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Quartic CartanM unzner polynomials and Casimir operators FUJII, Shinobu National Institute of Technology (KOSEN), Oshima College Workshop on the isoparametric theory At Beijing Normal University June 6th., 2019 FUJII, Shinobu ( NIT

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Quartic Cartan–M¨ unzner polynomials and Casimir operators

FUJII, Shinobu

National Institute of Technology (KOSEN), Oshima College

Workshop on the isoparametric theory At Beijing Normal University June 6th., 2019

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 1 / 34

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Contents of this talk

1 Introduction 2 Casimir elements and Casimir operators 3 Main Theorem : Quartic Cartan–M¨

unzner polynomials and Casimir operators

4 Squared-norms of moment maps and Casimir operators 5 Summary and Problems

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 2 / 34

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Introduction

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 3 / 34

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Quartic Cartan–M¨ unzner polynomials

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Quartic Cartan–M¨ unzner polynomials

Definition f ∈ R[x1, . . . , x2n]: Quartic Cartan–M¨ unzner polynomials

def

⇐ ⇒ f is homogeneous of degree four, ∥grad f(P )∥2 = 16 ∥P ∥6 for P ∈ R2n, ∆f(P ) = 8c ∥P ∥2 for P ∈ R2n & ∃c ∈ R.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 4 / 34

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Quartic Cartan–M¨ unzner polynomials

Definition f ∈ R[x1, . . . , x2n]: Quartic Cartan–M¨ unzner polynomials

def

⇐ ⇒ f is homogeneous of degree four, ∥grad f(P )∥2 = 16 ∥P ∥6 for P ∈ R2n, ∆f(P ) = 8c ∥P ∥2 for P ∈ R2n & ∃c ∈ R. FACTS The restriction of f to S2n−1 defines an isoparametric hypersurface in S2n−1 with four distinct principal curvatures and with the multiplicities (m1, m2).

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 4 / 34

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Examples of Quartic Cartan–M¨ unzner polynomials

The followings are Quartic Cartan–M¨ unzner polynomials: Example (cf. Nomizu (1973)) For x1, x2 ∈ Rn, we define F (x1, x2) := ( ∥x1∥2 − ∥x2∥2)2 + 4 ⟨x1, x2⟩2 .

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 5 / 34

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Examples of Quartic Cartan–M¨ unzner polynomials

The followings are Quartic Cartan–M¨ unzner polynomials: Example (cf. Nomizu (1973)) For x1, x2 ∈ Rn, we define F (x1, x2) := ( ∥x1∥2 − ∥x2∥2)2 + 4 ⟨x1, x2⟩2 . Example (cf. Ozeki & Takeuchi (1976)) For K ∈ {R, C, H} and X ∈ Mn,2(K), we define F (X) := 3 ( Tr(tXX) )2 − 4 Tr ( (tXX)2) .

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 5 / 34

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Examples of Quartic Cartan–M¨ unzner polynomials

The followings are Quartic Cartan–M¨ unzner polynomials: Example (cf. Nomizu (1973)) For x1, x2 ∈ Rn, we define F (x1, x2) := ( ∥x1∥2 − ∥x2∥2)2 + 4 ⟨x1, x2⟩2 . Example (cf. Ozeki & Takeuchi (1976)) For K ∈ {R, C, H} and X ∈ Mn,2(K), we define F (X) := 3 ( Tr(tXX) )2 − 4 Tr ( (tXX)2) . Remark The above examples are invariant polynomials for some SO–actions.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 5 / 34

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Examples of Quartic Cartan–M¨ unzner polynomials

{P0, P1, . . . , Pm}: symmetric Clifford system, i.e., Pi ∈ Sym2n(R), PiPj + PjPi = 2δijI2n.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 6 / 34

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Examples of Quartic Cartan–M¨ unzner polynomials

{P0, P1, . . . , Pm}: symmetric Clifford system, i.e., Pi ∈ Sym2n(R), PiPj + PjPi = 2δijI2n. Definition We define F : R2n → R as F (x) := ∥x∥4 − 2

m

∑

i=0

⟨x, Pix⟩2 . We call F (x) a Cartan–M¨ unzner polynomial of OT–FKM type.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 6 / 34

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Examples of Quartic Cartan–M¨ unzner polynomials

{P0, P1, . . . , Pm}: symmetric Clifford system, i.e., Pi ∈ Sym2n(R), PiPj + PjPi = 2δijI2n. Definition We define F : R2n → R as F (x) := ∥x∥4 − 2

m

∑

i=0

⟨x, Pix⟩2 . We call F (x) a Cartan–M¨ unzner polynomial of OT–FKM type. Remark We define H := ⟨ PiPj

  • 0 ≤ i < j ≤ m

⟩ ⊊ SO(2n). Then, the above F (x) is H–invariant.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 6 / 34

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Our problems and Main Theorems

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 7 / 34

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Our problems and Main Theorems

Problem

1 Study properties of quartic CM polynomials as invariant

polynomials for some group actions,

2 Characterize quartic CM polynomials from a view of

invariant theory.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 7 / 34

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Our problems and Main Theorems

Problem

1 Study properties of quartic CM polynomials as invariant

polynomials for some group actions,

2 Characterize quartic CM polynomials from a view of

invariant theory.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 7 / 34

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Our problems and Main Theorems

Problem

1 Study properties of quartic CM polynomials as invariant

polynomials for some group actions,

2 Characterize quartic CM polynomials from a view of

invariant theory.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 7 / 34

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Our problems and Main Theorems

Problem

1 Study properties of quartic CM polynomials as invariant

polynomials for some group actions,

2 Characterize quartic CM polynomials from a view of

invariant theory. Main Results (rough version )

1 Some quartic CM polynomials can be written by Casimir

  • perators,

2 Casimir operator approach is related to moment map

approach.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 7 / 34

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Casimir elements and Casimir operators

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 8 / 34

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Notations

Assume that g: a semisimple Lie algebra, V : a finite dimensional R–vector space, σ : g → gl(V ): a representation of g, Bg: the Killing form of g, βσ : g × g → R: symmetric quadratic form on g defined by (X, Y ) − → Tr(σ(X)σ(Y )), (βσ: trace form associated to σ) FACTS βσ: Ad(G)-invariant, where G is a Lie group whose Lie algebra is g.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 9 / 34

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Casimir elements

Definition {Xi}: basis of g, { X∗

i

} : Bg-dual basis of g, i.e., Bg(Xi, X∗

j ) = δi,j,

Then, we define Cg ∈ gl(V ) as follows: Cg := ∑

i

XiX∗

i .

We call Cg a Casimir element of g.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 10 / 34

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Casimir elements

Definition {Xi}: basis of g, { X∗

i

} : Bg-dual basis of g, i.e., Bg(Xi, X∗

j ) = δi,j,

Then, we define Cg ∈ gl(V ) as follows: Cg := ∑

i

XiX∗

i .

We call Cg a Casimir element of g. Remark The Casimir element Cg does not depend on the choices of basis of g and dual basis.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 10 / 34

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Casimir elements

Definition Cg := ∑

i XiX∗ i is called a Casimir element of g.

Remark The Casimir element Cg does not depend on the choices of basis of g and dual basis. FACTS Cg ∈ U(g), where U(g) is the universal enveloping algebra of g, For all X ∈ g, [Cg, X] = 0.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 11 / 34

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Casimir operators

n := Ker σ, n∗ := { X ∈ g

  • ∀Y ∈ n, Bg(X, Y ) = 0

} , FACTS g = n ⊕ n∗. σ|n∗ : n∗ → gl(V ): faithful representation, FACTS βσ|n∗: non–degenerated on n∗, where βσ is the trace form associated to σ.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 12 / 34

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Casimir operators

Definition {Xi}: basis of n∗, { X∗

i

} : βσ|n∗-dual basis of n∗, i.e., βσ|n∗(Xi, X∗

j ) = δi,j,

Then, we define Cσ ∈ gl(V ) as follows: Cσ := ∑

i

σ(Xi)σ(X∗

i ).

We call Cg a Casimir operator associated to σ.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 13 / 34

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Casimir operators

Definition {Xi}: basis of n∗, { X∗

i

} : βσ|n∗-dual basis of n∗, i.e., βσ|n∗(Xi, X∗

j ) = δi,j,

Then, we define Cσ ∈ gl(V ) as follows: Cσ := ∑

i

σ(Xi)σ(X∗

i ).

We call Cg a Casimir operator associated to σ. Remark The Casimir operator Cσ does not depend on the choices of basis of n∗ and dual basis.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 13 / 34

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Casimir operators

Definition Cσ := ∑

i σ(Xi)σ(X∗ i ) is called a Casimir operator

associated to σ. Remark The Casimir operator Cσ does not depend on the choices of basis of n∗ and dual basis.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 14 / 34

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Casimir operators

Definition Cσ := ∑

i σ(Xi)σ(X∗ i ) is called a Casimir operator

associated to σ. Remark The Casimir operator Cσ does not depend on the choices of basis of n∗ and dual basis. FACTS For all ξ ∈ U(g), [Cσ, ξ] = 0, Cσ = σ(Cg).

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 14 / 34

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Main Theorem : Quartic Cartan–M¨ unzner polynomials and Casimir operators

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 15 / 34

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Our problems and expectations

Problem Study properties of quartic CM polynomials as invariant polynomials for some group actions. Expectation There are some relations between quartic CM polynomials and Casimir operators. Definition (reminder) f ∈ R[x1, . . . , x2n]: Quartic Cartan–M¨ unzner polynomials

def

⇐ ⇒ f is homogeneous of degree four, ∥grad f(P )∥2 = 16 ∥P ∥6 for P ∈ R2n, ∆f(P ) = 8c ∥P ∥2 for P ∈ R2n & ∃c ∈ R.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 16 / 34

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Our notations

Assume that G/K: compact irreducible Hermitian symmetric space of rank two, g = k ⊕ p: the Cartan decomposition of g, Bg: the Killing form of g, ⟨−, −⟩: inner product on p defined by ⟨X, Y ⟩ = −Bg(X, Y ), J: complex structure on p, ω: symplectic form on p defined by ω(X, Y ) = ⟨J(X), Y ⟩, K ↷ p: Hamiltonian action, σ : k → sp(p, ω): symplectic representation of k.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 17 / 34

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Our notations

Then, we obtain the following commutative diagram: k

σ

  • sp(p, ω)

≃

  • Sym(p)

≃

  • O(p)2
  • U(k)

U(sp(p, ω)) U(Sym(p)) U(O(p)2)

Here, Sym(p) = { symmetric transformation on p} with {X, Y } = 2(XJY − Y JX) for X, Y ∈ Sym(p), O(p)2 = { real quadratic form on p}, sp(p, ω) ∋ ξ → 1 2Jξ ∈ Sym(p): isomorphic.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 18 / 34

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Isotropy reps. of symmetric spaces of rank two

G/K Herm.

  • f OT–FKM type

O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ × E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ SO(5) × SO(5)/ SO(5) × ×

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 19 / 34

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Isotropy reps. of symmetric spaces of rank two

G/K Herm.

  • f OT–FKM type

O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ × E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ SO(5) × SO(5)/ SO(5) × ×

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Isotropy reps. of symmetric spaces of rank two

G/K Herm.

  • f OT–FKM type

O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ × E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ SO(5) × SO(5)/ SO(5) × ×

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 19 / 34

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Isotropy reps. of symmetric spaces of rank two

G/K Herm.

  • f OT–FKM type

O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ × E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ SO(5) × SO(5)/ SO(5) × ×

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 19 / 34

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

Main Results (F. (in preparation)) G/K: compact irreducible Hermitian symmetric spaces of rank two and of classical type, : O(2 + n)/ O(2) × O(n), U(2 + n)/ U(2) × U(n), SO(10)/ U(5). Then, Quartic CM polynomials f(P ) obtained from the isotropy reps. of each G/K can be wriiten by Casimir

  • perators:

G/K f(P )

O(2 + n)/ O(2) × O(n)

8n Cad |o(2) − 64n Cad |o(n)

U(2 + n)/ U(2) × U(n)

−96 Cad |u(2) + 56n Cad |u(n)

SO(10)/ U(5)

−60 Cad |u(1) + 2048 Cad |su(5)

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 20 / 34

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

Outline of the proof:

1 Since G/K is Hermitian, we consider a direct sum

decomposition k = o(2) ⊕ k′ as Lie algebras.

2 we compute the Casimir operators of o(2) and k′

respectively by using the following diagram: k

σ

  • sp(p, ω)

≃

  • Sym(p)

≃

  • O(p)2
  • U(k)

U(sp(p, ω)) U(Sym(p)) U(O(p)2)

3 we compute coefficients a and b satisfying

a

  • Cad |o(2) + b

Cad |k′ is a quartic CM polynomial.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 21 / 34

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

Remark Our main theorem essentially coincide with the followings: Ozeki & Takeuchi (1976),

  • F. (2010) (⊂ F. & Tamaru (2015)),
  • R. Miyaoka (2014).

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 22 / 34

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

Remark Our main theorem essentially coincide with the followings: Ozeki & Takeuchi (1976),

  • F. (2010) (⊂ F. & Tamaru (2015)),
  • R. Miyaoka (2014).

The differences among them are how to compute Cartan–M¨ unzner polynomials of degree four.

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Squared-norms of moment maps and Casimir operators

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Moment maps and Quartic CM polynomials

We study originally the followings: Question Is there any relation between quartic CM polynomials and moment maps? More precisely, Question Quartic CM polynomials = squared–norms of moment maps ?

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

G/K Herm. moment maps O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ ○ E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ SO(5) × SO(5)/ SO(5) ×

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 25 / 34

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

G/K Herm. moment maps O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ ○ E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ SO(5) × SO(5)/ SO(5) ×

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 25 / 34

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

G/K Herm. moment maps O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ ○ E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ (in prepararion) SO(5) × SO(5)/ SO(5) × ?

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 25 / 34

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

G/K Herm. moment maps O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ ○ E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ (in prepararion) SO(5) × SO(5)/ SO(5) × ?

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 25 / 34

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Isotropy reps. of cpt. irred. Herm. symm. sps. of rank 2

G/K Herm. moment maps O(2 + n)/ O(2) × O(n) ○ ○ U(2 + n)/ U(2) × U(n) ○ ○ SO(10)/ U(5) ○ ○ E6/ U(1) · Spin(10) ○ ○ Sp(2 + n)/ Sp(2) × Sp(n) × ○ (in prepararion) SO(5) × SO(5)/ SO(5) × ? Question Is there any relation between representations of quartic CM polynomials by Casimir operators and ones by moment maps?

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Main results

Assume that (V, ω): R–symplectic vector space of dim 2n, G: Lie group, g: Lie algebra of G, ρ : G → Sp(V, ω) : Hamiltonian action, µ : V → g∗: moment map for G–action, σ : g → sp(V, ω) : differential representation of ρ, FACTS sp(V, ω)

Φ

≃ O(R2n)2 as Lie algebras. Here, O(R2n)2 = {real quadratic forms with 2n vaiables}.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 26 / 34

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Main results

Moreover, we assume that βσ: trace form associated to σ, Cσ: Casimir operator associated to σ, g × g ∋ (X, Y ) → ⟨X, Y ⟩σ := −βσ(X, Y ) ∈ R, g∗ ≃ g by ⟨−, −⟩σ, µ : V ∋ P → µ(P ) ∈ g∗ ≃ g: moment map, Main Results (F.) ∥µ(P )∥2

σ =

Φ(Cσ). Here, Φ : U(sp(V, ω)) → U(O(R2n)2).

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 27 / 34

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Main results

Main Results (F.) ∥µ(P )∥2

σ =

Φ(Cσ). Here, Φ : U(sp(V, ω)) → U(O(R2n)2).

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Main results

Main Results (F.) ∥µ(P )∥2

σ =

Φ(Cσ). Here, Φ : U(sp(V, ω)) → U(O(R2n)2). Schur’s lemma asserts that

∃c ∈ R \ {0} s.t., ∥µ(P )∥2 σ = c ∥µ(P )∥2 ,

where the norm of the right-hand side is defined by the Killing form.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 28 / 34

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Main results

Main Results (F.) ∥µ(P )∥2

σ =

Φ(Cσ). Here, Φ : U(sp(V, ω)) → U(O(R2n)2). Schur’s lemma asserts that

∃c ∈ R \ {0} s.t., ∥µ(P )∥2 σ = c ∥µ(P )∥2 ,

where the norm of the right-hand side is defined by the Killing form. Remark For construction of quartic CM polynomials, Casimir approach

essentially

= moment map approach.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 28 / 34

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Summary and Problems

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 29 / 34

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Summary

Problem

1 Study properties of quartic CM polynomials as invariant

polynomials for some group actions,

2 Characterize quartic CM polynomials from a view of

invariant theory. Main Results

1 Some quartic CM polynomials can be written by Casimir

  • perators,

2 Casimir operators of symplectic representations can be

written as squared–norms of some moment maps.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 30 / 34

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Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

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SLIDE 55

Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

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SLIDE 56

Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian. Sorry, I can’t explain the reason.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

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SLIDE 57

Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian. Sorry, I can’t explain the reason. By using Casimir operators, we may find the reason:

  • Cad |sp(n) ̸= 0,

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

slide-58
SLIDE 58

Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian. Sorry, I can’t explain the reason. By using Casimir operators, we may find the reason:

  • Cad |sp(n) ̸= 0,

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

slide-59
SLIDE 59

Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian. Sorry, I can’t explain the reason. By using Casimir operators, we may find the reason:

  • Cad |sp(n) ̸= 0,

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

slide-60
SLIDE 60

Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian. Sorry, I can’t explain the reason. By using Casimir operators, we may find the reason:

  • Cad |sp(n) ̸= 0,

By using a direct sum decomposition sp(2) = u(2) ⊕ Sym2(C)j, we obtain

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

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SLIDE 61

Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian. Sorry, I can’t explain the reason. By using Casimir operators, we may find the reason:

  • Cad |sp(n) ̸= 0,

By using a direct sum decomposition sp(2) = u(2) ⊕ Sym2(C)j, we obtain

  • Cad |u(2) ̸= 0 and
  • Cad |Sym2(C)j = 0,

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

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SLIDE 62

Summary

We have some problems for our moment map approach: For the case of Sp(2 + n)/ Sp(2) × Sp(n), the isotropy representation is not a Hamiltonian action. However, the restriction of the action to U(1) × Sp(1) × Sp(n) is Hamiltonian. Sorry, I can’t explain the reason. By using Casimir operators, we may find the reason:

  • Cad |sp(n) ̸= 0,

By using a direct sum decomposition sp(2) = u(2) ⊕ Sym2(C)j, we obtain

  • Cad |u(2) ̸= 0 and
  • Cad |Sym2(C)j = 0,

u(2) = u(1) ⊕ sp(1).

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 31 / 34

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Problems

Conjecture For every isoparametric hypersurface M in S2n−1 with four distinct principal curvatures, ∃K: a Lie group s.t. the corresponding quartic CM polynomial is invariant for an SO(2) × K–action on R2n,

  • Cρ|so(2) ̸= 0 and

Cρ|k ̸= 0, where k is the Lie algebra of K,

∃a, ∃b s.t. a

Cρ|so(2) + b Cρ|k is a quartic CM polynomial which defines M.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 32 / 34

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Further Problems

Problem Find some relations between theory of quartic CM polynomial (i.e., isoparametric theory) and Stanley–Reisner theory, Study some combinatorial properties of quartic CM polynomials.

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 33 / 34

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Further Problems

Problem Find some relations between theory of quartic CM polynomial (i.e., isoparametric theory) and Stanley–Reisner theory, Study some combinatorial properties of quartic CM polynomials.

For researchers of algebra, (quartic) CM polynomials are very interesting objects!

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 33 / 34

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SLIDE 66

Thank you for your attention!

FUJII, Shinobu (NIT (KOSEN), Oshima) Quartic CM polynomials and Casimir ops 2019/06/06 34 / 34