Basic relative invariants of homogeneous convex cones Hideto - PowerPoint PPT Presentation

Basic relative invariants of homogeneous convex cones Hideto Nakashima Kyushu university (JSPS Research Fellow) 2014/6/25 RIMS, Kyoto university Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 1 / 29

Background Ω ⊂ V : homogeneous convex cone ∆ 1 ( x ) , . . . , ∆ r ( x ): basic relative invariants of Ω Ω = { x ∈ V ; ∆ 1 ( x ) > 0 , . . . , ∆ r ( x ) > 0 } . . Theorem (Vinberg 1963) . Homogeneous convex domains ⇔ Clans Homogeneous convex cones ⇔ Clans with unit . R x y := y △ x : right multiplication operator Det R x = ∆ 1 ( x ) n 1 · · · ∆ r ( x ) n r ( n j ≥ 1) . Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 2 / 29

Contents 1. Preliminary clans, basic relative invariants, representations, ... 2. Inductive structure of a clan and of the basic relative invariants 3. Introduce the multiplier matrix and ε -representations 4. Explicit formula of the basic relative invariants Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 3 / 29

Contents 1. Preliminary clans, basic relative invariants, representations, ... 2. Inductive structure of a clan and of the basic relative invariants 3. Introduce the multiplier matrix and ε -representations 4. Explicit formula of the basic relative invariants Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 3 / 29

Clans (compact normal left symmetric algebras) V : finite-dimensional real vector space △ : bilinear product in V Definition ( V, △ ) is a clan ⇔ the following three conditions are satisfied: (C1) [ L x , L y ] = L x △ y − y △ x , (left symmetric algebra) (C2) ∃ s ∈ V ∗ s.t. s ( x △ y ) is an inner product, (compactness) (C3) L x has only real eigenvalues. (normality) ( L x y := x △ y : left multiplication operator)  non-associative,   In general, clans are non-commutative,  no unit element.  Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 4 / 29

Examples V = Herm( r, K ) , ( K = R , C , or H ) . x △ y := x y + y ( x ) ∗ ( x, y ∈ V ) . 1   0 · · · 0 2 x 11 . ... . 1   . x 21 2 x 22   x := . .  ... ...  .  . 0    1 x r 1 x r 2 · · · 2 x rr Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 5 / 29

Normal decomposition V : clan with unit element e 0 , c 1 , . . . , c r : complete system of orthogonal primitive idempotents ( c i △ c j = δ ij c i , c 1 + · · · + c r = e 0 ) ⊕ Normal decomposition : V = V kj , where 1 ≤ j ≤ k ≤ r { V jj = R c j ( j = 1 , . . . , r ) , x ∈ V ; L c i x = 1 { } V kj = 2 ( δ ij + δ ik ) x, R c i x = δ ij x . In the case of V = Sym( r, R ) , c j = E jj , V kj = R ( E kj + E jk ) . Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 6 / 29

Basic relative invariants h := { L x ; x ∈ V } (split solvable Lie algebra). H := exp h . Ω := H · e 0 ⇒ homogeneous cone. In particular, H � Ω: simply transitively. Definition . . . 1 f : Ω → R : relatively H -invariant ⇔ ∃ χ : H → R : 1-dim. representation s.t. f ( hx ) = χ ( h ) f ( x ) . . . 2 ∆ j ( x ): relatively H -invariant irreducible polynomials ( j = 1 , . . . , r ) ⇒ the basic relative invariants Remark . ( Ishi 2001 , Ishi–Nomura 2008 ) ∀ p ( x ): relatively H -invariant polynomial ⇒ p ( x ) = (const)∆ 1 ( x ) m 1 · · · ∆ r ( x ) m r ( m 1 , . . . , m r ∈ Z ≥ 0 ) . If p ( x ) = Det R x , then we have m k ≥ 1 ( k = 1 , . . . , r ) . Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 7 / 29

Dual clan Definition △ ( V, ): the dual clan of V △ ⟨ ⟩ y | z = ⟨ y | x △ z ⟩ ( x, y, z ∈ V ) . x Ω ∗ homogeneous cone Ω ← → ↕ dual ↕ △ clan ( V, △ ) ← → ( V, ) △ Relation between △ and : △ △ x △ y + x y = y △ x + y x. Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 8 / 29

Examples V = Herm( r, K ) , ( K = R , C , or H ) . x △ y := x y + y ( x ) ∗ ( x, y ∈ V ) .  1  2 x 11 0 · · · 0 . ... . 1   x 21 2 x 22 .   x := . .  ... ...  .   . 0   1 x r 1 x r 2 · · · 2 x rr Corresponding cone: Ω = { x ∈ V ; positive definite } . ( k ) ( x ) . basic relative invariants: ∆ k ( x ) = det Det R x = ∆ 1 ( x ) d · · · ∆ r − 1 ( x ) d ∆ r ( x ) ( d = dim K ) . Dual clan product: △ y = ( x ) ∗ y + y x ( x, y ∈ V ) . x Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 9 / 29

Examples V = Herm( r, K ) , ( K = R , C , or H ) . x △ y := x y + y ( x ) ∗ ( x, y ∈ V ) .  1  2 x 11 0 · · · 0 . ... . 1   x 21 2 x 22 .   x := . .  ... ...  .   . 0   1 x r 1 x r 2 · · · 2 x rr Corresponding cone: Ω = { x ∈ V ; positive definite } . ( k ) ( x ) . basic relative invariants: ∆ k ( x ) = det Det R x = ∆ 1 ( x ) d · · · ∆ r − 1 ( x ) d ∆ r ( x ) ( d = dim K ) . Dual clan product: △ y = ( x ) ∗ y + y x ( x, y ∈ V ) . x Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 9 / 29

Examples V = Herm( r, K ) , ( K = R , C , or H ) . x △ y := x y + y ( x ) ∗ ( x, y ∈ V ) .  1  2 x 11 0 · · · 0 . ... . 1   x 21 2 x 22 .   x := . .  ... ...  .   . 0   1 x r 1 x r 2 · · · 2 x rr Corresponding cone: Ω = { x ∈ V ; positive definite } . ( k ) ( x ) . basic relative invariants: ∆ k ( x ) = det Det R x = ∆ 1 ( x ) d · · · ∆ r − 1 ( x ) d ∆ r ( x ) ( d = dim K ) . Dual clan product: △ y = ( x ) ∗ y + y x ( x, y ∈ V ) . x Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 9 / 29

Examples V = Herm( r, K ) , ( K = R , C , or H ) . x △ y := x y + y ( x ) ∗ ( x, y ∈ V ) .  1  2 x 11 0 · · · 0 . ... . 1   x 21 2 x 22 .   x := . .  ... ...  .   . 0   1 x r 1 x r 2 · · · 2 x rr Corresponding cone: Ω = { x ∈ V ; positive definite } . ( k ) ( x ) . basic relative invariants: ∆ k ( x ) = det Det R x = ∆ 1 ( x ) d · · · ∆ r − 1 ( x ) d ∆ r ( x ) ( d = dim K ) . Dual clan product: △ y = ( x ) ∗ y + y x ( x, y ∈ V ) . x Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 9 / 29

Examples V = Herm( r, K ) , ( K = R , C , or H ) . x △ y := x y + y ( x ) ∗ ( x, y ∈ V ) .  1  2 x 11 0 · · · 0 . ... . 1   x 21 2 x 22 .   x := . .  ... ...  .   . 0   1 x r 1 x r 2 · · · 2 x rr Corresponding cone: Ω = { x ∈ V ; positive definite } . ( k ) ( x ) . basic relative invariants: ∆ k ( x ) = det Det R x = ∆ 1 ( x ) d · · · ∆ r − 1 ( x ) d ∆ r ( x ) ( d = dim K ) . Dual clan product: △ y = ( x ) ∗ y + y x ( x, y ∈ V ) . x Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 9 / 29

Representations of clans E : a real Euclidean vector space with ⟨·|·⟩ E Definition Let ϕ : V → L ( E ) = { Linear maps on E } . △ ( ϕ, E ): a selfadjoint representation of the dual clan ( V, ) : ϕ ( x ) ∗ = ϕ ( x ) and ϕ ( e 0 ) = id E , △ ϕ ( x y ) = ϕ ( x ) ϕ ( y ) + ϕ ( y ) ϕ ( x ) , where ϕ ( x ) (resp. ϕ ( x ) ) is lower (resp. upper) triangular part of ϕ ( x ) . △ △ i.e. ϕ : ( V, ) → (Sym( E ) , ) is a homomorphism of a clan. Definition . Q : E × E → V : bilinear map associated with ϕ : ⟨ Q ( ξ, η ) | x ⟩ = ⟨ ϕ ( x ) ξ | η ⟩ E ( ξ, η ∈ E, x ∈ V ) . Q [ ξ ] := Q ( ξ, ξ ) and Q [ E ] := { Q [ ξ ]; ξ ∈ E } . Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 10 / 29

Contents 1. Preliminary clans, basic relative invariants, representations, ... 2. Inductive structure of a clan and of the basic relative invariants 3. Introduce the multiplier matrix and ε -representations 4. Explicit formula of the basic relative invariants Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 11 / 29

Inductive structure of V Ω: Homogeneous cone V : clan associated with Ω V = ⊕ j ≤ k V kj : normal decomposition ⊕ ⊕ Put E = V k 1 , W = V kj . k ≥ 2 2 ≤ j ≤ k ≤ r Note that W is a subclan of V . V is decomposed as ( R c 1 t E ) V = V 11 ⊕ E ⊕ W = . E W We denote general elements x of V by x = λc 1 + ξ + w ( λ ∈ R , ξ ∈ E, w ∈ W ) . Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 12 / 29

Inductive structure of V Proposition Define a linear map ϕ : W → L ( E ) by △ ϕ ( w ) ξ := ξ ( w ∈ W, ξ ∈ E ) . w △ Then ( ϕ, E ) is a selfadjoint representation of ( W, ) . With respect to this decomposition, the multiplication is described as x △ y = ( λµ ) c 1 + ( µξ + 1 2 λη + ϕ ( w ) η ) + ( Q ( ξ, η ) + w △ v ) , where y = µc 1 + η + v . Calculate Det R x and express by using Det R W w Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 13 / 29

Right multiplication operators R :Right multiplication operator of V  0 0  λ 1  , Then we have R λc 1 + ξ + w = 2 ξ λ id E R ξ  R W 0 R ξ w where R W is right multiplication operator of W . Proposition Det R λc 1 + ξ + w = λ 1+dim E − dim W Det R W λw − 1 2 Q [ ξ ] Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 14 / 29