The slide is used in ALTReT 2019. This result will appear in Journal of Lie Theory in the near future.

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