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Functional equation of an enhanced zeta distribution the case of positive symmetric cone joint work in progress with Bent rsted & Akihito Wachi Kyo Nishiyama Aoyama Gakuin University WORKSHOP Dirac

  1. Functional equation of an enhanced zeta distribution — the case of positive symmetric cone joint work in progress with Bent Ørsted & Akihito Wachi 西山 享 Kyo Nishiyama 青山学院大学 Aoyama Gakuin University WORKSHOP ”Dirac operators and representation theory” Zagreb University, June 18-22, 2018 K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 1 / 23

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. Zeta distributions/integrals Tate’s zeta integral ż Z T p ϕ, s q “ ϕ p z q| z | s dz p s P C , ϕ P S p R qq R K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 3 / 23

  8. Zeta distributions/integrals Tate’s zeta integral ż Z T p ϕ, s q “ ϕ p z q| z | s dz p s P C , ϕ P S p R qq R Convergence: Re s ą ´ 1 K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 3 / 23

  9. Zeta distributions/integrals Tate’s zeta integral ż Z T p ϕ, s q “ ϕ p z q| z | s dz p s P C , ϕ P S p R qq R Convergence: Re s ą ´ 1 Meromorphic continuation to C K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 3 / 23

  10. Zeta distributions/integrals Tate’s zeta integral ż Z T p ϕ, s q “ ϕ p z q| z | s dz p s P C , ϕ P S p R qq R Convergence: Re s ą ´ 1 Meromorphic continuation to C Functional equation: Z T p ϕ, s q “ π ´ s ´ 1 2 Γ pp s ` 1 q{ 2 q Z T p p ϕ, ´ s ´ 1 q Γ p´ s { 2 q K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 3 / 23

  11. Zeta distributions/integrals Tate’s zeta integral ż Z T p ϕ, s q “ ϕ p z q| z | s dz p s P C , ϕ P S p R qq R Convergence: Re s ą ´ 1 Meromorphic continuation to C Functional equation: Z T p ϕ, s q “ π ´ s ´ 1 2 Γ pp s ` 1 q{ 2 q Z T p p ϕ, ´ s ´ 1 q Γ p´ s { 2 q ¨ ¨ ¨ K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 3 / 23

  12. Zeta distributions/integrals Tate’s zeta integral ż Z T p ϕ, s q “ ϕ p z q| z | s dz p s P C , ϕ P S p R qq R Convergence: Re s ą ´ 1 Meromorphic continuation to C Functional equation: Z T p ϕ, s q “ π ´ s ´ 1 2 Γ pp s ` 1 q{ 2 q Z T p p ϕ, ´ s ´ 1 q Γ p´ s { 2 q ¨ ¨ ¨ just copied it from Leticia’s paper (JFA 2004) [Bar04] K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 3 / 23

  13. Zeta distributions/integrals Tate’s zeta integral ż Z T p ϕ, s q “ ϕ p z q| z | s dz p s P C , ϕ P S p R qq R Convergence: Re s ą ´ 1 Meromorphic continuation to C Functional equation: Z T p ϕ, s q “ π ´ s ´ 1 2 Γ pp s ` 1 q{ 2 q Z T p p ϕ, ´ s ´ 1 q Γ p´ s { 2 q ¨ ¨ ¨ just copied it from Leticia’s paper (JFA 2004) [Bar04] Related to Riemann zeta ζ p s q K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 3 / 23

  14. Zeta distributions/integrals Godement-Jacquet zeta integral A generalization to p GL n p R q ˆ GL n p R q , M n p R qq : ż Z GJ p ϕ, s q “ ϕ p z q| det z | s dz M n p R q Again we have K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 4 / 23

  15. Zeta distributions/integrals Godement-Jacquet zeta integral A generalization to p GL n p R q ˆ GL n p R q , M n p R qq : ż Z GJ p ϕ, s q “ ϕ p z q| det z | s dz M n p R q Again we have ¨ ¨ ¨ K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 4 / 23

  16. Zeta distributions/integrals Godement-Jacquet zeta integral A generalization to p GL n p R q ˆ GL n p R q , M n p R qq : ż Z GJ p ϕ, s q “ ϕ p z q| det z | s dz M n p R q Again we have ¨ ¨ ¨ copied it from Leticia’s paper (JFA 2004) [Bar04] Convergence: Re s ą ´ 1 with meromorphic continuation to C Functional equation: 1 π ´ n 2 n p s ` n q 2 s π 2 q Z GJ p ϕ, s q “ Γ n p´ s { 2 q Z GJ p p ϕ, ´ s ´ n q Γ n p s ` n Related to zeta function ζ p L , s q for a lattice L ¨ ¨ ¨ important in NT K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 4 / 23

  17. Zeta distributions/integrals Godement-Jacquet zeta integral A generalization to p GL n p R q ˆ GL n p R q , M n p R qq : ż Z GJ p ϕ, s q “ ϕ p z q| det z | s dz M n p R q Again we have ¨ ¨ ¨ copied it from Leticia’s paper (JFA 2004) [Bar04] Convergence: Re s ą ´ 1 with meromorphic continuation to C Functional equation: 1 π ´ n 2 n p s ` n q 2 s π 2 q Z GJ p ϕ, s q “ Γ n p´ s { 2 q Z GJ p p ϕ, ´ s ´ n q Γ n p s ` n Related to zeta function ζ p L , s q for a lattice L ¨ ¨ ¨ important in NT D further generalizat’n to p GL n p R q , Sym n p R qq (Shintani, Satake-Faraut, ...) K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 4 / 23

  18. Zeta distributions/integrals Godement-Jacquet zeta integral A generalization to p GL n p R q ˆ GL n p R q , M n p R qq : ż Z GJ p ϕ, s q “ ϕ p z q| det z | s dz M n p R q Again we have ¨ ¨ ¨ copied it from Leticia’s paper (JFA 2004) [Bar04] Convergence: Re s ą ´ 1 with meromorphic continuation to C Functional equation: 1 π ´ n 2 n p s ` n q 2 s π 2 q Z GJ p ϕ, s q “ Γ n p´ s { 2 q Z GJ p p ϕ, ´ s ´ n q Γ n p s ` n Related to zeta function ζ p L , s q for a lattice L ¨ ¨ ¨ important in NT D further generalizat’n to p GL n p R q , Sym n p R qq (Shintani, Satake-Faraut, ...) Question : What is a right frame work? K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 4 / 23

  19. 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

  20. 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

  21. Zeta distributions/integrals Prehomogeneous Vector Space Frame work brought by Shintani and Mikio Sato ( „ 70’s, [SS74]): Setting (roughly): 1 p G , V q : PV { C , i.e., D an open orbit K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 5 / 23

  22. Zeta distributions/integrals Prehomogeneous Vector Space Frame work brought by Shintani and Mikio Sato ( „ 70’s, [SS74]): Setting (roughly): 1 p G , V q : PV { C , i.e., D an open orbit 2 P p z q P C r V s : fundamental relative invariant with character χ P K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 5 / 23

  23. Zeta distributions/integrals Prehomogeneous Vector Space Frame work brought by Shintani and Mikio Sato ( „ 70’s, [SS74]): Setting (roughly): 1 p G , V q : PV { C , i.e., D an open orbit 2 P p z q P C r V s : fundamental relative invariant with character χ P 3 V R zt P “ 0 u “ Ť ℓ i “ 1 O i : decomposition to open orbits K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 5 / 23

  24. Zeta distributions/integrals Prehomogeneous Vector Space Frame work brought by Shintani and Mikio Sato ( „ 70’s, [SS74]): Setting (roughly): 1 p G , V q : PV { C , i.e., D an open orbit 2 P p z q P C r V s : fundamental relative invariant with character χ P 3 V R zt P “ 0 u “ Ť ℓ i “ 1 O i : decomposition to open orbits Definition 2.1 (local zeta integral) ż Z p G , V q ϕ p z q| P p z q| s dz p ϕ, s q “ i O i K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 5 / 23

  25. Zeta distributions/integrals Fundamental Theorem of PV Theorem 2.2 (Sato-Shintani [SS74]) Assume G is reductive and D P p z q only one fundamental rel inv K. Nishiyama (AGU) Enhanced Zeta Distribution 2018/06/18 6 / 23

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