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- 1. An algebraic approach to Duﬂo’s polynomial conjecture in the nilpotent case Yoshinori Tanimura Graduate School of Mathematical Sciences, The University of Tokyo May 26, 2019 Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 1 / 50
- 2. Contents 1 Introduction 2 Strict Formulation of Problems 3 Our Method: Split Symmetrization Map 4 Strategy, Bottlenecks and Packaged Tools 5 Demonstration 6 Future Works Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 2 / 50
- 3. Contents 1 Introduction 2 Strict Formulation of Problems 3 Our Method: Split Symmetrization Map 4 Strategy, Bottlenecks and Packaged Tools 5 Demonstration 6 Future Works Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 3 / 50
- 4. Motivation Fact (well-known) Properties of the Laplacian ∆: ∆ is invariant under isometries. ∆ is commutative with other invariant diﬀerential operators. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 4 / 50
- 5. Motivation Problem How many diﬀerential operators D satisﬁes the following properties? D is invariant under isometries. D is commutative with other invariant diﬀerential operators. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 4 / 50
- 6. Main Conjectures (naive description) Duﬂo’s polynomial conjecture (1986) The center of the algebra of all invariant diﬀerential operators is isomorphic to a subalgebra of a polynomial algebra. Corwin-Greenleaf’s polynomial conjecture (1992) If the algebra of all invariant diﬀerential operators is commutative, then it is isomorphic to a subalgebra of a polynomial algebra. speciﬁc case Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 5 / 50
- 7. Previous Research Our method algebraic, rustic, clear Duﬂo’s polynomial conjecture CG’s polynomial conjecture Benoist-Corwin-Greenleaf-Fujiwara’s approach analytic, representation theoretic, diﬃcult ⃝ 2-step nilpotent, ⃝ special case ⃝ “common polarization”, × 2-step nilpotent speciﬁc case Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 6 / 50
- 8. Our Result Our method algebraic, rustic, clear Duﬂo’s polynomial conjecture CG’s polynomial conjecture Benoist-Corwin-Greenleaf-Fujiwara’s approach analytic, representation theoretic, diﬃcult ⃝ 2-step nilpotent, ⃝ special case ⃝ “common polarization”, × 2-step nilpotent speciﬁc case Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 7 / 50
- 9. Historical Background Duﬂo Benoist Rouvi`ere : the symmetric case (before 1986) Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 8 / 50
- 10. Historical Background Duﬂo Benoist Rouvi`ere : the symmetric case (before 1986) Corwin-Greenleaf (1992) Oshima-Kobayashi (1991) Fujiwara (1998) analytic representation theoretic Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 8 / 50
- 11. Historical Background Duﬂo Benoist Rouvi`ere : the symmetric case (before 1986) Corwin-Greenleaf (1992) Oshima-Kobayashi (1991) Fujiwara (1998) analytic representation theoretic Our method (2019) F-method geometric algebraic Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 8 / 50
- 12. Contents 1 Introduction 2 Strict Formulation of Problems 3 Our Method: Split Symmetrization Map 4 Strategy, Bottlenecks and Packaged Tools 5 Demonstration 6 Future Works Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 9 / 50
- 13. Basic Notations K : a ﬁeld, char K = 0, g : a Lie algebra over K, h ⊂ g : a Lie subalgebra, λ: h → K : a character Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 10 / 50
- 14. Basic Notations S(g): the symmetric algebra, U(g): the universal enveloping algebra hλ := {X − λ(X) | X ∈ h} Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 11 / 50
- 15. Basic Notations S(g): the symmetric algebra, U(g): the universal enveloping algebra hλ := {X − λ(X) | X ∈ h} ˜Sλ(g) := S(g)/S(g)hλ, ˜Uλ(g) := U(g)/U(g)hλ Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 11 / 50
- 16. Poisson Algebra Deﬁnition (Poisson algebra) A K-algebra A with a Lie bracket [, ] is called a Poisson algebra if (A, [, ]) satisﬁes the Leibniz rule: [x, yz] = [x, y]z + y[x, z] for all x, y, z ∈ A. Example A K-algebra A is a Poisson algebra with the commutator [x, y] = xy − yx (x, y ∈ A). For a symplectic manifold M, the algebra of all smooth functions C∞(M) is a Poisson algebra with the Poisson bracket {, }. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 12 / 50
- 17. Poisson Algebra Deﬁnition (Poisson algebra) A K-algebra A with a Lie bracket [, ] is called a Poisson algebra if (A, [, ]) satisﬁes the Leibniz rule: [x, yz] = [x, y]z + y[x, z] for all x, y, z ∈ A. Example For a Lie algebra g, The Lie bracket [, ] of g is extended to the whole S(g) by the Leibniz rule. S(g) with this bracket [, ] becomes a Poisson algebra. U(g) is regarded as a Poisson algebra by the commutator: [A, B] = AB − BA (A, B ∈ U(g)). Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 12 / 50
- 18. Basic Notations Example The Lie bracket [, ] of g is extended to the whole S(g) by the Leibniz rule. S(g) with this bracket [, ] becomes a Poisson algebra. U(g) is regarded as a Poisson algebra by the commutator: [A, B] = AB − BA (A, B ∈ U(g)). Observation The representation h ↷ S(g), U(g) is deﬁned by the Poisson algebra structures. The representation h ↷ ˜Sλ(g), ˜Uλ(g) is induced by the above rep. The h-invariant spaces ˜Sλ(g)h and ˜Uλ(g)h have the induced Poisson algebra structures. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 13 / 50
- 19. Main Conjecture (strict description) Duﬂo’s polynomial conjecture (1986) Z( ˜Sλ+ρ(g)h) is ring isomorphic to Z( ˜Uλ(g)h) for ρ := 1/2 trg/h : h → K. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 14 / 50
- 20. Main Conjecture (strict description) Duﬂo’s polynomial conjecture (1986) Z( ˜Sλ+ρ(g)h) is ring isomorphic to Z( ˜Uλ(g)h) for ρ := 1/2 trg/h : h → K. Deﬁnition (Poisson center) For a Poisson bracket A, the abelian K-subalgebra Z(A) := {x ∈ A | yx = xy, [x, y] = 0 (y ∈ A)} is called a Poisson center. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 14 / 50
- 21. Geometrical and Analytical Background In this page, we consider the K = R or C case. . . Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 15 / 50
- 22. Geometrical and Analytical Background In this page, we consider the K = R or C case. . . G: the 1-connected Lie group corresponding with g, H ⊂ G: the closed connected subgroup of h ⊂ g, χ: H → K: the character such that dχ = −λ. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 15 / 50
- 23. Geometrical and Analytical Background In this page, we consider the K = R or C case. . . G: the 1-connected Lie group corresponding with g, H ⊂ G: the closed connected subgroup of h ⊂ g, χ: H → K: the character such that dχ = −λ. Then ˜Uλ(g)h ∼= the algebra of all G-invariant diﬀerential operators of G ×(H,χ) K ↓ G/H . ˜Sλ(g)h ∼= K[Γλ]H where Γλ := {u ∈ g∗ | u|h = λ}. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 15 / 50
- 24. Main Conjecture Duﬂo’s polynomial conjecture (1986) Z( ˜Sλ+ρ(g)h) is ring isomorphic to Z( ˜Uλ(g)h) for ρ := 1/2 trg/h : h → K. If K = R or C, ˜Uλ(g)h ∼= the algebra of all G-invariant diﬀerential operators of G ×(H,χ) K ↓ G/H . ˜Sλ(g)h ∼= K[Γλ]H where Γλ := {u ∈ g∗ | u|h = λ}. Duﬂo’s polynomial conjecture (naive description) The center of the algebra of all invariant diﬀerential operators is isomorphic to a certain subalgebra of a polynomial algebra. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 16 / 50
- 25. To the nilpotent case. . . In the following, we suppose that g is nilpotent. Duﬂo’s polynomial conjecture in the nilpotent case If g is nilpotent, then Z( ˜Sλ(g)h) is ring isomorphic to Z( ˜Uλ(g)h). Corwin-Greenleaf’s polynomial conjecture (1992), algebraic ver. If g is nilpotent and ˜Uλ(g)h is commutative, then ˜Sλ(g)h is ring isomorphic to ˜Uλ(g)h. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 17 / 50
- 26. Contents 1 Introduction 2 Strict Formulation of Problems 3 Our Method: Split Symmetrization Map 4 Strategy, Bottlenecks and Packaged Tools 5 Demonstration 6 Future Works Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 18 / 50
- 27. Motivation Duﬂo’s polynomial conjecture did not give a candidate map expected to be isomorphic. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 19 / 50
- 28. Motivation Duﬂo’s polynomial conjecture did not give a candidate map expected to be isomorphic. → At ﬁrst, we have to give a candidate! Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 19 / 50
- 29. Basic Notations K : a ﬁeld, char K = 0, g : a nilpotent Lie algebra over K, h ⊂ g : a Lie subalgebra, λ: h → K : a character Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 20 / 50
- 30. Split Symmetrization Map Deﬁnition (T, 2019) For a linear complement q of h ⊂ g, we deﬁne the split symmetrization map σq : S(g) → U(g) by σq(P Q) = σ(P ) · σ(Q) (P ∈ S(q), Q ∈ S(h)). Here, σ : S(g) → U(g) is the symmetrizaton map: σ(X0 · · · Xn−1) = 1 n! ∑ s∈Sn Xs(0) · · · Xs(n−1) (X0, . . . , Xn−1 ∈ g). Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 21 / 50
- 31. Split Symmetrization Map Proposition For any linear complement q of h ⊂ g, there exists the induced map ˜σq : ˜Sλ(g) → ˜Uλ(g) such that ⟲ U(g)S(g) ˜Sλ(g) ˜Uλ(g) σq ˜σq Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 22 / 50
- 32. Split Symmetrization Map Proposition For any linear complement q of h ⊂ g, there exists the induced map ˜σq : ˜Sλ(g) → ˜Uλ(g) such that ⟲ U(g)S(g) ˜Sλ(g) ˜Uλ(g) σq ˜σq We suggest this induced map ˜σq as a candidate. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 22 / 50
- 33. Two Properties Deﬁnition (T, 2019) The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property if: ˜Sλ(g)h is linear isomorphic to ˜Uλ(g)h under ˜σq. Z( ˜Sλ(g)h) is ring isomorphic to Z( ˜Uλ(g)h) under ˜σq. The quadruple (g, h, λ, q) satisﬁes the split Corwin-Greenleaf property if ˜Uλ(g)h is non-commutative or ˜Sλ(g)h is ring isomorphic to ˜Uλ(g)h under ˜σq. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 23 / 50
- 34. Two Properties Deﬁnition (T, 2019) The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property if: ˜Sλ(g)h is linear isomorphic to ˜Uλ(g)h under ˜σq. Z( ˜Sλ(g)h) is ring isomorphic to Z( ˜Uλ(g)h) under ˜σq. The quadruple (g, h, λ, q) satisﬁes the split Corwin-Greenleaf property if ˜Uλ(g)h is non-commutative or ˜Sλ(g)h is ring isomorphic to ˜Uλ(g)h under ˜σq. split Duﬂo propertyDuﬂo’s polynomial conjecture CG’s polynomial conjecture split CG property Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 23 / 50
- 35. Two Properties Deﬁnition (T, 2019) The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property if: ˜Sλ(g)h is linear isomorphic to ˜Uλ(g)h under ˜σq. Z( ˜Sλ(g)h) is ring isomorphic to Z( ˜Uλ(g)h) under ˜σq. The quadruple (g, h, λ, q) satisﬁes the split Corwin-Greenleaf property if ˜Uλ(g)h is non-commutative or ˜Sλ(g)h is ring isomorphic to ˜Uλ(g)h under ˜σq. Conjecture (T, 2019) For the above g, h, λ, there exists a linear complement q of h ⊂ g such that the quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 23 / 50
- 36. Easy Cases Conjecture (T, 2019) For the above g, h, λ, there exists a linear complement q of h ⊂ g such that the quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. Proposition The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property if: g is abelian, or h = 0 (Dixmier, 1959). Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 24 / 50
- 37. Main Result Theorem (T, 2019) g: a nilpotent Lie algebra over K, h ⊂ g: a Lie subalgebra, λ: h → K: a character, q: a linear complement of h ⊂ g. Then, the quadruple (g, h, λ, q) satisﬁes the split Duﬂo property if: g is 2-step nilpotent and g = q ⊕ h is compatible with [g, g] ⊂ g; g = K ⋉ Kn (i.e. g is special) and g = q ⊕ h is compatible with 0 ⋉ Kn ⊂ g; (g, h, θ) is a symmetric pair, q = g−θ, λ([q, q]) = 0; or there exists an abelian ideal a ⊂ g containing q. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 25 / 50
- 38. Main Result Theorem (T, 2019) g := 0 ∗ ∗ ∗ 0 0 ∗ ∗ 0 0 0 ∗ 0 0 0 0 ∈ gl4(K) , h ⊂ g: any Lie subalgebra, λ: h → K: any character. dim h − dim h ∩ [g, g] ̸= 1 =⇒ ∃q s.t. (g, h, λ, q) satisﬁes the split Duﬂo property. dim h − dim h ∩ [g, g] = 1 =⇒ ∃q s.t. (g, h, λ, q) satisﬁes the split CG property. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 26 / 50
- 39. Contents 1 Introduction 2 Strict Formulation of Problems 3 Our Method: Split Symmetrization Map 4 Strategy, Bottlenecks and Packaged Tools 5 Demonstration 6 Future Works Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 27 / 50
- 40. Basic Strategy We are considering the nilpotent case. Nilpotency is characterized by the ﬁltrations. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 28 / 50
- 41. Basic Strategy We are considering the nilpotent case. Nilpotency is characterized by the ﬁltrations. → The basic strategy is induction on dimension. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 28 / 50
- 42. Bottlenecks Three bottlenecks to judge the split Duﬂo property. 1 Does ˜Uλ(g)h coincide the image of ˜Sλ(g)h under ˜σq? 2 Does Z( ˜Uλ(g)h) coincide the image of Z( ˜Sλ(g))h under ˜σq? 3 Does ˜σq preserve the multiplication on Z( ˜Sλ(g)h)? Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 29 / 50
- 43. Bottlenecks Three bottlenecks to judge the split Duﬂo property. 1 Does ˜Uλ(g)h coincide the image of ˜Sλ(g)h under ˜σq? 2 Does Z( ˜Uλ(g)h) coincide the image of Z( ˜Sλ(g))h under ˜σq? 3 Does ˜σq preserve the multiplication on Z( ˜Sλ(g)h)? The 3rd bottleneck appear even if h = 0. This bottleneck is solved by induction on dimension. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 29 / 50
- 44. The 1st Bottleneck The 1st bottleneck Does ˜Uλ(g)h coincide the image of ˜Sλ(g)h under ˜σq? The 1st bottleneck is caused by that ˜σq is not necessarily h-equivariant. Proposition If [h, q] ⊂ q ⊕ Ch(q), then ˜σq is h-equivariant. Here, Ch(q) := {Y ∈ h | [Y, q] = 0}. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 30 / 50
- 45. The 2nd Bottleneck The 2nd bottleneck Does Z( ˜Uλ(g)h) coincide the image of Z( ˜Sλ(g))h under ˜σq? The 2nd bottleneck is caused by that ˜Sλ(g)h is not necessarily generated by 1-degree elements. Proposition Suppose that: ˜Uλ(g)h coincides the image of ˜Sλ(g)h under ˜σq, ˜Sλ(g)h is generated by U := {X ∈ q | [h, X] ⊂ hλ}, and [U, q] ⊂ q ⊕ Ch(q). Then, Z( ˜Uλ(g)h) coincides the image of Z( ˜Sλ(g)h) under ˜σq. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 31 / 50
- 46. Packaged Tools For induction on dim g, there are some packaged techniques to reduce dimension of g. a packaged reduction its obstruction the h-quotient reduction Nothing the q-quotient reduction the 2nd bottleneck the (h, q)-subreduction the 1st bottleneck (partially) the h-subreduction Nothing the (q, q)-subreduction the 2nd bottleneck the special h-subredution the 1st bottleneck Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 32 / 50
- 47. The h-quotient reduction Theorem (T, 2019) Let a ⊂ g be an ideal included in Ker λ. Let π : g → g/a be the quotient map, ¯g := g/a, ¯h := h/a, ¯q := π(q), and ¯λ: ¯h → K the induced map of λ. Then, the following are equivalent. The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. The quadruple (¯g, ¯h, ¯λ, ¯q) satisﬁes the split Duﬂo property. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 33 / 50
- 48. the (h, q)-subreduction Theorem (T, 2019) Take Y ∈ h such that ˜σq( ˜Sλ(g)ad(Y ) ) = ˜Uλ(g)ad(Y ) , dim[Y, g] − dim[Y, g] ∩ Ker λ = 1. Moreover, we suppose that g′ := ad(Y )−1(Ker λ) is a subalgebra and let q′ := q ∩ g′. Then, the following are equivalent. The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. The quadruple (g′, h, λ, q′) satisﬁes the split Duﬂo property. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 34 / 50
- 49. Contents 1 Introduction 2 Strict Formulation of Problems 3 Our Method: Split Symmetrization Map 4 Strategy, Bottlenecks and Packaged Tools 5 Demonstration 6 Future Works Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 35 / 50
- 50. Demonstration: Strictly Upper Triangular Matrices Theorem (T, 2019) g := 0 ∗ ∗ ∗ 0 0 ∗ ∗ 0 0 0 ∗ 0 0 0 0 ∈ gl4(K) , h ⊂ g: any Lie subalgebra, λ: h → K: any character. dim h − dim h ∩ [g, g] ̸= 1 =⇒ ∃q s.t. (g, h, λ, q) satisﬁes the split Duﬂo property. dim h − dim h ∩ [g, g] = 1 =⇒ ∃q s.t. (g, h, λ, q) satisﬁes the split CG property. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 36 / 50
- 51. Demonstration: Strictly Upper Triangular Matrices g := 0 ∗ ∗ ∗ 0 0 ∗ ∗ 0 0 0 ∗ 0 0 0 0 ∈ gl4(K) Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 37 / 50
- 52. Demonstration: Strictly Upper Triangular Matrices g := 0 ∗ ∗ ∗ 0 0 ∗ ∗ 0 0 0 ∗ 0 0 0 0 ∈ gl4(K) T = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 , X0 = 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , X1 = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 , Y0 = 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 , Y1 = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 , Z = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 37 / 50
- 53. Demonstration: Strictly Upper Triangular Matrices g = ⟨T, X0, X1, Y0, Y1, Z⟩K [T, X0] = Y0, [T, X1] = Y1, [X0, Y1] = Z, [X1, Y0] = Z Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 38 / 50
- 54. Demonstration: Strictly Upper Triangular Matrices g = ⟨T, X0, X1, Y0, Y1, Z⟩K [T, X0] = Y0, [T, X1] = Y1, [X0, Y1] = Z, [X1, Y0] = Z [g, g] = ⟨Y0, Y1, Z⟩, C(g) = [g, [g, g]] = ⟨Z⟩ Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 38 / 50
- 55. Demonstration: Strictly Upper Triangular Matrices Case 1: Z ∈ Ker λ Case 2: h ⊂ C(g) and Z /∈ Ker λ Case 3: h ∩ [g, g] ̸⊂ C(g) and Z /∈ Ker λ Case 4: dim h − dim h ∩ [g, g] ≥ 2 Case 5: dim h − dim h ∩ [g, g] = 1 (the split CG property) Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 39 / 50
- 56. Demonstration: Strictly Upper Triangular Matrices Case 1: Z ∈ Ker λ Case 2: h ⊂ C(g) and Z /∈ Ker λ Case 3: h ∩ [g, g] ̸⊂ C(g) and Z /∈ Ker λ Case 4: dim h − dim h ∩ [g, g] ≥ 2 Case 5: dim h − dim h ∩ [g, g] = 1 (the split CG property) Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 39 / 50
- 57. We already know. . . Theorem g: a nilpotent Lie algebra over K, h ⊂ g: a Lie subalgebra, λ: h → K: a character, q: a linear complement of h ⊂ g. Then, the quadruple (g, h, λ, q) satisﬁes the split Duﬂo property if: g is 2-step nilpotent and g = q ⊕ h is compatible with [g, g] ⊂ g; g = K ⋉ Kn (i.e. g is special) and g = q ⊕ h is compatible with 0 ⋉ Kn ⊂ g; (g, h, θ) is a symmetric pair, q = g−θ, λ([q, q]) = 0; or there exists an abelian ideal a ⊂ g containing q. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 40 / 50
- 58. We already know. . . Theorem g: a nilpotent Lie algebra over K, h ⊂ g: a Lie subalgebra, λ: h → K: a character, q: a linear complement of h ⊂ g. Then, the quadruple (g, h, λ, q) satisﬁes the split Duﬂo property if: g is 2-step nilpotent and g = q ⊕ h is compatible with [g, g] ⊂ g; g = K ⋉ Kn (i.e. g is special) and g = q ⊕ h is compatible with 0 ⋉ Kn ⊂ g; (g, h, θ) is a symmetric pair, q = g−θ, λ([q, q]) = 0; or there exists an abelian ideal a ⊂ g containing q. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 40 / 50
- 59. Case 1: Z ∈ Ker λ Theorem (the h-quotient reduction) Let a ⊂ g be an ideal included in Ker λ. Let π : g → g/a be the quotient map, ¯g := g/a, ¯h := h/a, ¯q := π(q), and ¯λ: ¯h → K the induced map of λ. Then, the following are equivalent. The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. The quadruple (¯g, ¯h, ¯λ, ¯q) satisﬁes the split Duﬂo property. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 41 / 50
- 60. Case 1: Z ∈ Ker λ Theorem (the h-quotient reduction) Let a ⊂ g be an ideal included in Ker λ. Let π : g → g/a be the quotient map, ¯g := g/a, ¯h := h/a, ¯q := π(q), and ¯λ: ¯h → K the induced map of λ. Then, the following are equivalent. The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. The quadruple (¯g, ¯h, ¯λ, ¯q) satisﬁes the split Duﬂo property. a = C(g), ¯g = ⟨T ⟩ ⋉ ⟨X0, X1, Y0, Y1⟩: special We know that ∃¯q s.t. (¯g, ¯h, ¯λ, ¯q) satisﬁes the split Duﬂo property. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 41 / 50
- 61. Case 2: h ⊂ C(g) and Z /∈ Ker λ h = 0 h = C(g) = ⟨Z⟩ Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 42 / 50
- 62. Case 2: h ⊂ C(g) and Z /∈ Ker λ h = 0: a known case h = C(g) = ⟨Z⟩: It is enough to show. . . Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 42 / 50
- 63. Case 2’: h = C(g) = ⟨Z⟩ and λ ̸= 0 Theorem (the (h, q)-subreduction) Take Y ∈ h such that ˜σq( ˜Sλ(g)ad(Y ) ) = ˜Uλ(g)ad(Y ) , dim[Y, g] − dim[Y, g] ∩ Ker λ = 1. Moreover, we suppose that g′ := ad(Y )−1(Ker λ) is a subalgebra and let q′ := q ∩ g′. Then, the following are equivalent. The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. The quadruple (g′, h, λ, q′) satisﬁes the split Duﬂo property. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 43 / 50
- 64. Case 2’: h = C(g) = ⟨Z⟩ and λ ̸= 0 Theorem (the (h, q)-subreduction) Take Y ∈ h such that ˜σq( ˜Sλ(g)ad(Y ) ) = ˜Uλ(g)ad(Y ) , dim[Y, g] − dim[Y, g] ∩ Ker λ = 1. Moreover, we suppose that g′ := ad(Y )−1(Ker λ) is a subalgebra and let q′ := q ∩ g′. Then, the following are equivalent. The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. The quadruple (g′, h, λ, q′) satisﬁes the split Duﬂo property. Y = Y0 g′ = ⟨X0⟩ ⋉ ⟨T, Y0, Y1, Z⟩: special Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 43 / 50
- 65. Case 3: h ∩ [g, g] ̸⊂ C(g) and Z /∈ Ker λ Theorem (the (h, q)-subreduction) Take Y ∈ h such that ˜σq( ˜Sλ(g)ad(Y ) ) = ˜Uλ(g)ad(Y ) , dim[Y, g] − dim[Y, g] ∩ Ker λ = 1. Moreover, we suppose that g′ := ad(Y )−1(Ker λ) is a subalgebra and let q′ := q ∩ g′. Then, the following are equivalent. The quadruple (g, h, λ, q) satisﬁes the split Duﬂo property. The quadruple (g′, h, λ, q′) satisﬁes the split Duﬂo property. Y = aY0 + bY1 + cZ ∈ h ∩ [g, g] − C(g) g′ = ⟨aX0 − bX1⟩ ⋉ ⟨T, Y0, Y1, Z⟩: special Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 44 / 50
- 66. Contents 1 Introduction 2 Strict Formulation of Problems 3 Our Method: Split Symmetrization Map 4 Strategy, Bottlenecks and Packaged Tools 5 Demonstration 6 Future Works Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 45 / 50
- 67. Future Works Our method algebraic, rustic, clear Duﬂo’s polynomial conjecture CG’s polynomial conjecture Benoist-Corwin-Greenleaf-Fujiwara’s approach analytic, representation theoretic, diﬃcult ⃝ 2-step nilpotent, ⃝ special case ⃝ “common polarization”, × 2-step nilpotent speciﬁc case Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 46 / 50
- 68. Future Works Our method algebraic, rustic, clear Duﬂo’s polynomial conjecture CG’s polynomial conjecture Benoist-Corwin-Greenleaf-Fujiwara’s approach analytic, representation theoretic, diﬃcult ⃝ 2-step nilpotent, ⃝ special case ⃝ “common polarization”, × 2-step nilpotent speciﬁc case Question Show Duﬂo’s polynomial conjecture in the “common polarization” case. Is there some relationship between our method and BCGF’s approach? Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 46 / 50
- 69. Future Works Question Extend our method to the solvable case. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 47 / 50
- 70. Future Works Question Extend our method to the solvable case. Fact (well-known) There exists a solvable Lie algebra g such that σ : Z(S(g)) → Z(U(g)) is NOT ring isomorphic. Fact (Duﬂo, 1970) If g is solvable or semi-simple, then there exists a certain endmorphism D : S(g) → S(g) such that σ ◦ D : Z(S(g)) → Z(U(g)) is ring isomorphic. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 47 / 50
- 71. Future Works Theorem g: a nilpotent Lie algebra over K, h ⊂ g: a Lie subalgebra, λ: h → K: a character, q: a linear complement of h ⊂ g. Then, the quadruple (g, h, λ, q) satisﬁes the split Duﬂo property if g is 2-step nilpotent and g = q ⊕ h is compatible with [g, g] ⊂ g. Question The proof of the above theorem is so complicated and technical. Please make the proof more elegant by expert knowledge about algebra. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 48 / 50
- 72. References T. “An algebraic approach to Duﬂo’s polynomial conjecture in the nilpotent case” Journal of Lie Theory, To appear M. Duﬂo. “Open problems in representation theory of lie groups.” edited by T. Oshima, Katata in Japan, pages 1–5, 1986. Jacques Dixmier. “Sur l’alg`ebre enveloppante d’une alg`ebre de Lie nilpotente.” Arch. Math., 10:321–326, 1959. L. Corwin and F. P. Greenleaf. “Commutativity of invariant diﬀerential operators on nilpotent homogeneous spaces with ﬁnite multiplicity.” Comm. Pure Appl. Math., 45(6):681–748, 1992. Michel Duﬂo. “Caract`eres des groupes et des alg`ebres de Lie r´esolubles.” Ann. Sci. ´Ecole Norm. Sup. (4), 3:23–74, 1970. Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 49 / 50
- 73. Our Result Our method algebraic, rustic, clear Duﬂo’s polynomial conjecture CG’s polynomial conjecture Benoist-Corwin-Greenleaf-Fujiwara’s approach analytic, representation theoretic, diﬃcult ⃝ 2-step nilpotent, ⃝ special case ⃝ “common polarization”, × 2-step nilpotent speciﬁc case Thank you for your attentions! Yoshinori Tanimura An algebraic approach to Duﬂo’s polynomial conjecturein the nilpotent case May 26, 2019 50 / 50

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