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 Rxy := y △ x: right multiplication operator Det Rx = ∆1(x)n1 · · · ∆r(x)nr (nj ≥ 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) [Lx, Ly] = Lx △ y−y △ x, (left symmetric algebra) (C2) ∃s ∈ V ∗ s.t. s(x △ y) is an inner product, (compactness) (C3) Lx has only real eigenvalues. (normality) (Lxy := x △ y : left multiplication operator) In general, clans are      non-associative, 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 ). x :=      

1 2x11

· · · x21

1 2x22

... . . . . . . ... ... xr1 xr2 · · ·

1 2xrr

      .

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 5 / 29

Normal decomposition

V : clan with unit element e0, c1, . . . , cr : complete system of orthogonal primitive idempotents (ci △ cj = δijci, c1 + · · · + cr = e0) Normal decomposition: V = ⊕

1≤j≤k≤r

Vkj, where { Vjj = Rcj (j = 1, . . . , r), Vkj = { x ∈ V ; Lcix = 1

2(δij + δik)x, Rcix = δijx

} . In the case of V = Sym(r, R), cj = Ejj, Vkj = R(Ekj + Ejk).

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 6 / 29

Basic relative invariants

h := {Lx; x ∈ V } (split solvable Lie algebra). H := exp h. Ω := H · e0 ⇒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)m1 · · · ∆r(x)mr (m1, . . . , mr ∈ Z≥0). If p(x) = Det Rx, then we have mk ≥ 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 ⟨ x △ y |z ⟩ = ⟨y |x △ z ⟩ (x, y, z ∈ V ). homogeneous cone Ω ← → 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 ). x :=      

1 2x11

· · · x21

1 2x22

... . . . . . . ... ... xr1 xr2 · · ·

1 2xrr

      . Corresponding cone: Ω = {x ∈ V ; positive definite} . basic relative invariants:∆k(x) = det

(k)(x).

Det Rx = ∆1(x)d · · · ∆r−1(x)d∆r(x) (d = dim K). Dual clan product: x △ y = (x)∗y + y x (x, y ∈ V ).

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 ). x :=      

1 2x11

· · · x21

1 2x22

... . . . . . . ... ... xr1 xr2 · · ·

1 2xrr

      . Corresponding cone: Ω = {x ∈ V ; positive definite} . basic relative invariants:∆k(x) = det

(k)(x).

Det Rx = ∆1(x)d · · · ∆r−1(x)d∆r(x) (d = dim K). Dual clan product: x △ y = (x)∗y + y x (x, y ∈ V ).

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 ). x :=      

1 2x11

· · · x21

1 2x22

... . . . . . . ... ... xr1 xr2 · · ·

1 2xrr

      . Corresponding cone: Ω = {x ∈ V ; positive definite} . basic relative invariants:∆k(x) = det

(k)(x).

Det Rx = ∆1(x)d · · · ∆r−1(x)d∆r(x) (d = dim K). Dual clan product: x △ y = (x)∗y + y x (x, y ∈ V ).

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 ). x :=      

1 2x11

· · · x21

1 2x22

... . . . . . . ... ... xr1 xr2 · · ·

1 2xrr

      . Corresponding cone: Ω = {x ∈ V ; positive definite} . basic relative invariants:∆k(x) = det

(k)(x).

Det Rx = ∆1(x)d · · · ∆r−1(x)d∆r(x) (d = dim K). Dual clan product: x △ y = (x)∗y + y x (x, y ∈ V ).

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 ). x :=      

1 2x11

· · · x21

1 2x22

... . . . . . . ... ... xr1 xr2 · · ·

1 2xrr

      . Corresponding cone: Ω = {x ∈ V ; positive definite} . basic relative invariants:∆k(x) = det

(k)(x).

Det Rx = ∆1(x)d · · · ∆r−1(x)d∆r(x) (d = dim K). Dual clan product: x △ y = (x)∗y + y x (x, y ∈ V ).

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(e0) = idE, 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 Vkj : normal decomposition

Put E = ⊕

k≥2

Vk1, W = ⊕

2≤j≤k≤r

Vkj. Note that W is a subclan of V . V is decomposed as V = V11 ⊕ E ⊕ W = (Rc1

tE

E W ) . We denote general elements x of V by x = λc1 + 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 ∈ W, ξ ∈ E). Then (ϕ, E) is a selfadjoint representation of (W, △ ). With respect to this decomposition, the multiplication is described as x △ y = (λµ)c1 + (µξ + 1

2λη + ϕ(w)η) + (Q(ξ, η) + w △ v),

where y = µc1 + η + v. Calculate Det Rx and express by using Det RW

w

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 13 / 29

Right multiplication operators

R:Right multiplication operator of V Then we have Rλc1+ξ+w =   λ

1 2ξ

λidE Rξ Rξ RW

w

  , where RW is right multiplication operator of W .

Proposition

Det Rλc1+ξ+w = λ1+dim E−dim W Det RW

λw− 1

2 Q[ξ] Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 14 / 29

Right multiplication operators

The basic relative invariants of V are exhausted by { λ, irreducible factors of ∆W

j (λw − 1 2Q[ξ])

(j = 2, . . . , r), where ∆W

2 (w), . . . , ∆W r (w) are the basic relative invariants of W .

.

Theorem

. . ∆1(x), . . . , ∆r(x): the basic relative invariants of V . There exist non-negative integers α2, . . . , αr s.t. ∆j(x) = { λ (j = 1), λ−αj∆W

j (λw − 1 2Q[ξ])

(j = 2, . . . , r).

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 15 / 29

Right multiplication operators

The basic relative invariants of V are exhausted by { λ, irreducible factors of ∆W

j (λw − 1 2Q[ξ])

(j = 2, . . . , r), where ∆W

2 (w), . . . , ∆W r (w) are the basic relative invariants of W .

.

Theorem

. . ∆1(x), . . . , ∆r(x): the basic relative invariants of V . There exist non-negative integers α2, . . . , αr s.t. ∆j(x) = { λ (j = 1), λ−αj∆W

j (λw − 1 2Q[ξ])

(j = 2, . . . , r).

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 15 / 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 16 / 29

In order to determine αj, we introduce . .

1 multiplier matrix,

. .

2 ε-representation. Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 17 / 29

Multiplier matrix

H : split solvable (= lower triangular) Lie group hjj > 0: diagonal component of h ∈ H f : relatively H-invariant (∃χ s.t. f(hx) = χ(h)f(x)) ⇒ ∃τj ∈ R s.t. χ(h) = (h11)2τ1 . . . (hrr)2τr. τ := (τ1, . . . , τr): multiplier of f.

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 18 / 29

Multiplier matrix

Definition

σj = (σj1, . . . , σjr): multiplier of ∆j (∆j(hx) = (h11)2σj1 · · · (hrr)2σjr∆j(x)) σ : the multiplier matrix ⇔ σ =    σ1 . . . σr    = (σjk)1≤j,k≤r

• Remark. (Ishi 2001)

σ is (lower) triangular, σjj = 1. i.e. σ =       1 · · · σ21 1 ... . . . . . . ... ... σr1 σr2 · · · 1       .

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 19 / 29

ε-representation

(ϕ, E): representation of (V, △ ) ε = t(ε1, . . . , εr) ∈ {0, 1}r, cε := ε1c1 + · · · + εrcr.

Definition

(ϕ, E): ε-representation ⇔ Q[E] = H · cε.

• Remark. Put Oε := H · cε. Then one has

Ω = ⊔

ε∈{0,1}r

Oε (Ishi 2000).

Proposition (Graczyk–Ishi)

(ϕ, E): any representation ∃1 ε = ε(ϕ) ∈ {0, 1}r s.t. ϕ is an ε-representation.

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 20 / 29

ε-representation

.

Calculation of ε(ϕ)

. . Put dkj := dim Vkj. l(1) := t(dim E1, . . . , dim Er) (Ej := ϕ(cj)E), l(k) := { l(k−1) − t(0, . . . , 0, dk,k−1, . . . , dr,k−1) l(k−1) (l(k−1)

k−1

> 0), (otherwise). Then ε(ϕ) = t(ε1, . . . , εr) is defined by εk = { 1 (if l(k)

k

> 0), (otherwise).

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 21 / 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 22 / 29

Determination of αj

.

Theorem

. . V = Rc1 ⊕ E ⊕ W : clan of rank r with W : subclan of V (ϕ, E): representation of (W, △ ) (ϕ(w)ξ = ξ △ w). ∆j(x): basic relative invariants of V (j = 1, . . . , r). ∆j(λc1 + ξ + w) = λ−αj∆W

j (λw − 1 2Q[ξ])

(j = 2, . . . , r). Let σW be the multiplier matrix of W and ϕ an ε-representation. Then    α2 . . . αr    = σW    1 − ε2 . . . 1 − εr    where ε =    ε2 . . . εr    ∈ {0, 1}r−1. σV : multiplier matrix of V ⇒ σV = ( 1 σW ε σW ) = (1 σW ) (1 ε Ir−1 ) .

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 23 / 29

Calculation of multiplier matrix

For k = 1, 2, . . . , r − 1, we put E[k] := ⊕

m>k

Vmk, V [k] := ⊕

k<l≤k≤r

Vml.       Rck

tE[k]

E[k] V [k]       V [k] is a subclan of V [k−1]. (R[k], E[k]) is a representation of (V [k], △ ): R[k](x)ξ := ξ △ x (x ∈ V [k], ξ ∈ E[k]). Assume that R[k] is an ε[k]-representation (ε[k] ∈ {0, 1}r−k).

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 24 / 29

Calculation of multiplier matrix

Put Ek :=   Ik−1 1 ε[k] Ir−k   . Recalling σV = (1 σW ) (1 ε Ir−1 ) = (1 σW ) E1, we obtain the following theorem. .

Theorem

. . σV = Er−1Er−2 · · · E1.

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 25 / 29

Vinberg’s polynomials

∥x∥2 := ⟨x|x⟩ . Definition of Vinberg’s polynomials Dj(x) are as follows: Define x(j) = ∑r

k=j x(j) kk ck + ∑ m>k≥j X(j) mk ∈ V [j−1] by

x(1) := x, x(j+1)

kk

:= x(j)

jj x(j) kk −

1 2s0(ck)∥X(j)

kj ∥2

(j < k ≤ r), X(j+1)

mk

:= x(j)

jj X(j) mk − X(j) mj △ X(j) kj

(j < k < m ≤ r). Then Dj(x) := x(j)

jj ∈ R.

Note that each Dj(x) is a relatively H-invariant polynomial.

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 26 / 29

Vinberg’s polynomials

D1(x), . . . , Dr(x) appear in the solution h ∈ H of the equation he0 = x (given x ∈ Ω). The diagonal components are calculated as h2

11 = D1(x),

h2

jj = D1(x)−1 · · · Dj−1(x)−1Dj(x)

(j ≥ 2). This implies Ω = {x ∈ V ; D1(x) > 0, . . . , Dr(x) > 0} = {x ∈ V ; ∆1(x) > 0, . . . , ∆r(x) > 0} . Recalling ∆j(x) are described as ∆j(he0) = (h11)2σj1 · · · (hjj)2σjj, we obtain the main theorem.

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 27 / 29

Explicit expression

.

Main theorem

. . Let σV = (σjk) be the multiplier matrix of V. Then one has ∆1(x) = D1(x), ∆j(x) = Dj(x) ∏

i<j Di(x)τji ,

where τji = −σji + σj,i+1 + · · · + σjj ∈ Z≥0. To determine ∆j(x): Divide Dj(x) by ∆1(x), . . . , ∆j−1(x) until not-divisible ↓ Divide Dj(x) by Di(x) τji-times.

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 28 / 29

Thank you for your attention!

Hideto Nakashima (Kyushu univ.) Basic relative invariants 2014/6/25 29 / 29