重複度１の表現と複素多様体上の

可視的な作用

Multip

licity-fre

ere

pre

sentatio

ns

and

visib

leactio

ns

on

com

ple

xm

anifold

s

ToshiyukiK

obayashi

(RIM

S,K

yoto

Univers

ity)

Sym

posiu

mon

Repre

sentatio

nT

heory

2005

Kakegawa,Shizuoka

Novem

ber15

-18,2005

アブストラクト：

　　正則ベクトル束に群作用が与えられているときに、そ

の正則な大域切断の空間に実現された表現を考える。　この

表現は、

1)底空間が非コンパクトならば一般に無限次元表現

であり、

2)底空間における群軌道が無限個ならば一般に既約

とは程遠い表現になる。

　　しかし、ある種の幾何的な条件（『可視的な作用』）が

満たされていれば、重複度が１という性質がファイバーへの

表現⇒大域切断における表現に伝播することが証明できる。

　　この概説講演では、有限次元および無限次元の表現に

おける重複度１の表現のたくさんの例を紹介し、それが『可

視的な作用』という幾何的な条件からどのように理解できる

かを説明する予定です。

[1]

Mult

iplici

ty-fre

eth

eore

min

bra

nch

ing

pro

ble

ms

ofuni-

tary

hig

hes

tw

eigh

tm

odule

s,P

roc.

Sym

pos

ium

onR

ep-

rese

nta

tion

Theo

ryhel

dat

Sag

a,K

yush

u19

97(e

d.

K.

Mim

achi)

,(1

997)

,9–

17.

[2]

Mult

iplici

tyon

eth

eore

min

the

orbit

met

hod

(wit

hN

as-

rin),

inm

emor

yof

Kar

pel

evic

“Lie

Gro

upsan

dSym

met

-

ric

Spac

es”,

(eds.

S.

Gin

dik

in),

161–

169

Am

er.

Mat

h.

Soc.

2003

[3]

Geo

met

ryof

Mult

iplici

ty-fre

ere

pre

senta

tion

sof

GL(n

),

vis

ible

acti

ons

onflag

vari

etie

s,an

dtr

iunity,

Act

aA

ppl.

Mat

h.81

(200

4),12

9-14

6.

[4]

Mult

iplici

ty-fre

ere

pre

senta

tion

san

dvis

ible

acti

ons

on

com

ple

xm

anifol

ds,

Publ.

RIM

S41

(200

5),49

7-54

9.

表現論シンポジウム講演集，2005pp.33-63

－ 33 －

Eg.1

(Eig

enspace

decom

positio

n)

H:Vectorsp./

C,d

im<

∞

A∈

End

C(H

)1©

s.t.

Ais

dia

gonalizable

,

all

eig

envalu

es

are

distin

ct.

⇒H

=Ce 1

⊕···⊕

Ce n

≃C

n(canonical)

1©&

detA

6=0

⇓

πA

:Z

−→

GL

C(H

)is

MF

∈

∈

n7−→

An

Eg.2

(Fourierseries

expansio

n)

L2(S

1)

≃∑

⊕

n∈

Z

Cei

nx

f(x

)=

∑

n∈

Z

an

einx

Tra

nsla

tio

n(⇒

rep.ofthe

gro

up

S1 )

f(·)

7→f(·−

c)(c

∈S

1≃

R/2π

Z)

S1

yL

2(S

1)

isM

F

－ 34 －

π:

G→

GL

C(H

)

gro

up

Def.

(naive)

(π,H

)is

MF

multip

licity-fre

e

ifdim

Hom

G(τ,π)

≦1

(∀τ

:irre

d.re

p.of

G).

L2(S

1)

∃1

←ein

x

⇔dim

Hom

S1(τ,L

2(S

1))

=1

(∀τ

:irre

d.re

p.

of

S1)

⇒S

1y

L2(S

1)

isM

F

－ 35 －

Eg.3

(Taylo

rexpansio

n,

Laure

nt

expansio

n)

f(z1,...,zn)=

∑

α=

(α1,...,α

n)a

αzα1

1···z

αn

n

Poin

t(too

obvio

us)

∃1

aα∈

Cfo

reach

α

⇑

dim

Hom

(S

1)n(τ

,O

(0)

≦1

(∀τ

:irre

d.re

p.of

(S1)n

)

i.e.

MF

Eg.4

(Peter-W

eyl)

¡¡

¡

irre

d.re

p.of

G×

G

G:com

pact

(Lie

)gro

up

L2(G)≃

∑⊕

τ∈

G

τ⊠

τ∗

Tra

nsla

tio

n(⇒

rep.of

G×

G)

f(·)

7→f(g

−1

1·g2)

⇒G

×G

yL

2(G

)is

MF

－ 36 －

Eg.5

(Sphericalharm

onics)

Hl:=

f∈

C∞

(S

n−1):

∆S

n−1f

=−

l(l+

n−

2)f

L2(S

n−1)≃

∞ ∑

⊕

l=0

Hl

O(n

)irre

d.

⇓

O(n)

yL

2(S

n−1)

isM

F

⊗-p

roduct

rep.

SL

2(C)

πk

y

irre

d.S

k(C2)

(k=

0,1,2,...)

Eg.6

(Cle

bsch-G

ord

an)

πk⊗

πl≃

πk+

l⊕

πk+

l−2⊕

···⊕

π|k−

l|

↑

MF

－ 37 －

Notatio

n

Hig

htest

weig

ht

λ=

(λ1

...,λ

n)∈

Zn,λ1≥

λ2≥

···≥

λn

⇓

πG

Ln

λ≡

πλ

:irred.

rep.

of

GL

n(C)

Eg.

λ=

(k,0,...,0)

↔G

Ln(C)

yS

k(C

n)

λ=

(1,...,1

︸︷︷

︸

k

,0,...,0)

↔G

Ln(C)

yΛ

k(C

n)

⊗-p

roduct

rep.

(G

Ln-c

ase)

Eg.7

(Pie

ri’s

law)

π(λ

1,...,λ

n)⊗

π(k,0

,...,0

)

≃⊕

µ1≥

λ1≥···≥

µn≥

λn

∑(µ

i−λ

i)=

k

π(µ1,...,µ

n)

↑

MF

as

aG

Ln-m

odule

.

－ 38 －

⊗-p

roduct

rep.

(contin

ued)

Eg.

(countere

xam

ple

)π(2

,1,0

)⊗

π(2

,1,0

)is

NO

TM

F

as

aG

L3(C)-m

odule

.

⊗-p

roduct

rep.

(contin

ued)

λ=

(a,···,a

︸︷︷

︸

p,

b,···,b

︸︷︷

︸

q)∈

Zn,

a≥

b

Eg.

8.

(Stem

bridge

2001,K

–)

πλ

⊗π

νis

MF

as

aG

Ln(C

)-m

odule

if

1)

min

(a−

b,p,q)=

1(a

nd

νis

any),

or 2)

min

(a−

b,p,q)=

2and

νis

ofth

efo

rmν

=(x···x

︸︷︷

︸

n1

y···y

︸︷︷

︸

n2

z···z

︸︷︷

︸

n3

)(x≥

y≥

z),

or 3)

min

(a−

b,p,q)≥

3,

&m

in(x−

y,

y−

z,n1,n2,n3)=

1.

⋆

⋆

－ 39 －

Eg.9

(G

Ln↓

GL

n−1)

πG

Ln

(λ1,...,λ

n)| G

Ln−1

≃⊕

λ1≥

µ1≥···≥

µn−1≥

λn

πG

Ln−1

(µ1,...,µ

n−1)

↑M

Fas

aG

Ln−1(C

)-m

odule

⇓Applicatio

n

Gelfand-T

setlin

basis

GL

n

πλ

yV

Fin

da

‘good’basis

of

V

Rem

ark.

Vis

not

necessarily

MF

as

a(C

×)n-m

odule

.

Idea

ofG

elfand-T

setlin

basis

GL

n(C)

∪

y

MF

C×

×G

Ln−1(C)

∪

y

MF

(C×)2

×G

Ln−2(C)

∪

y

MF

. . .. . .

∪

y

MF

(C×)n

－ 40 －

Infinite

dim

ensio

nalcase

(cf.

Eg.9

)

Eg.1

0.

(U(p,q)↓

U(p−

1,q))

∀π:

irre

d.

unitary

rep.

of

U(p,q)

with

hig

est

weig

ht

⇒re

strictio

nπ| U

(p−1,q)is

MF

as

aU(p

−1,q)-

module

Eg.1

1.

(G

L−

GL

duality

)

N=

mn

⇒G

Lm

×G

Lny

S(C

N)

≃

S(M

(m

,n;C))

This

rep.

isM

F

•Explicit

form

ula

···

GL

m−

GL

nduality

•G

enera

lizatio

n

(Bra

nchin

glaw

ofholo

.disc.

w.r.t.

sym

metric

pair)

Hua-K

ostant-S

chm

id-K

-¢ ¢ ¢ ¢

£ £ £ £

each

com

ponent···

finite

dim

∞−

dim

－ 41 －

Anothergenera

lizatio

n

Eg.1

2(K

ac’s

MF

space)

S(C

N)

isstill

MF

as

aG

Lm−1×

GL

nm

odule

We

recall:

π:

Ggro

up→

GL

C(H

)

Def.

(naive)

(π,H

)is

MF

multip

licity-fre

eif

dim

Hom

G(τ,π)≤

1(∀τ:

irre

d.

rep.

of

G).

－ 42 －

Observ

atio

n

n≤

1⇔

End(C

n)

iscom

mutative.

(π,H

):

unitary

rep.of

G

Def.

(π,H

)is

MF

if

End

G(H

)is

com

mutative.

More

genera

lly,(

,W

):top.re

p.

Def.

(,W

)is

MF

ifany

unitary

subre

p.

(π,H

)is

MF

(∃G-inj.

cont.hom

.H

→W

)

Eg.2

(Fourierseries

expansio

n)

L2(S

1)

≃∑

⊕

n∈

Z

Cei

nx

f(x

)=

∑

n∈

Z

an

einx

einx

(cf.

centrifu

galsepara

tor)

Tra

nsla

tio

n(⇒

rep.ofthe

gro

up

S1 )

f(·)

7→f(·−

c)(c

∈S

1≃

R/2π

Z)

S1

yL

2(S

1)

isM

F

－ 43 －

Eg.1

3(Fouriertra

nsfo

rm)

L2(R)≃

∫ ⊕ RC

eiζ

xdζ

(direct

integra

lofHilbert

sp.)

f(x

)=∫ R

f(ζ

)eiζ

xdζ

Tra

nsla

tio

n(⇒

regularre

pre

sentatio

non

L2(R))

f(·)

7→f(·−

c)

Ry

L2(R

)is

“M

F”

. . .contin

uous

spectru

m

Sym

metric

Space

Eg.1

4.

G/K

:Rie

mannia

nSym

m.Space

⇒G

yL

2(G

/K

)is

MF.

Eg.1

5.

(counterexam

ple

)

G/H

:Sem

isim

ple

Sym

m.Space

⇒G

yL

2(G

/H

)

isNO

Talw

ays

MF.

－ 44 －

Eg.1

4-A.

G/K

=SL(n,R

)/SO

(n)

L2(G

/K

)≃

∫⊕ λ

1≥···≥

λn

Σλ

i=0

Hλ

dλ

cont.spec.

MF

Hλ:∞

-dim

,irred.rep.of

G

Eg.1

5-A.

G/H

=SL(n,R

)/SO

(p,q

)

Multip

licity

ofm

ost

cont.spec.

inL

2(G

/H

)

=n!

p!q

!>

1if

p,q

>0.

⇒NO

TM

F

Vectorbundle

case

Eg.1

5-B.

G/K

=G

L(n

,R)/O

(n)

Vτ:=

G× K

τ→

G/K

τ:

unitary

rep.of

K

⇒G

yL

2(V

τ)

unitary

Gy

L2(V

τ)

isNO

Talw

ays

MF.

but

itis

MF

ifτ≃

Λk(C

n)

(0≤

k≤

n)

－ 45 －

Exam

ple

sofM

ultip

licity-F

ree

reps

•Peter-W

eyltheore

m

•Cart

an-H

elg

ason

theore

m

•Bra

nchin

glaws:

GL

n↓

GL

n−1,

On↓

On−1

•Cle

bsch-G

ord

an

form

ula

•Pie

ri’s

law

•G

Lm-G

Ln

duality

•Pla

nchere

lfo

rmula

for

L2(G

/K

)(G

/K

:Rie

mannia

nsy

mm

etr

icsp

aces)

•(G

elfand-G

raev-V

ers

hik

)canonical

repre

sentatio

ns

•Hua-K

ostant-S

chm

idK

-type

form

ula

•(K

ac)

linearm

ultip

licity-fre

espaces

•(Panyushev)

sphericalnilpotent

orb

its

•(Stem

bridge)

multip

licity-fre

etensor

pro

duct

repre

sentatio

ns

of

GL

n

etc.

Accord

ingly,various

techniq

ues

can

be

applied

ineach

MF

case.

Forexam

ple

,one

can

1)

look

foran

open

orb

it

ofa

Bore

lsubgro

up.

2)

apply

Little

wood-R

ichard

son

rule

sand

variants.

3)

use

com

putat’l

com

bin

atorics.

4)

em

plo

ythe

com

mutativity

ofthe

Hecke

alg

ebra

.

5)

apply

Schur-W

eylduality

and

Howe

duality

.

－ 46 －

Aim

···

To

give

asim

ple

prin

cip

le

that

expla

ins

the

property

MF

ofall

these

exam

ple

s,

and

more.

§2M

Ftheorem

multip

licity

free

H,K

:Lie

groups

D:

com

ple

xm

fd.

P→

D:

H-equiv.

prin

cip

alK

-b’d

le

µ:

K→

GL

C(V

)

⇓ Settin

g1

H-equiv.

holo

.vectorb’d

le:

V:=

P×

KV

→D

⇓ HyO

(D,V)=

holo

.sectio

ns

－ 47 －

Settin

g2

σ1

yP

diff

eo.

σσ2

yK

auto.

σ3

yH

auto.

s.t.

σ(h

pk)=

σ(h)σ

(p)σ(k)

∋∋∋

HP

K

σy

D≃

P/K

anti-holo

morphic

Recall

Hy

Px

K

↓ D

B B B B B B B B B B

BM

∩∩

Hy

Px

Kµ

−→

GL

C(V

)

↓ D

B⊂

Pσ

:=p∈

P:σ(p)=

p

subset

⇓

M≡

MB

:=k∈

K:bk

∈H

b(∀b∈

B)

Assum

ptio

n

(a)

HB

K=

P

(b)

µ| M

isM

F

say,

µ| M

≃l ⊕

i=1

νi

(c)

µ

σ≃

µ∗

as

K-m

odule

s

νi

σ≃

ν∗ i

as

M-m

odule

s(∀i)

－ 48 －

Assum

ptio

n

(a)

HB

Kcontain

san

interior

poin

tof

P

(b)

µ| M

isM

F

say,

µ| M

≃l ⊕

i=1

νi

(c)

µ

σ≃

µ∗

as

K-m

odule

s

νi

σ≃

ν∗ i

as

M-m

odule

s(∀i)

Poin

tofAssum

ptio

ns

(a)

···

base

sp.

(b)

···

fiber

(c)

···

oft

en

autom

atic

Theore

m(M

Ftheore

m)

Assum

e∃σ

and

∃B

⊂P

σ

satisfy

ing

(a)∼(c).

⇒H

yO

(D

,V)

isM

F.

－ 49 －

Poin

t

•propagatio

nofM

Fproperty

fiberM

F⇒

sectio

ns

MF

•geom

etry

ofbase

space

···

‘visib

leactio

n’

yV

H↓

y

D

⇒H

y

O(D

,V)

Assum

ptio

n(c)

µ

σ≃

µ∗

as

K-m

odule

s

hold

sif

σis

aW

eylin

volu

tio

n

i.e. σ∈

Aut(

K),

σ2

=id

s.t.

σ(g)=

g−1

on

som

em

ax

toru

sof

K

e.g

.K

=U(n),

σ(g)=

g

－ 50 －

§3Visib

leactio

n

holo

morp

hic

Hy

Dcom

ple

xm

fd,connected

Def.

The

actio

nis

(stro

ngly)visib

le

∃D

′⊂

Dopen

subset

if∃σ

yD

anti-holo

morp

hic

∃N

⊂D

totally

real

s.t.

σ| N

=id

Nm

eets

every

H-o

rbit

inD

′

σstabilizes

every

H-o

rbit.

Assum

ptio

n(a)

+(stro

ngly)

visib

le

actio

n

Assum

ptio

n(a):

HB

K=

P

forsom

eB

⊂P

σ

⇒N

:=

PσK

/K

meets

every

H-o

rbit

on

D:=

P/K

⇒H

yD

visib

le

(σ:

involu

sio

n⇒

N:

totally

real)

－ 51 －

Exam

ple

ofVisib

leactio

ns

T=

a∈

C:|a|=

1

(≃

S1)

Eg.

Ty

C⊃

R

∈

az7→

az

Rm

eets

every

T-orbit

R

⇒T-actio

non

Cis

visib

le.

holo

morphic

Hy

(D

,J)com

ple

xm

fd,connected

Def.

Actio

nis

visib

leif

∃D

′⊂

D,

open

∃N

⊂D

s.t.

totally

real

Nm

eets

every

H-orbit

Jx(T

xN

)⊂

Tx(H

·x)

(x∈

N)

－ 52 －

sym

ple

ctic

Hy

(D

,ω)

sym

ple

ctic

mfd

Def.

(G

uille

min

-Stern

berg

,

Huckle

berry-W

urz

bacher)

Actio

nis

coisotro

pic

(orm

ultip

licity-fre

e)

if

Tx(H

·x)⊥

ω⊂

Tx(H

·x)

(x∈

D)

isom

etric

Hy

(D,g)

Rie

mannia

nm

fd

Def.

(Podesta-T

horb

erg

sson)

Actio

nis

polarif

∃N

⊂D

s.t.

Nm

eets

every

H-o

rbit.

TxN

⊥T

x(H

·x)

(x∈

N)

－ 53 －

Hy

D

com

pact

com

pact,K

ahle

r

Rie

mannia

n

Polar

Visib

le&

dim

S=

dim

M

Coisotro

pic

(m

ultip

licity-

free)

Com

ple

xSym

ple

ctic

Stro

ngly

Visib

le

Com

ple

x

Exam

ple

sofVisib

leactio

ns

Eg.

Ty

Cis

visib

le.

∪ R⇓

Eg.

Tn

yC

nis

visib

le.

∪ Rn

⇓

Eg.

Tn

yPn−1

Cis

visib

le.

∪

Pn−1

R⇓

Eg.

U(1

)×

U(n−

1)

yB

n

(fu

llflag

variety

)is

visib

le.

Eg.

U(n)

yPn−1

C×

Bn

isvisib

le.

－ 54 －

Unders

tandin

gofvisib

leactions

HL

⊃

⊃G ∪ Gσ

:=

Tn

U(1

)×

U(n−

1)

⊃

⊃U(n)

∪O

(n)

Geom

etr

y² ±

¯ °P

n−1R

meets

every

Tn-o

rbit

on

Pn−1C

‖G

σ/G

σ∩

L‖ H

‖G

/L

(visib

leaction)

←→

Gro

up

® ­

© ªG

=H

GσL

⇒Eg.3

l

Gro

up

® ­

© ªG

=L

GσH

⇒Eg.9

l

Gro

up

² ±

¯ °(G

×G)=

dia

g(G)(

Gσ×

Gσ)(H

×L)

⇒Eg.7

⇓T

heore

m

Various

kin

ds

ofM

Fre

sult

inclu

din

g

•(Fourierseries)

Ty

L2(T)

Eg.2

•(Taylo

rseries)

Tn

yO

(C

n)

Eg.3

•(G

Ln↓

GL

n−1)Restrictio

nπ| G

Ln−1Eg.9

•(Pie

ri)

π⊗

Sk(C

n)

Eg.7

•(K

ac)

GL

m−1×

GL

ny

S(C

mn)

Eg.1

2

－ 55 －

Eg.

G/K

com

pact

sym

m.sp.

⇒G

yG

C/K

Cis

visib

le.

⇓T

heore

m

Eg.1

6.

Gy

L2(G

/K

)is

MF.

Eg.1

7.(vectorbundle

case)

Vk=

U(n)

×O

(n)Λ

k(C

n)→

U(n

)/O

(n)

U(n

)y

L2(V

k)

isM

F.

Eg.

G/K

non-c

om

pact

sym

m.sp.

⇒G

yΩ

⊂cro

wn

GC/K

Cis

visib

le.

⇓T

heore

m

Eg.1

6′ .

Gy

L2(G

/K

)is

MF.

Eg.1

7′ .

(vectorbundle

case)

Vk=

GL(n,R

)×

O(n

)Λk(C

n)→

U(n

)/O

(n)

GL(n

,R)

yL

2(V

k)

isM

F.

－ 56 －

G/K

Herm

itia

nsym

m.space

Eg.

G=

SL(2

,R)

K=

SO

(2)

H=

a0

0a−1

:a

>0

G/K

≃z∈

C:|z|<

1

Ky

G/K

visib

leH

yG

/K

visib

le

K-orbits

H-orbits

Theorem

H⊂

G⊃

K

Assum

e

G/K

Herm

itia

nsym

m.sp.

(G

,H)

sym

metric

pair

⇒H

yG

/K

isvisib

le

⇓T

heorem

Eg.18

πλ,π

µ:

hig

hest

wt.m

odule

s

ofscalartype

⇒π

λ⊗

πµ

isM

F

Eg.19

πλ

:hig

hest

wt.m

odule

ofscalartype

(G,H

):

sym

metric

pair

⇒π

λ| H

isM

F

－ 57 －

Also,fo

rfinite

dim

ensio

nalcase

⇓T

heore

m

Eg.2

0(O

kada,1998)

gC

=gl(

n,C)

λ=

(a,···,a

︸︷︷

︸

p

,b,···,b

︸︷︷

︸

n−

p

)∈

Zn,a≥

b

πλ| h

Cis

MF

if

hC

=

gl(

k,C)+

gl(

n−

k,C)

(1≤

k≤

n)

o(n,C)

sp(n 2,C)

(n:even)

G:

non-com

pact,sim

ple

Lie

gp.,

G/K

Herm

itia

n

Eg.

SU(p,q

),S

O(n,2

),S

p(n,R

),

SO

∗(2

n),E

6(−14),E

7(−25)

gC

=k C

+p+

+p−

Def.

(π,V

)∈

Gunitary

hig

hest

wt

rep

⇔υ∈

V∞

:dπ(X

)υ

=0

(∀X

∈p+)

6=0

µ KW

rite

π=

πG(µ)

(µ∈

K)

Def.

π:

holo

morphic

discrete

serie

s

⇔Hom

G(π,L

2(G))

6=0

π:

scalartype

⇔dim

µ=

1

－ 58 －

Def.

(G,H

)sym

metric

pair,

holo

morphic

type

⇔∃τ∈

Aut(

G),

τ2

=id

s.t.

(G

τ) 0

⊂H

⊂G

τ

τy

G/K

holo

morphic

gC

=k C

+p++

p−

∪∪

∪∪

hC

=gτ C

=kτ C

+pτ ++

pτ −

tτ⊂

kτ

Cartan

∩∩

Cartan

t⊂

k

ν1,.

..,ν

k

maxim

alset

of

strongly

orth.roots

in∆

(p−

τ+

,tτ)

Note:

k=

R-rank

G/H

Def.

(G,H

)sym

metric

pair,

holo

morphic

type

⇔∃τ∈

Aut(

G),

τ2

=id

s.t.

(G

τ) 0

⊂H

⊂G

τ

τy

G/K

holo

morphic

Eg.

τ=

θ,

H=

K

Eg.

G=

Sp(n,R

)(n=

p+

q)

H=

Sp(p,R

)×

Sp(q

,R)

H=

U(p,q

)

H=

U(n)

(=K

)

cf.

H=

GL(n

,R)

not

holo

.type

－ 59 －

Theore

m

πG(µ)∈

G:holo

.disc.series,

scalarty

pe

(G

,H

):sym

metric

pair,

holo

morp

hic

type

⇒

πG(µ

)H

≃∑

⊕

a1≥···≥

ak≥0

aj∈

N

πH

µ

∣ ∣ ∣

tτ−

k∑ j=

1a

jνj

H=

K···

Schm

id(1969)

GL(n

,C)

yN

⊂nilpotent

orb

itM

(n,C)

Eg.2

1(Panyushev

1994)

O(N

)is

MF

as

GL(n

,C)-m

odule

⇔N

=N

(2p,1

n−2p)

(0≤

∃p≤

n 2)

N(2

p,1

n−2p)∋

01

00

p

···︷

︸︸

︷0

10

0

0···n−

2p

0

︷

︸︸

︷

－ 60 －

Eg.

0≤

2p≤

n

1)

U(n)

yN

(2p,1

n−2p)is

visib

le.

2)(P

anyushev)

O(N

2p,1

n−2p)

isM

F

N(2

p,1

n−2p)∋

01

00

p

···

︷

︸︸

︷0

10

0

0···n−

2p

0

︷︸︸

︷

Sketch

ofpro

of

G=

U(n)

H=

U(p)×

U(q)

(p+

q=

n)

G× H

M(p,q;

C)→

N(2

p,1

n−2p)

Hy

M(p

,q;

C)

visib

le

⇒G

yN

(2p,1

n−2p)

visib

le

More

exam

ple

sofvisib

leactio

ns

Eg.

n1+

n2+

n3

=p+

q=

n

Grp(R

n)

meets

every

orb

itof

U(n

1)×

U(n2)×

U(n3)y

Grp(C

n)

(⇒

visib

leactio

n)

⇔(U(n1)×

U(n2)×

U(n3))×

O(n)×(U(p)×

U(q))

։U(n)

⇔m

in(n

1+

1,n2+

1,n3+

1,p,q)

≦2

－ 61 －

⇓T

heorem

MF

property

ofthe

followin

g

•G

Lm

×G

Ln

yS(C

mn)

Eg.1

1

•G

Lm−1×

GL

ny

S(C

mn)

Eg.1

2

•the

Stem

brid

ge

list

of

πλ⊗

πν

Eg.8

•G

Ln↓(G

Lp×

GL

q)

Eg.2

2

•G

Ln↓(G

Ln1×

GL

n2×

GL

n3)

Eg.2

3

•∞

-dim

ensio

nalversio

ns

...

⊗-p

roduct

rep.

(contin

ued)

λ=

(a,···,a

︸︷︷

︸

p,

b,···,b

︸︷︷

︸

q)∈

Zn,

a≥

b

Eg.

8.

(Stem

bridge

2001,K

–)

πλ

⊗π

νis

MF

as

aG

Ln(C

)-m

odule

if

1)

min

(a−

b,p,q)=

1(a

nd

νis

any),

or 2)

min

(a−

b,p,q)=

2and

νis

ofth

efo

rmν

=(x···x

︸︷︷

︸

n1

y···y

︸︷︷

︸

n2

z···z

︸︷︷

︸

n3

)(x≥

y≥

z),

or 3)

min

(a−

b,p,q)≥

3,

&m

in(x−

y,

y−

z,n1,n2,n3)=

1.

⋆

⋆

－ 62 －

Eg.2

2.

(G

Ln↓(G

Lp×

GL

q))

n=

p+

q

πG

Ln

(x,...,x

,y,...,y

,z,...,z

)| G

Lp×

GL

qis

MF

(

n1

(

n2

(

n3

ifm

in(p,q)≤

2or

ifm

in(n1,n2,n3,x−

y,y−

z)≤

1(K

ostant

n3

=0;K

rattenthale

r1998)

Eg.2

3.

(G

Ln↓(G

Ln1×

GL

n2×

GL

n3))

n=

n1+

n2+

n3

πG

Ln

(a,...,a

,b,...,b)| G

Ln1×

GL

n2×

GL

n3

isM

F

(

p

(

q

ifm

in(n1,n2,n3)≤

1or

ifm

in(p,q,a−

b)≤

2

Orb

itm

ethod

GAd

yg∗

⊃O

G

∪↓pr

HAd∗

yh∗

⊃O

H

H⊂

G⊃

Ksym

m.

pair

Herm

itia

nsym

m.pair

g⊃

k⊃

center

⇔R·z

⊂g∗

6=

0

Theore

m(Nasrin

&K

-)

IfO

G∩

R·z6=

∅,then

#(O

G∩

pr−

1(O

H))/H

≦1

forany

OH.

－ 63 －