HAL Id: hal-00325594

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HAL Id: hal-00325594

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Poisson Homology in Degree 0 for some Rings of Symplectic Invariants

Frédéric Butin

To cite this version:

Frédéric Butin. Poisson Homology in Degree 0 for some Rings of Symplectic Invariants. 2008. �hal-

00325594�

(2)

Poisson Homology in Degree

0

for some Rings of Sympleti Invariants

FrédériBUTIN 1

Abstrat

Let

g

^be ^anite-dimensionalsemi-simpleLiealgebra,

h

^a^Cartan ^subalgebra ^of

g

^,^and

W

^its^Weyl ^group. ^The

group

W

^ats^diagonally^on

V := h ⊕ h ^∗

^,âs^wellâsôn

C[V ]

^.^The^purposeôf^thisârtile îs^to^study^the^Poisson

homologyofthealgebraofinvariants

C[V ] ^W

^endowed^with^the^standard^sympleti^braket.

Tobegin with,wegivegeneralresults aboutthePoissonhomologyspaeindegree

0

^,^denoted^by

HP 0 (C[V ] ^W )

^,

intheasewhere

g

^is^of^type

B n − C n

^or

D n

^,^results^whih^supportÂlev'sônjeture.^Then^weâre^fousing^the

interestonthepartiularasesofranks

2

^and

3

^,^by^omputing^the^Poisson^homology^spaeⁱⁿ^degree

0

ⁱⁿ^the^ases

where

g

^is^of^type

B 2

⁽

so 5

^),

D 2

⁽

so 4

^),^then

B 3

⁽

so 7

^),^and

D 3 = A 3

⁽

so 6 ≃ sl 4

^).Înôrder^to^do^this,^we^makeûse

ofafuntionalequationintroduedbyY.Berest,P.EtingofandV.Ginzburg.Wereover,byadierentmethod,

the resultestablished by J.Alevand L. Foissy,aording to whihthe dimensionof

HP 0 (C[V ] ^W )

^equals

2

^for

B 2

^.^Then^weâlulate^the^dimensionôf^this^spaeând^we^show^thatîtîsêqual^to¹^for

D 2

^.^Weâlsoâlulateît

fortherank

3

âses,^we^show^thatîtîsêqual^to

3

^for

B 3 − C 3

^and

1

^for

D 3 = A 3

^.

Key-words

Alev'sonjeture;Pfa;Poissonhomology;Weylgroup;invariants;Berest-Etingof-Ginzburgequation.

1 Introdution

Let

G

^be^a^nite^subgroup^of^the^sympleti^group

Sp(V )

^,^where

V

^is^a

C−

^vetor^spae^of^dimension

2n

^.^Then

the algebraof polynomialfuntions on

V

^, ^denoted ^by

C[V ]

^, îs â ^Poisson âlgebra ^for ^the ^standard ^sympleti

braket,and as

G

îs â^subgroupôf ^the^sympleti^group,^the âlgebraôfinvariants,denotedby

C[V ] ^G

^,îsâlso â

Poissonalgebra.

Severalartiles weredevotedtotheomputation ofPoissonhomologyandohomology ofthealgebraofinvari-

ants

C[V ] ^G

^. În ^partiular, ^Y. ^Berest, ^P. Êtingof ând ^V. ^Ginzburg, ⁱⁿ ^[BEG04℄, ^prove ^that ^the

0−

^th ^spae ^of

Poissonhomologyof

C[V ] ^G

^isnite-dimensional.

Aftertheirworks[AL98 ℄and[AL98℄,J.Alev,M.A.Farinati,T.LambreandA.L.Solotarestablishafundamental

resultin[AFLS00 ℄ :theyomputeall thespaes ofHohshildhomology andohomologyof

A n (C) ^G

^for ^every

nitesubgroup

G

^of

Sp _2n C

^.

Besides, J.Alevand L.Foissyshowin[AF06 ℄thatthe dimensionof thePoissonhomologyspaeindegree

0

^of

C[h ⊕ h ^∗ ] ^W

îsêqual ^to ^theôneôf ^the ^Hohshild^homology ^spae ⁱⁿ^degree

0

^of

A 2 (C) ^W

^, ^where

h

^is ^a^Cartan

subalgebra ofasemi-simpleLiealgebraofrank

2

^with^Weyl^group

W

^.

Inthe following,givenanite-dimensionalsemi-simpleLiealgebra

g

^,^a^Cartan^subalgebra

h

^of

g

^,^and ^its^Weyl

group

W

^, ^we âre înterested ⁱⁿ ^the ^Poisson ^homology ôf

C[V ] ^G

ⁱⁿ ^the ^ase ^where

V

^is ^the ^sympleti ^spae

V := h ⊕ h ^∗

^and

G := W

^.

Thegroup

W

^ats ^diagonally^on

V

^,^this înduesânâtion ôf

W

^on

C[V ]

^.^We^denote ^by

C[V ] ^W

^the^algebra^of

invariants underthis ation.Endowedwith thestandard sympletibraket, thisalgebra is aPoisson algebra.

Theationofthegroup

W

^on

V

âlsoînduesânâtionôf

W

^on^the^Weyl^algebra

A n (C)

^.

Tobeginwith,wewill givegeneralresultsaboutthePoissonhomologyspaeindegree

0

^of

C[V ] ^W

^for^the^types

B n − C n

^and

D n

^,^results^whih^supportÂlev'sônjetureândêstablishâ^framework^forâ^possible^proof.^Setions

2.3,2.4 and2.5ontainthemainresultsofthisstudy.

Nextwewilluse theseresults inorderto ompletelyalulatethePoisson homologyspae indegree

0

^,^denoted

by

HP 0 (C[V ] ^W )

^,ⁱⁿ^the^ase^where

g

^is

so 5

^(i.e.

B 2

⁾^so^we^reover,^by^a^dierent^method,^the^resultestablished byJ.AlevandL.Foissyfor

so 5

ⁱⁿ^the^artile^{[AF06 ℄,}^namely

dim HP 0 (C[V ] ^W ) = 2

^thenⁱⁿ^the^ase ^where

g

is

so 4

^(i.e.

D 2 = A 1 × A 1

⁾^by^showing^that

dim HP 0 (C[V ] ^W ) = 1

^.^Finally^we^will ^prove^the^important^property

forrank

3

^:

Proposition1 (Poissonhomologyindegree

0−

^for

g

^of^rank

3

⁾

Let

HP 0 (C[V ] ^W )

^be^the^Poisson^homology^spaeⁱⁿ^degree

0

^of

C[V ] ^W

^and

HH 0 A n (C) ^W

theHohshildhomol-

ogyspae indegree

0

^of

A n (C) ^W

^.

For

g

^of^type

B 3

⁽

so 7

^),^we ^have

dim HP 0 C[V ] ^W

= dim HH 0 A n (C) ^W

= 3

^.

For

g

^of^type

D 3 = A 3

⁽

so 6 ≃ sl 4

^),^we^have

dim HP 0 C[V ] ^W

= dim HH 0 A n (C) ^W

= 1

^.

1

UniversitédeLyon,UniversitéLyon1,CNRS,UMR5208,InstitutCamilleJordan,

43blvddu11novembre1918,F-69622Villeurbanne-Cedex,Frane

email:butinmath.univ-lyon1.fr

(3)

above.

2 Results about

B _n − C _n

and

D _n

Asindiatedabove,weareinterestedinthePoissonhomologyof

C[h ⊕ h ^∗ ] ^W

^,^where

h

îsâ^Cartan^subalgebraôfâ

nite-dimensionalsemi-simpleLiealgebra

g

^,^and

W

^its^Weyl^group.^We^will^study^the^types

B n

^and

D n

^.^Reall

thattherootsystemoftype

C n

^is^dual^to^the^root^system^of^type

B n

^.^So^their^Weyl^groupsâreîsomorphi,ând

thestudyofthease

C n

îs^redued^to^the^studyôf^theâse

B n

^.

2.1 Denitions and notations

•

^Set

S := C[x, y] = C[x 1 , . . . , x n , y 1 , . . . , y n ]

^.

For

m ∈ N

^,^we^denote^by

S (m)

^the^elements^of

S

^of^degree

m

^.

For

B n

^,^we^have

W = (±1) ⁿ ⋊ S n = (±1) ⁿ ·S n

(permutationsofthevariablesandsignhangesofthevariables).

For

D n

^,^we^have

W = (±1) ⁿ⁻¹ ⋊ S n = (±1) ⁿ⁻¹ · S n

(permutationsofthevariablesandsignhangesofaneven numberofvariables).

Everyelement

(a 1 , . . . , a n ) ∈ (±1) ⁿ

^is^identied^with^the^diagonal^matrix

Diag(a 1 , . . . , a n )

^,ândêveryêlement

σ ∈ S n

^is^identied^with^the^matrix

(δ i,σ(j) ) (i,j)∈[[1, n]]

^.^We^will^denote^by

s j

^the

j−

^th^sign^hange,^i.e.

s j (x k ) = x k

^if

k 6= j

^,

s j (x j ) = −x j

^,^and

s j (y k ) = y k

^if

k 6= j

^,

s j (y j ) = −y j

^.

Asfortheelementsof

(±1) ⁿ⁻¹

^,^theyâreîdentied^with^the^matriesôf^the^form

Diag((−1) ⁱ ¹ , (−1) ⁱ ¹ ⁺ⁱ ² , (−1) ⁱ ² ⁺ⁱ ³ , . . . , (−1) ⁱ ⁿ⁻ ² ⁺ⁱ ⁿ⁻ ¹ , (−1) ⁱ ⁿ⁻ ¹ ),

with

i k ∈ {0, 1}

^.^We^will^denote^by

s i,j

^the^sign^hangeôf^the^variablesôfîndies

i

^and

j

^.

So,all theseelementsare in

O n C

^,^and^byidentifying

g ∈ W

^with

g 0 0 g

,weobtain

W ⊂ Sp _2n C

^.

•

^The^(right)^ation^of

W

^on

S

^is^dened^for

P ∈ S

^and

g ∈ W

^by

g · P (x, y) := P X n j=1

g 1j x j , . . . , X n j=1

g nj x j , X n j=1

g 1j y j , . . . , X n j=1

g nj y j

! .

Sowehave

h · (g · P ) = (gh) · P

^.

Inthepartiularasewhere

σ ∈ S n

^,^we^have

σ · P (x, y) = P (x σ − 1 (1) , . . . , x σ − 1 (n) , y σ − 1 (1) , . . . , y σ − 1 (n) )

^.

•

^On

S

^,^we^dene^the^Poisson^braket

{P, Q} := h∇P, ∇Qi = ∇P · (J ∇Q) = ∇ x P · ∇ y Q − ∇ y P · ∇ x Q,

where

h·, ·i

^is^the^standard^sympleti^produt,^assoiated^to^the^matrix

J =

0 I n

−I n 0

.

Asthe group

W

îs â^subgroupôf

Sp(h·, ·i) = Sp _2n C

^,^theâlgebraôfînvariants

S ^W

îsâ^Poisson âlgebra^for^the

braketdenedabove.

•

^We^dene^the^Reynolds^operator^as^the^linear^map

R n

^from

S

^to

S

^determined^by

R n (P ) = 1

|W | X

g∈W

g · P.

Weset

A = C[z, t] = C[z 1 , . . . , z n , t 1 , . . . , t n ]

^and

S ^′ := A[x, y]

^,^and^we^extend^the^map

R n

^as^a

A−

^linear^map

from

S ^′

^to

S ^′

^.

Remark 2

Intheaseof

B n

^,êvery êlementôf

S ^W

^hasânêven^degree.^(It îs^false^for

D n

^).

2.2 Poisson homology

•

^Let

A

^be^a ^Poisson^algebra. ^We ^denote^by

Ω ^p (A)

^the

A−

^module ^of^Kählerdierentials, i.e. thevetorspae spannedbytheelementsoftheform

F 0 dF 1 ∧ · · · ∧ dF p

^,^where^the

F j

^belong^to

A

^,^and

d : Ω ^p (A) → Ω ^p+1 (A)

^is

(4)

Weonsidertheomplex

. . . ^∂ ⁵ / / _Ω ⁴ _(A)

∂ ₄

/ / _Ω ³ _(A)

∂ ₃

/ / _Ω ² _(A)

∂ ₂

/ / _Ω ¹ _(A)

∂ ₁

/ / _Ω ⁰ _(A)

withBrylinsky-Koszulboundaryoperator

∂ p

^(See^{[B88 ℄)}^:

∂ p (F 0 dF 1 ∧ · · · ∧ dF p ) = X p j=1

(−1) ^j+1 {F 0 , F j } dF 1 ∧ · · · ∧ dF d j ∧ · · · ∧ dF p

+ X

1≤i<j≤p

(−1) ^i+j F 0 d{F i , F j } ∧ dF 1 ∧ · · · ∧ d dF i ∧ · · · ∧ d dF j ∧ · · · ∧ dF p .

ThePoissonhomologyspaeindegree

p

^is^given^by^the^formula

HP p (A) =

^Ker

∂ p /

^Im

∂ p+1 .

Inpartiular,wehave

HP 0 (A) = A / {A, A}.

•

^In^the^sequel,^we^will^denote

HP 0 (C[V ] ^W )

^by

HP 0 (W )

^and

HH 0 (C[V ] ^W )

^by

HH 0 (W )

^.

2.3 Vetors of highest weight

0

Theaimofthis setionis toshowthat

S ^W (2)

^is ^isomorphi^to

sl 2

^and ^that^the^vetors^whih^do^not^belong^to

{S ^W , S ^W }

^,âreâmong^the^vetorsôf^highest^weight

0

^of^the

sl 2 −

^module

S ^W

^,^anobservationwhihwillsimplify thealulations.

Proposition3

For

B n

⁽

n ≥ 2

⁾^and

D n

⁽

n ≥ 3

^), ^the^subspae

S ^W (2)

^is ^isomorphi^to

sl 2

^. ^More ^preisely

S ^W (2) = hE, F, Hi

with therelations

{H, E} = 2E, {H, F } = −2F

^and

{E, F } = H

^, ^where ^the^elements

E, F, H

^are ^expliitly

givenby

E = ⁿ ₂ R n (x ² ₁ ) = ¹ ₂ x · x F = − ⁿ ₂ R n (y ² ₁ ) = − ₂ ¹ y · y H = −n R n (x 1 y 1 ) = −x · y.

For

D 2

^,^we^have

S ^W (2) = hE, F, H i ⊕ hE ^′ , F ^′ , H ^′ i

^(diret^sumôf^two^Lieâlgebrasîsomorphi^to

sl 2

^).

Sothespaes

S

^and

S ^W

^are

sl 2 −

^modules.

Proof:

Wedemonstratethepropositionfor

B n

^,^the^proof^being^analogous^for

D n

^.

•

^As

x ² ₁

îsînvariantûnder^sign^hanges,^we^may^write

R n (x ² ₁ ) = _n! ¹ P

σ∈ S _n σ · x ² ₁

^.^Moreover,^we^have^the^partition

S n = ` n

j=1 A j

^,^where

A j := {σ ∈ S n / σ(1) = j}

^has^ardinality

(n − 1)!

^.

Thus

R n (x ² ₁ ) = ^(n−1)! _n! P n

j=1 x ² j

^.^We^proeed^likewise^with

R n (y ₁ ² )

^and

R n (x 1 y 1 )

^.

•

^We^obviously^have

S ^W (2) ⊃ hE, F, Hi

^.

Moreover,

R(x ² j ) = R(x ² 1 )

^,

R(y j ² ) = R(y 1 ² )

^and

R(x j y j ) = R(x 1 y 1 )

^.^Last, ^if

i 6= j

^, ^then

x i x j

^,

y i y j

^and

x i y j

^are

mappedtotheiroppositebythe

i−

^th^sign^hange

s i

^,^and

W = s i · hs 1 , . . . , s i−1 , s i+1 , . . . , s n i · S n ⊔ hs 1 , . . . , s i−1 , s i+1 , . . . , s n i · S n

^,

thus

R n (x i x j ) = R n (y i y j ) = R n (x i y j ) = 0

^.^Hene

S ^W (2) ⊂ hE, F, Hi

^.

•

^We^have

∇E = x

0 , ∇F = 0

−y

and

∇H = −y

−x

,

so

{E, F } = H, {H, E} = 2E

^and

{H, F } = −2F

^.

Wedenoteby

S sl ₂

^the^set^of^vetors^of^highest^weight

0

^,î.e.^the^setôfêlementsôf

S

^whih^areannihilatedbythe ationof

sl 2

^.

For

j ∈ N

^,^we^denote^by

S(j) ^sl 2

^the^elements^of

S ^sl 2

^of^degree

j

^.

As the ation of

W

ând ^the âtion ôf

sl 2

^ommute, ^we ^may ^write

(S ^W ) sl ₂ = (S sl ₂ ) ^W = S sl ^W ₂

^and ^likewise

for

S ^W (j) sl ₂

^.

Proposition4

•

^If

S ^W

ôntains^noêlementôf^degree

1

^,^then^the^vetors^of^highest^weight

0

^of^degree

0

^do^not^belong^to

{S ^W , S ^W }

^.

•

^Let

W

^be ^of^type

B n

⁽

n ≥ 2

⁾^or

D n

⁽

n ≥ 3

^).^If

S ^W

ôntains^noêlement ôf ^degree

1, 3,

^, ^then^the^vetors ^of

highestweight

0

^of^degree

4

^do^not ^belong^to

{S ^W , S ^W }

^.

(5)

•

^The^Poisson^braket^beinghomogeneousofdegree

−2

^,^we^have

S ^W (0) ∩ {S ^W , S ^W } = {S ^W (2), S ^W (0)} = {0}

^.

•

^Similarly,

S ^W (4) ∩ {S ^W , S ^W } = {S ^W (0), S ^W (6)} + {S ^W (2), S ^W (4)} = {sl 2 , S ^W (4)}

^,^aording ^to^Proposi-

tion3.Withthedeompositionofthe

sl 2 −

^modules,^we^have

S ^W (4) = L

m∈ N V (m)

^,^with

{sl 2 , V (m)} = V (m)

^if

m ∈ N ^∗

^and

{sl 2 , V (0)} = {0}

^.^So^if

a ∈ S ^W (4) ∩ {S ^W , S ^W }

^,^then

a ∈ L

m∈ N ∗ V (m)

^.

Proposition5

Forevery monomial

M = x ⁱ y ^j

^,^we^have

{H, M } = (|i| − |j|)M = (

^deg

_x (M ) −

^deg

_y (M ))M {E, M } = P n

k=1 j k x ⁱ ₁ ¹ . . . x ⁱ _k−1 ^k−1 x ⁱ _k ^k ⁺¹ x ⁱ _k+1 ^k+1 . . . x ⁱ n ⁿ y ₁ ^j ¹ . . . y _k−1 ^j ^k−1 y ^j _k ^k ⁻¹ y _k+1 ^j ^k+1 . . . y ^j n ⁿ , {F, M } = P n

k=1 i k x ⁱ ₁ ¹ . . . x ⁱ _k−1 ^k−1 x ⁱ _k ^k ⁻¹ x ⁱ _k+1 ^k+1 . . . x ⁱ n ⁿ y ^j ₁ ¹ . . . y ^j _k−1 ^k−1 y _k ^j ^k ⁺¹ y _k+1 ^j ^k+1 . . . y ^j n ⁿ .

Inpartiular,everyvetorofhighestweight

0

îsôfêven^degree.

Proof:Thisresultsfromasimplealulation.

Remark 6

Let

j ∈ N

^and

P ∈ S ^W (j)

^. ^Aording ^to ^thedeomposition of

sl 2 −

^module

S ^W (j)

ⁱⁿ^weight ^subspaes, ^we ^may

write

P = P m

k=−m P k

^with

{H, P k } = k P k

^. ^So^we ^have

S ^W (j)

{S ^W , S ^W }∩S ^W (j) = ^S

W (j) sl 2 {S ^W , S ^W }∩S ^W (j) sl 2

.

Thusthevetors whihdonot belongto

{S ^W , S ^W }

âre ^to^be^found âmong^the^vetorsôf ^highest^weight

0

^.

ThefollowingpropertyisageneralizationofProposition3provedbyJ.AlevandL.Foissyin[AF06 ℄.Itenables

ustoknowthePoinaréseriesofthealgebra

S ^sl 2

^.

Proposition7

For

l ∈ N

^,^we ^have

S ^sl ₂ (2l + 1) = {0}

^and

dim S ^sl ₂ (2l) = (C _l+n−1 ⁿ⁻¹ ) ² − C _l+n ⁿ⁻¹ C _l+n−2 ⁿ⁻¹

^.

ThefollowingresultisimportantforsolvingtheequationofBerest-Etingof-Ginzburg,beauseitgivesadesrip-

tionof thespaeof vetorsof highestweight

0

^,^spae ⁱⁿ^whih^we^will ^searh^the^solutionsôf ^thisêquation.În

theproofofthisproposition,weusetheartiles[DCP76 ℄and[GK04 ℄onerningthepfaanalgebras.

Proposition8

For

i 6= j

^, ^set

X i,j := x i y j − y i x j

^.^Then^the^algebra

C[x, y] ^sl 2

^is^the^algebra ^generated ^by^the

X i,j

^'s^for

(i, j) ∈ [[1, n]] ²

^.^We^denote^this^algebra^by

ChX i,j i

^.

Thisalgebraisnotapolynomialalgebrafor

n ≥ 4

^(e.^g.

X 1,2 X 3,4 − X 1,3 X 2,4 + X 2,3 X 1,4 = 0

^).

Proof:

•

^The^inlusion

ChX i,j i ⊂ C[x, y] sl ₂

^being ôbvious,âll^that^we^have^to^doîs^to^show^that^the^Poinaré^seriesôf

bothspaesareequal,knowingthattheonefor

C[x, y] sl ₂

^is^already^given^byProposition7.

•

^Consider ^the ^vetors

u j :=

x j

y j

for

j = 1 . . . n

^, ⁱⁿ^the ^sympleti ^spae

C ²

^endowed^with ^the ^standard

sympletiform

h·i

^dened^by^the^matrix

J :=

0 1

−1 0

.Let

{T i,j / 1 ≤ i < j ≤ n}

^be^a^set^ofindeterminates, andlet

T e

^be^theantisymmetrimatrixthegeneraltermofwhihis

T i,j

^if

i < j

^.^Then,^aording ^to^setion⁶^of

[DCP76 ℄,theideal

I 2

^of^relations^between^the

hu i , u j i

^(i.e.^between^the

X i,j

⁾^is^generated^by^the^pfaan^minors

of

T e

^of^size

4 × 4

^.^Set

P F := ChX i,j i

^.^So^we^have

P F ≃ C[(T i,j ) _i<j ] / I 2 =: P F 0

^,^i.e.^the^algebra

P F

^is^isomorphi

tothepfaanalgebra

P F 0

^.Îts^Poinaré^seriesîs^givenⁱⁿ^setion ⁴ôf^{[GK04 ℄}^by

dim P F 0 (m) = (C _m+n−2 ^m ) ² − C _m+n−2 ^m−1 C _m+n−2 ^m+1 .

Sowehave

dim P F (2l) = (C ^l _l+n−2 ) ² − C ^l−1 _l+n−2 C _l+n−2 ^l+1

^.^We^verify^that

dim P F (2l) = (C _l+n−1 ⁿ⁻¹ ) ² − C _l+n ⁿ⁻¹ C _l+n−2 ⁿ⁻¹ = dim C[x, y] ^sl ₂ (2l).

Wehaveobviously

dim P F(2l + 1) = 0 = dim C[x, y] sl ₂ (2l + 1)

^.^Hene^the^equality^of^the^Poinaré^series.

(6)

WestudythefuntionnalequationintroduedbyY.Berest,P.EtingofandV.Ginzburgin[BEG04℄.Thepoint

isthatsolvingthisequation,inthespae

S sl ^W ₂

^,^is^equivalent^to^thedeterminationofthequotient

S ^W

{S ^W , S ^W }

^,^that

istosaytheomputationofthePoissonhomologyspaeindegree

0

^of

S ^W

^.

Lemma 9 (Berest-Etingof-Ginzburg)

Weonsider

C ²ⁿ

^,^endowed^with^its^standard^sympleti ^form,^denoted ^by

h·, ·i

^.^Let

j ∈ N

^.

Let

S := C[x, y] = C[z]

^,^and^let

L j :=

S ^W (j) {S ^W , S ^W }∩S ^W (j)

∗

bethelineardualof

S ^W (j) {S ^W , S ^W }∩S ^W (j)

^.

Then

L j

îsîsomorphi^to^the^vetor^spaeôf polynomials

P ∈ C[w] ^W (j)

^satisfying ^the^following^equation^:

∀ w, w ^′ ∈ C ²ⁿ , X

g∈W

hw, gw ^′ i P (w + gw ^′ ) = 0

⁽¹⁾

Proof:

•

^For

w = (u, v) ∈ C ²ⁿ

^and

z = (x, y) ∈ C ²ⁿ

^,^we^set

L w (z) := P

g∈W e ^{hw, gzi}

^.

Sowehave

{L w (z), L w ^′ (z)} = ∇ x L w (z) · ∇ y L w ^′ (z) − ∇ y L w (z) · ∇ x L w ^′ (z)

Wededuetheformula

{L w (z), L w ^′ (z)} = X

g∈W

hw, gw ^′ iL w+gw ^′ (z).

⁽²⁾

•

^Moreover,

L w (z)

^is^a^power^seriesⁱⁿ

w

^,^the^oeients^of^whih^generate

S ^W

^:

L w (z) =

X ∞ p=0

|W | p! R n

" _n X

i=1

y i u i − x i v i

! p #

(3)

Theoeientsoftheseriesaretheimagesby

R n

ôf^theêlementsôf^theânonial^basisôf

S

^.

Remark:intheaseof

B n

^,^thereîs^noînvariantôfôdd^degree,^so^we^have

L w (z) = P

g∈W

^h

(hw, gzi)

^.

⊲

^Set

M p (z) = {z ⁱ / |i| = p}

^and

M p (w) = {w ⁱ / |i| = p}

^.

Foramonomial

m = x ⁱ y ^j ∈ M p (z)

^,^let

m e = u ^j v ⁱ

^.

Similarly,foramonomial

m = u ⁱ v ^j ∈ M p (w)

^,^let

m = x ^j y ⁱ

^.

So,for

m ∈ M p (z)

^,^we^have

m e = m

^,^and^for

m ∈ M p (w)

^,^we^have

m e = m

^.

Theseries

L w (z)

^may^then^be^written^as

L w (z) = |W | +

X ∞ j=1

X

m _j ∈M j (w)

α m _j R n (m j )m j = |W | + X ∞ j=1

X

m _j ∈M j (z)

α _g m _j R n (m j ) m f j = |W | + X ∞ j=1

L ^j w (z),

⁽⁴⁾

with

α m _j ∈ Q ^∗

^.

⊲

^Now

P n

i=1 y i u i − x i v i p

= P

|a|+|b|=p (−1) ^|b| C p â,b x ^b y â u â v ^b ,

^where

C p ^a,b = _a ^p!

1 !...a n !b 1 !...b n !

^is^themultinomial oeient,thereforeaordingtoformula (3),wehave

L w (z) = |W | + X ∞ p=1

X

|a|+|b|=p

(−1) ^|b| |W | p! C p ^a,b R n

x ^b y ^a

u ^a v ^b

⁽⁵⁾

Byolletingtheformulae(4)and(5),weobtain

α _u ^a _v ^b = (−1) ^|b| |W |

p! C ^a,b p .

⁽⁶⁾

•

^We^identify

L j

^with^the^vetor^spae^of^linear^forms^on

S ^W (j)

^whih^vanish^on

{S ^W , S ^W } ∩ S ^W (j)

^.

Denethemap

π : L j → {P ∈ C[w] ^W (j) / ∀ w, w ^′ ∈ C ²ⁿ , X

g∈W

hw, gw ^′ i P (w + gw ^′ ) = 0}

f 7→ π f := f(L ^j _w ).

(7)

Then

π

^is ^well^dened^:^indeed

L ^j w

^is^a^polynomialⁱⁿ

z

^of^degree

j

^with^oeientsⁱⁿ

C[w]

^,^and ^expliitly,^we

have

f(L ^j w ) = X

m _j ∈M j (z)

α _g m _j f(R n (m j )) m f j ∈ C[w].

⁽⁸⁾

(7)

⊲

^If ^two ^monomials

m j , m ^′ j ∈ M j (z)

^belong ^to â ^same ôrbit ûnder ^the âtion ôf

S n

^, ^then ^the ^oeients

α m _g _j f(R n (m j ))

^and

α _g

m ^′ _j f(R n (m ^′ _j ))

^of

m j

^and

m ^′ _j

^are^the^same,^thus

f(L ^j _w )

îsînvariantûnder

W

^.

⊲

^Besides,

π f

^is ^solution ^of

E n (P ) = 0

^: ^indeed, ^we ^may ^extend

f

âs â ^linear ^map ^dened ôn

_{S W ^S , S ^W ^W } = L ∞

i=0

S ^W (i)

{S ^W , S ^W }∩S ^W (i)

^,^by^setting

f = 0

^on

_{S _W _{, S} ^S ^W _W ⁽ⁱ⁾ _}∩S _W _(i)

^for

i 6= j

^.

Then,aordingto(2),wehavetheequality

0 = f ({L w , L w ^′ }) = P

g∈W hw, gw ^′ if (L w+gw ^′ )

^,

hene

P

g∈W hw, gw ^′ if L ^j _w+gw ′

= 0

^.^So,^the^polynomial

f(L ^j w ) ∈ C[w]

^satises^equation^(1).

•

^Dene^the^map

ϕ : {P ∈ C[w] ^W (j) / ∀ w, w ^′ ∈ C ²ⁿ , X

g∈W

hw, gw ^′ i P (w + gw ^′ ) = 0} → L j

P = X

m _j ∈M j (w)

β m _j m j 7→

ϕ P : R n (m j ) 7→ _α ^β ^mj

mj

.

⁽⁹⁾

⊲

^For

f ∈ L j

^,^we^have

ϕ π _f (R n (m j )) = ^α ^mj ^f ( ^R n (m _j ) )

α _mj = f (R n (m j ))

^,

thus

ϕ π _f = f

^.

⊲

^For

P = X

m _j ∈M _j (w)

β m _j m j ∈ C[w] ^W (j)

^,^we^have

P = X

m _j ∈M _j (z)

β _g m _j m f j

^,^so^if

m j ∈ M j (z)

^,^then

ϕ P (R n (m j )) = _α ^β ^mj ^g

g mj

.

Consequently,

π ϕ _P = P

m _j ∈M j (z) α _g m _j ϕ P (R n (m j )) m f j = P

m _j ∈M j (z) α _g m _j β _mj _g

α _mj _g m f j = P.

So

π

îs^bijetiveândîtsînverseîs

ϕ

^.

⊲

^All^we^have^to^do^is^to^show^that

ϕ P

^vanishes^on

{S ^W , S ^W } ∩ S ^W (j)

^.

Let

P ∈ C[w] ^W (j)

^beâ^solutionôfêquation^(1).^Thenâs,

π ϕ _P = P

^,^we^have^for

k + l = j

^,

0 = P

g∈W hw, gw ^′ iP (w + gw ^′ ) = ϕ P {L ^k w , L ^l _w ′ }

But

{L ^k _w , L ^l _w ′ } = P

m _k ∈M k (w)

P

µ _l ∈M l (w ^′ ) α m _k α µ _l m k µ l {R n (m k ), R n (µ l )},

sothat

P

m _k ∈M _k (w)

P

µ _l ∈M _l (w ^′ ) α m _k α µ _l m k µ l ϕ P ({R n (m k ), R n (µ l )}) = 0.

Thislastequalityisequivalentto

∀ k + l = j, ϕ P ({R n (m k ), R n (µ l )}) = 0,

whihshows that

ϕ P

^vanishes^on

{S ^W , S ^W } ∩ S ^W (j)

^.

ThefollowingorollaryenablesustomaketheequationofBerest-Etingof-Ginzburgmoreexpliit.

Corollary 10

Weintrodue

2n

indeterminates,denotedby

z 1 , . . . , z n , t 1 , . . . , t n

^,ând^weêxtend^the^Reynoldsôperatorⁱⁿâ^map

from

C[x, y, z, t]

^to^itself^whih^is

C[z, t]−

^linear.^Then^the^vetor^spae

L j

îsîsomorphi^to^the^vetor^spaeôf

polynomials

P ∈ S ^W (j)

^satisfying^the^following^equation ^:

R n

X ⁿ

i=1

z i y i − t i x i

P (z 1 + x 1 , . . . , z n + x n , t 1 + y 1 , . . . , t n + y n )

= 0

⁽¹⁰⁾

i.e.

E n (P ) := R n

(z · y − t · x) P (x + z, y + t)

= 0

⁽¹¹⁾

Proof:

Wehave

hw, w ^′ i = w · (Jw ^′ ) = P n

i=1 (w i w ^′ n+i − w n+i w ^′ i )

^.^Thenêquation⁽¹⁰⁾îsêquivalent^to

∀ w, w ^′ ∈ C, P

g∈W

P n

i=1 (w i P n

j=1 g ij w _n+j ^′ − w n+i P n

j=1 g ij w ^′ j ) P

w 1 + P n

j=1 g 1j w j ^′ , . . . , w n + P n

j=1 g nj w j ^′ , w n+1 + P n

j=1 g 1j w ^′ n+j , . . . , w 2n + P n

j=1 g nj w ^′ n+j

= 0

(12)

(8)

X

g∈W

X n i=1

(z i

X n j=1

g ij y j − t i

X n j=1

g ij x j ) P z 1 +

X n j=1

g 1j x j , . . . , z n + X n j=1

g nj x j , t 1 + X n j=1

g 1j y j , . . . , t n + X n j=1

g nj y j

(13)

iszero.

Thisisequivalentto

X n i=1

(z i g · y i − t i g · x i ) X

g∈W

g ·

P(z 1 + x 1 , . . . , z n + x n , t 1 + y 1 , . . . , t n + y n )

= 0,

⁽¹⁴⁾

thatistosay

R n

X ⁿ

i=1

z i y i − t i x i

P (z 1 + x 1 , . . . , z n + x n , t 1 + y 1 , . . . , t n + y n )

= 0,

⁽¹⁵⁾

where

R n

îs^the^Reynoldsôperatorêxtendedⁱⁿâ

C[z, t]−

^linear^map.

Remark 11

•

^Case^of

B n

^: ^for^a ^monomial

M ∈ C[x, y]

^,

⊲

êither ^therê êxistsâ^sign^hange^whih^sends

M

^toîtsôpposite,ând^then

R n (M ) = 0

⊲

^or

M

îsînvariantûnder êvery^sign^hangeând^then

R n (M ) = P

σ∈ S _n σ · M

^.

If

Q = R n (P )

^with

P ∈ C[x, y]

^,^then^we ^mayâlwaysâssume ^thatêah ^monomialôf

P

^,ⁱⁿ^partiular

P

^itself,^is

invariantunderthesignhanges.

•

^Case^of

D n

^:^we^have^the^same^result,^byônsidering^this^time^the^sign^hangesôfânêven^numberôf^variables.

The aimof Proposition12and itsorollary is to reduedrastially the spae inwhihwe searhthe solutions

of equation(11) :indeed, insteadof searhingthe solutions in

S ^W

^,^we ^may^limit ^ourselves^to^the ^spae ^of^the

elementswhihareannihilatedbytheationof

sl 2

^.

Proposition12

Let

P ∈ C[x, y] ^W

^.^We^onsider^the^element

E n (P )

^dened^by^the^formula⁽¹¹⁾âsâ^polynomialⁱⁿ^theîndetermi-

nates

z, t

^and^with^oeientsⁱⁿ

C[x, y]

^.

Thentheoeientof

z 1 t 1

ⁱⁿ

E n (P )

^is

⁻¹ _n {H, P }

^,^that^of

t ² ₁

^is

⁻¹ _n {E, P }

^and^that^of

z ² ₁

^is

_n ¹ {F, P }

^.

Proof:

Wearryouttheprooffor

B n

^.^The^method^is^the^same^for

D n

^.

•

^We^denote ^by

c z ₁ t ₁ (P )

^the ^oeient ^of

z 1 t 1

ⁱⁿ

E n (P )

^. ^Sine^the ^maps

P 7→ c z ₁ t ₁ (P )

^and

P 7→ {H, P }

^are

linear, allwehavetodoistoprovethepropertyfor

P

^of^the^form

P = R n (M)

^,^where

M = x ⁱ y ^j

^is^a^monomial

whihwemayassumeinvariantunderthesignhangesthankstoremark11.

Thentheformula(11)maybewritten

|W | E n (M ) = |W | R n (z · y − t · x)(x + z) ⁱ (y + t) ^j

= X

c∈(±1) ⁿ

X

σ∈ S _n

c ·

"

(z 1 y _σ −1 (1) + · · · + z n y _σ −1 (n) ) Y n k=1

(z k + x _σ −1 (k) ) ⁱ ^k (t k + y _σ −1 (k) ) ^j ^k

#

− X

c∈(±1) ⁿ

X

σ∈ S _n

c ·

"

(t 1 x σ − 1 (1) + · · · + t n x σ − 1 (n) ) Y n k=1

(z k + x σ − 1 (k) ) ⁱ ^k (t k + y σ − 1 (k) ) ^j ^k

#

•

^So^the^oeient^of

z 1 t 1

^is^given^by

|W | c z 1 t 1 (M ) = X

c∈(±1) ⁿ

X

σ∈ S _n

c · h y _σ − 1 (1)

Y n k=1

x ⁱ _σ ^k ₋₁ _(k)

!

j 1 y _σ ^j ¹ ₋ ⁻¹ 1 (1)

Y n k=2

y _σ ^j ^k ₋₁ _(k)

!

−x _σ −1 (1) i 1 x ⁱ _σ ¹ ₋ ⁻¹ 1 (1)

Y n k=2

x ⁱ _σ ^k ₋ 1 (k)

! _n Y

k=1

y _σ ^j ^k ₋ 1 (k)

! i

= |(±1) ⁿ | X

σ∈ S _n

(j 1 − i 1 ) Y n k=1

x ⁱ _σ ^k −1 (k) y _σ ^j ^k −1 (k)

!

= |W | (j 1 − i 1 ) R n (M ).

(9)

Sine

P = _n! ¹ P

σ∈ S _n

Q n

k=1 x ⁱ _k ^σ(k) y ^j _k ^σ(k)

^,^we^dedue^that

n! c z ₁ t ₁ (P ) = X

σ∈ S _n

c z ₁ t ₁

Y n k=1

x ⁱ _k ^σ(k) y _k ^j ^σ(k)

!

= X

σ∈ S _n

(j _σ(1) − i _σ(1) ) R n

Y n k=1

x ⁱ _k ^σ(k) y _k ^j ^σ(k)

!

= X

σ∈ S _n

(j _σ(1) − i _σ(1) ) R n (M) = (n − 1)!

X n k=1

(j k − i k ) R n (M)

= (n − 1)! (

^deg

_y (M ) −

^deg

_x (M)) R n (M ) = −(n − 1)! {H, P }.

•

^We^proeed^as^for

z 1 t 1

^,^by^denoting^by

c _t 2

1 (P)

^the^oeient^of

t ² ₁

ⁱⁿ

E n (P)

^.^Then^we^have

|W | c t ² ₁ (M ) = − X

c∈(±1) ⁿ

X

σ∈ S _n

c · h x σ − 1 (1)

Y n k=1

x ⁱ _σ ^k −1 (k)

!

j 1 y _σ ^j ¹ −1 ⁻¹ (1)

Y n k=2

y _σ ^j ^k −1 (k)

! i

= −|(±1) ⁿ | j 1

X

σ∈ S _n

x ⁱ _σ ¹ ₋ ⁺¹ 1 (1) y _σ ^j ¹ ₋ ⁻¹ 1 (1)

Y n k=2

x ⁱ _σ ^k ₋ 1 (k) y _σ ^j ^k ₋ 1 (k)

!

= −|W | j 1 R n x ⁱ ₁ ¹ ⁺¹ y ₁ ^j ¹ ⁻¹ Y n k=2

x ⁱ _k ^k y ^j _k ^k

!!

.

Thus

n! c _t 2

1 (P ) = X

σ∈ S _n

c _t 2 1

Y n k=1

x ⁱ _k ^σ(k) y _k ^j ^σ(k)

!

= − X

σ∈ S _n

j _σ(1) R n x ⁱ ₁ ^σ(1) ⁺¹ y ₁ ^j ^σ(1) ⁻¹ Y n k=2

x ⁱ _k ^σ(k) y ^j _k ^σ(k)

!!

= −

X n p=1

X

σ∈ S _n σ(1)=p

j _σ(1) R n x ⁱ ₁ ^σ(1) ⁺¹ y ^j ₁ ^σ(1) ⁻¹ Y n k=2

x ⁱ _k ^σ(k) y _k ^j ^σ(k)

!!

= −(n − 1)!

X n p=1

j p R n

x ⁱ ₁ ¹ . . . x ⁱ p ^p ⁺¹ . . . x ⁱ n ⁿ y ^j ₁ ¹ . . . y ^j p ^p ⁻¹ . . . y ^j n ⁿ

.

But

n! {E, P } = X

σ∈ S _n

{E, x ⁱ ₁ ^σ(1) . . . x ⁱ n ^σ(n) y ₁ ^j ^σ(1) . . . y ^j n ^σ(n) }

= X

σ∈ S _n

X n p=1

j _σ(p) x ⁱ ₁ ^σ(1) . . . x ⁱ p ^σ(p) ⁺¹ . . . x ⁱ n ^σ(n) y ₁ ^j ^σ(1) . . . y ^j p ^σ(p) ⁻¹ . . . y ^j n ^σ(n)

= X n p=1

X n q=1

X

σ∈ S _n σ(p)=q

j _σ(p) x ⁱ ₁ ^σ(1) . . . x ⁱ p ^σ(p) ⁺¹ . . . x ⁱ n ^σ(n) y ^j ₁ ^σ(1) . . . y p ^j ^σ(p) ⁻¹ . . . y n ^j ^σ(n)

= X n q=1

j q

X n p=1

X

σ∈ S _n σ(p)=q

x ⁱ ₁ ^σ(1) . . . x ⁱ p ^σ(p) ⁺¹ . . . x ⁱ n ^σ(n) y ₁ ^j ^σ(1) . . . y p ^j ^σ(p) ⁻¹ . . . y n ^j ^σ(n)

= n!

X n q=1

j q R n

x ⁱ ₁ ¹ . . . x ⁱ q ^q ⁺¹ . . . x ⁱ n ⁿ y ₁ ^j ¹ . . . y q ^j ^q ⁻¹ . . . y ^j n ⁿ

.

So

c _t 2

1 (P ) = ⁻¹ _n {E, P }

^.^Similarly^we^show^that

c _z 2

1 (P ) = ¹ _n {F, P }

^.

Corollary 13

•

^Let

P ∈ C[x, y] ^W

^.^If

P

^satises^equation^(11),^then

P

^isannihilatedby

sl 2

^,^i.e.

P ∈ S sl ^W ₂

^.

•

^Therefore ^the ^ve^tor ^spae

L j

îs îsomorphi ^to ^the ^vetor ^spae ôf ^the polynomials

P ∈ S sl ^W ₂ (j)

^satisfying

equation(11).

Thusthedeterminationof

S ^W {S ^W , S ^W }

= ^S

W sl 2 {S ^W , S ^W }∩S ^W _sl

2

isequivalenttotheresolution,in

S sl ^W ₂

^,^of^equation^(11).

Proof:

Let

P ∈ C[x, y] ^W

^satisfyingêquation^(11).^Thenâll^theôeientsôf^the^polynomial

E n (P ) ∈ (C[x, y])[z, t]

^are

zero.Inpartiular,aordingtoProposition12,wehave

{H, P } = {E, P } = {F, P } = 0

^.^Hene

P ∈ S sl ^W ₂

^.

TheseondpointresultsfromCorollary10andfromtherstpoint.

WeendthissetionbydeningtwovariantsoftheequationofBerest-Etingof-Ginzburg:thesearetehnialtools

whihenableustoeliminatesomevariablesandthustosolveequation(11)moreeasily.

(10)

Wedenethe (intermediate)map

s ⁿ int : C[x, y, z, t] → C[x, y, z, t]

P 7→ P (0 y z t 1 , 0),

andwe set

E ⁿ int (P ) := s ⁿ int (E n (P )).

⁽¹⁶⁾

Similarly,wedenethemap

s n : C[x, y, z, t] → C[x, y, z, t]

P 7→ P (0 y 1 , 0 z t 1 , 0),

andwe set

E n ^′ (P ) := s n (E n (P )).

⁽¹⁷⁾

Thislastequation isequation (11)afterthesubstitution

x 1 = · · · = x n = y 2 = · · · = y n = t 2 = · · · = t n = 0

^.

Remark 15

If

P

^satisesêquation^(11),ît^satisesôbviouslyêquation ^(17).

Intheasewhere

n

îsânôddînteger,^the^vetorsôf^highest^weight

0

ôfêven^degreeâre^the^same^for

B n

^and

D n

^,

and equations (17) are idential for bothtypes.Moreover, the linkbetweenequations (11) for

B n

^and

D n

^en-

ablesustoprovetheinequality

dim HP 0 (D n ) ≤ dim HP 0 (B n )

^.Îtîs^the^purposeôf^the^two^followingpropositions.

Proposition16

Byabuseofnotation,wedenoteby

S ^B ⁿ (2p)

^(resp.

S ^D ⁿ (2p)

⁾^the^setôfînvariantêlementsôf^degree

2p

ⁱⁿ^type

B n

(resp.

D n

^).^Then ^we ^have

S ^B ²ⁿ⁺¹ (2p) = S ^D ²ⁿ⁺¹ (2p)

^,

S ^B sl ²ⁿ⁺¹ ₂ (p) = S sl ^D ₂ ²ⁿ⁺¹ (p)

^,^and ^equations ⁽¹⁷⁾ ⁱⁿ

S sl ^B ²ⁿ⁺¹ ₂ = S sl ^D ₂ ²ⁿ⁺¹

^assoiated^to^both^types^are ^the^same.

Thisresult isfalsefortheevenindies:ounter-example :

dim S sl ^D ₂ ⁴ (6) = 1

^whereas

S sl ^B ⁴ ₂ (6) = {0}

^.

Proof:

•

^We^set

Φ 2n+1 (P ) = X

σ∈ S _2n+1

σ · P, Ψ ^B _2n+1 (P ) = X

g∈(±1) ²ⁿ⁺¹

g · P, Ψ ^D _2n+1 (P ) = X

g∈(±1) ²ⁿ

g · P,

sothat

R ^B _2n+1 (P) = 1

|B 2n+1 | Φ 2n+1 ◦ Ψ ^B _2n+1 ,

^and

R ^D _2n+1 (P ) = 1

|D 2n+1 | Φ 2n+1 ◦ Ψ ^D _2n+1 .

Weobviouslyhave

Ψ ^B 2n+1 (S(2p)) ⊂ Ψ ^D 2n+1 (S(2p))

^.

Conversely,sine

Ψ ^D _2n+1 (S(2p))

îs ^spanned ^by ^the êlementsôf ^the ^form

Ψ ^D _2n+1 (m)

^with

m ∈ S(2p)

^monomial,

all we havetodoistoshowthat

Ψ ^D _2n+1 (m)

^belongs^to

Ψ ^B _2n+1 (S(2p))

^,^i.e.

Ψ ^D _2n+1 (m)

îs învariantûnder^the^sign

hanges. Now

m = x ⁱ ₁ ¹ . . . x ⁱ _2n+1 ²ⁿ⁺¹ y ₁ ^j ¹ . . . y _2n+1 ^j ²ⁿ⁺¹

^with

P 2n+1

k=1 (i k + j k ) = 2p

^,^therefore ât^leastôneôf^the

i k + j k

^is

even.Let'sdenoteby

l

^theorrespondingindex.

So,forevery

k 6= l

^,^we^have

s k (m) = (−1) ⁱ ^k ^+j ^k m = s k,l (m)

^and

s l (m) = m

^.^But

Ψ ^D 2n+1 (m) =



 X

q 1 =0,1...q 2n+1 =0,1

(−1) ^q ¹ ^[(i ¹ ^+j ¹ ⁾⁺⁽ⁱ ² ^+j ² ^)]+q ² ^[(i ² ^+j ² ⁾⁺⁽ⁱ ³ ^+j ³ ^)]+···+q ²ⁿ ^[(i ²ⁿ ^+j ²ⁿ ⁾⁺⁽ⁱ ²ⁿ⁺¹ ^+j ²ⁿ⁺¹ ^)]





| {z }

a m

m,

therefore

s k Ψ ^D _2n+1 (m)

=

a m s k,l (m) = s k,l (a m m) = s k,l Ψ ^D _2n+1 (m)

si

k 6= l

a m m

^si

k = l

= Ψ ^D _2n+1 (m)

^.

•

^So^we^have

S ^D ⁿ (2p) = Φ 2n+1 Ψ ^D _2n+1 (S(2p))

= Φ 2n+1 Ψ ^B _2n+1 (S(2p))

= S ^B ⁿ (2p)

^.

Hene

S sl ^B ⁿ ₂ (2p) = S sl ^D ₂ ⁿ (2p)

^.^Besides,^aording^toProposition5,

S ^B sl ⁿ ₂ (2p + 1) = S sl ^D ₂ ⁿ (2p + 1) = {0}

^.

•

^For

P ∈ S ^B sl ⁿ ₂

^,^equation⁽¹⁷⁾^may^be^written

2n+1 X

i=1

z i

X

σ∈ S _n σ(1)=i



y 1 P (z 1 , . . . , z 2n+1 , t 1 , 0, . . . , 0, y 1

|{z}

i

, 0, . . . , 0) − y 1 P (z 1 , . . . , z 2n+1 , t 1 , 0, . . . , 0, −y 1

|{z}

i

, 0, . . . , 0)



 = 0.

Itisequation(17)for

P ∈ S sl ^D ₂ ⁿ

^.

(11)

Let

P

^be^invariant^by^sign^hanges.^If

P

îs^solution ôfêquation ⁽¹¹⁾^for

D n

^,^then

P

îs^solution ôfêquation⁽¹¹⁾

for

B n

^.

Inpartiular,wehave

dim HP 0 (D 2n+1 ) ≤ dim HP 0 (B 2n+1 )

^.

Proof:

Let

SB n

^(resp.

SD n

⁾^be^the^group^of^sign^hanges^of

B n

^(resp.

D n

^).^We^may^write

SB n = SD n ⊔ SD n · s 1

^.

Let

P

^be^invariant^by^sign^hanges.^So^we^have

P = R ^B n (P ) = R ^D n (P)

^,^and^equation⁽¹¹⁾^for

B n

^(resp.

D n

⁾^may

bewritten

E ^B n (P ) = R ^B n (Q)

^(resp.

E n ^D (P ) = R ^D n (Q)

^),^with

Q = (z · y − t · x) P(x + z y + t)

^.

If

P

îs^solutionôfêquation⁽¹¹⁾^for

D n

^,^then^we^have^:

R ^B n (Q) = X

h∈SB n

X

σ∈ S _n

(σh) · Q = X

h∈SB n

h · X

σ∈ S _n

σ · Q

!

= X

g∈SD n

g · X

σ∈ S _n

σ · Q

!

+ X

g∈SD n

(gs 1 ) · X

σ∈ S _n

σ · Q

!

= X

g∈SD n

g · X

σ∈ S _n

σ · Q

! + s 1 ·

"

X

g∈SD n

g · X

σ∈ S _n

σ · Q

!#

= R ^D n (Q) + s 1 · R n ^D (Q) = 0.

So,

P

îs^solutionôfêquation⁽¹¹⁾^for

B n

^.

Wededuethelaimedinequality,knowingthat,aordingtoProposition16,

S sl ^B ²ⁿ⁺¹ ₂ = S sl ^D ₂ ²ⁿ⁺¹

^.

2.5 Constrution of graphs attahed to the invariant polynomials

Letus realltheequalityofProposition8:

S sl ^W ₂ = R n (ChX i,j i)

^.^Moreover,^aording ^to^Corollary ^13,^the^om-

putationof

HP 0 (S ^W )

^an^be^redues^to^solving^Equation⁽¹¹⁾ⁱⁿ^the^spae

S ^W sl ₂

^.

Inordertohaveshorterandmorevisualnotations,werepresentthepolynomialsofthisspaebygraphs,bythe

methodexplainedindenition21.

Butbefore, letusquote,for thepartiularasethat weare interestedin, thefundamentalresultestablishedby

J.Alev,M.A.Farinati,T.LambreandA.L.Solotarin[AFLS00℄:

Theorem 18 (Alev-Farinati-Lambre-Solotar)

For

k = 0 . . . 2n

^,^the^dimension^of

HH k (A n (C) ^W )

îs^the^numberôfônjugay^lassesôf

W

^admitting^the^eigenvalue

1

^with^themultipliity

k

^.

Byspeializingtotheasesof

B n

^and

D n

^,^we^obtain^:

Corollary 19 (Alev-Farinati-Lambre-Solotar)

•

^For^type

B n

^,^the^dimension^of

HH 0 (A n (C) ^W )

^is^the^number^of ^partitions

π(n)

^of ^the^integer

n

^.

•

^For^type

D n

^, ^the^dimension^of

HH 0 (A n (C) ^W )

^is^the^number^of^partitions

e π(n)

^of^the^integer

n

^having^an^even

number ofparts.

TheonjetureofJ.Alevmaybesetforthasfollows :

Conjeture 20 (Alev)

•

^For^the^type

B n

^,^the^dimension^of

HP 0 (S ^W )

^equals^the^number^of^partitions

π(n)

^of^the^integer

n

^.

•

^For^the ^type

D n

^, ^the ^dimension^of

HP 0 (S ^W )

^equals ^the^number ^of ^partitions

e π(n)

^of ^the^integer

n

^having ^an

evennumberofparts.

Now,letusshowhowtoonstrut

π(n)

^solutionsôfêquation⁽¹¹⁾^for^theâseôf

B n

^.

Denition 21

For

i 6= j

^, ^we^note

X i,j = x i y j − y i x j

^.

Toeahelementof theform

M := Q n−1 i=1

Q n

j=i+1 X _i,j ^2a ^i,j

^,^we ^assoiate ^the(non-oriented)graph

g Γ M

^suh ^that

⊲

^the^set ^of^verties^of

g Γ M

îs^the^set ôfîndies

{k ∈ [[1, n]] / ∃ i ∈ [[1, n]] / a i,k 6= 0

^or

a k,i 6= 0}

^,

⊲

^two^verties

i, j

^of

Γ g M

âreônneted ^by^theêdge

i

a _i,j

j

^if

a i,j 6= 0

^.

•

^If

σ ∈ S n

^,^then^the^graph

Γ ] σ·M

îsôbtained^by^permuting^the^vertiesôf

g Γ M

^.

So,byreplaingeahvertexbythesymbol

•

^,^we^obtain^a^graph

Γ M

^suh^that^the^map

M 7→ Γ M

îsônstantônêv-

eryorbitundertheationof

B n

^(resp.

D n

^).^So^we^may^assoiate^this^graph^to^the^element

R n Q n−1 i=1

Q n

j=i+1 X _i,j ^2a ^i,j

.

HAL Id: hal-00325594