FUJII, Shinobu National Institute of Technology (KOSEN), Oshima - PowerPoint PPT Presentation

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

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

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

Quartic Cartan–M¨ unzner polynomials FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 4 / 34

Quartic Cartan–M¨ unzner polynomials Definition f ∈ R [ x 1 , . . . , x 2 n ] : Quartic Cartan–M¨ unzner polynomials def ⇐ ⇒ f is homogeneous of degree four, ∥ grad f ( P ) ∥ 2 = 16 ∥ P ∥ 6 for P ∈ R 2 n , ∆ f ( P ) = 8 c ∥ P ∥ 2 for P ∈ R 2 n & ∃ c ∈ R . FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 4 / 34

Quartic Cartan–M¨ unzner polynomials Definition f ∈ R [ x 1 , . . . , x 2 n ] : Quartic Cartan–M¨ unzner polynomials def ⇐ ⇒ f is homogeneous of degree four, ∥ grad f ( P ) ∥ 2 = 16 ∥ P ∥ 6 for P ∈ R 2 n , ∆ f ( P ) = 8 c ∥ P ∥ 2 for P ∈ R 2 n & ∃ c ∈ R . FACTS The restriction of f to S 2 n − 1 defines an isoparametric hypersurface in S 2 n − 1 with four distinct principal curvatures and with the multiplicities ( m 1 , m 2 ) . FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 4 / 34

Examples of Quartic Cartan–M¨ unzner polynomials The followings are Quartic Cartan–M¨ unzner polynomials: Example ( cf. Nomizu (1973) ) For x 1 , x 2 ∈ R n , we define ( ∥ x 1 ∥ 2 − ∥ x 2 ∥ 2 ) 2 + 4 ⟨ x 1 , x 2 ⟩ 2 . F ( x 1 , x 2 ) := FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 5 / 34

Examples of Quartic Cartan–M¨ unzner polynomials The followings are Quartic Cartan–M¨ unzner polynomials: Example ( cf. Nomizu (1973) ) For x 1 , x 2 ∈ R n , we define ( ∥ x 1 ∥ 2 − ∥ x 2 ∥ 2 ) 2 + 4 ⟨ x 1 , x 2 ⟩ 2 . F ( x 1 , x 2 ) := Example ( cf. Ozeki & Takeuchi (1976) ) For K ∈ { R , C , H } and X ∈ M n, 2 ( K ) , we define ( ) 2 − 4 Tr ( ( t XX ) 2 ) F ( X ) := 3 Tr( t XX ) . FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 5 / 34

Examples of Quartic Cartan–M¨ unzner polynomials The followings are Quartic Cartan–M¨ unzner polynomials: Example ( cf. Nomizu (1973) ) For x 1 , x 2 ∈ R n , we define ( ∥ x 1 ∥ 2 − ∥ x 2 ∥ 2 ) 2 + 4 ⟨ x 1 , x 2 ⟩ 2 . F ( x 1 , x 2 ) := Example ( cf. Ozeki & Takeuchi (1976) ) For K ∈ { R , C , H } and X ∈ M n, 2 ( K ) , we define ( ) 2 − 4 Tr ( ( t XX ) 2 ) F ( X ) := 3 Tr( t XX ) . 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

Examples of Quartic Cartan–M¨ unzner polynomials { P 0 , P 1 , . . . , P m } : symmetric Clifford system, i.e., P i ∈ Sym 2 n ( R ) , P i P j + P j P i = 2 δ ij I 2 n . FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 6 / 34

Examples of Quartic Cartan–M¨ unzner polynomials { P 0 , P 1 , . . . , P m } : symmetric Clifford system, i.e., P i ∈ Sym 2 n ( R ) , P i P j + P j P i = 2 δ ij I 2 n . Definition We define F : R 2 n → R as ∑ m F ( x ) := ∥ x ∥ 4 − 2 ⟨ x, P i x ⟩ 2 . i =0 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

Examples of Quartic Cartan–M¨ unzner polynomials { P 0 , P 1 , . . . , P m } : symmetric Clifford system, i.e., P i ∈ Sym 2 n ( R ) , P i P j + P j P i = 2 δ ij I 2 n . Definition We define F : R 2 n → R as ∑ m F ( x ) := ∥ x ∥ 4 − 2 ⟨ x, P i x ⟩ 2 . i =0 We call F ( x ) a Cartan–M¨ unzner polynomial of OT–FKM type. Remark ⟨ � ⟩ � 0 ≤ i < j ≤ m ⊊ SO(2 n ) . Then, We define H := P i P j the above F ( x ) is H –invariant. FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 6 / 34

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

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

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

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

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 operators, 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

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

Notations Assume that g : a semisimple Lie algebra, V : a finite dimensional R –vector space, σ : g → gl ( V ) : a representation of g , B g : the Killing form of g , β σ : g × g → R : symmetric quadratic form on g defined by ( X, Y ) �− → Tr( σ ( X ) σ ( Y )) , ( β σ : trace form associated to σ ) FACTS β σ : 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

Casimir elements Definition { X i } : basis of g , { } X ∗ : B g -dual basis of g , i.e., i B g ( X i , X ∗ j ) = δ i,j , Then, we define C g ∈ gl ( V ) as follows: ∑ X i X ∗ C g := i . i We call C g a Casimir element of g . FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 10 / 34

Casimir elements Definition { X i } : basis of g , { } X ∗ : B g -dual basis of g , i.e., i B g ( X i , X ∗ j ) = δ i,j , Then, we define C g ∈ gl ( V ) as follows: ∑ X i X ∗ C g := i . i We call C g a Casimir element of g . Remark The Casimir element C g 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

Casimir elements Definition C g := ∑ i X i X ∗ i is called a Casimir element of g . Remark The Casimir element C g does not depend on the choices of basis of g and dual basis. FACTS C g ∈ U ( g ) , where U ( g ) is the universal enveloping algebra of g , For all X ∈ g , [ C g , X ] = 0 . FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 11 / 34

Casimir operators n := Ker σ , � { } n ∗ := � ∀ Y ∈ n , B g ( X, Y ) = 0 X ∈ g , FACTS g = n ⊕ n ∗ . σ | n ∗ : n ∗ → gl ( V ) : faithful representation, FACTS β σ | n ∗ : non–degenerated on n ∗ , where β σ is the trace form associated to σ . FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 12 / 34

Casimir operators Definition { X i } : basis of n ∗ , { } X ∗ : β σ | n ∗ -dual basis of n ∗ , i.e., i β σ | n ∗ ( X i , X ∗ j ) = δ i,j , Then, we define C σ ∈ gl ( V ) as follows: ∑ σ ( X i ) σ ( X ∗ C σ := i ) . i We call C g a Casimir operator associated to σ . FUJII, Shinobu ( NIT (KOSEN), Oshima ) Quartic CM polynomials and Casimir ops 2019/06/06 13 / 34

Casimir operators Definition { X i } : basis of n ∗ , { } X ∗ : β σ | n ∗ -dual basis of n ∗ , i.e., i β σ | n ∗ ( X i , X ∗ j ) = δ i,j , Then, we define C σ ∈ gl ( V ) as follows: ∑ σ ( X i ) σ ( X ∗ C σ := i ) . i We call C g a Casimir operator associated to σ . Remark The Casimir operator C σ 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

Casimir operators Definition C σ := ∑ i σ ( X i ) σ ( X ∗ i ) is called a Casimir operator associated to σ . Remark The Casimir operator C σ 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