J. Funct. Anal. 274 (2018), no. 2, 573–604. MR 3724150 [Sat82a] Fumihiro Sat¯

• , Zeta functions in several variables associated with prehomogeneous

vector spaces. I. Functional equations, Tˆ

• hoku Math. J. (2) 34 (1982), no. 3,

437–483. MR 676121

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 22 / 23

References

References II

[Sat82b] , Zeta functions in several variables associated with prehomogeneous vector

• spaces. III. Eisenstein series for indefinite quadratic forms, Ann. of Math. (2) 116

(1982), no. 1, 177–212. MR 662121 [Sat83] , Zeta functions in several variables associated with prehomogeneous vector

• spaces. II. A convergence criterion, Tˆ
• hoku Math. J. (2) 35 (1983), no. 1, 77–99.

MR 695661 [SF84]

• I. Satake and J. Faraut, The functional equation of zeta distributions associated with

formally real Jordan algebras, Tohoku Math. J. (2) 36 (1984), no. 3, 469–482. MR 756029 [SKKO81] M. Sato, M. Kashiwara, T. Kimura, and T. ¯ Oshima, Microlocal analysis of prehomogeneous vector spaces, Invent. Math. 62 (1980/81), no. 1, 117–179. MR 595585 [SS74] Mikio Sato and Takuro Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131–170. MR 0344230 [Suz79] Toshiaki Suzuki, On zeta functions associated with quadratic forms of variable coefficients, Nagoya Math. J. 73 (1979), 117–147. MR 524011

• K. Nishiyama (AGU)

Enhanced Zeta Distribution 2018/06/18 23 / 23