HyperKähler Potentials via Finite-Dimensional Quotients

20 Pages·0.29 MB·Italian

by  Piotr Kobak

HYPERKÄHLER POTENTIALS VIA
0 FINITE-DIMENSIONAL QUOTIENTS
0
0
2 PIOTR KOBAK AND ANDREW SWANN

n
a
J Abstra
t. It is known that nilpotent orbits in a
omplex simple
Lie algebra admit hyperKählermetri
s with asingle fun
tion that
5
is a global potential for ea
h of the Kähler stru
tures (a hyper-

] Kähler potential). In an earlier paper the authors showed that
G
nilpotent orbits in
lassi
al Lie algebras
an be
onstru
ted as
D
(cid:28)nite-dimensionalhyperKählerquotientofa(cid:29)atve
torspa
e. This
. paperusesthatquotient
onstru
tionto
omputehyperKählerpo-
h
t tentials expli
itly for orbits of elements with small Jordan blo
ks.
a
m SItLi(sns,eCe)nthattheKählerpotentiaXls2of=B0iquardandGaudu
honfor
-orbits of elements with , are in fa
t hyperKähler
[
potentials.

1
v
7
2
0 1. Introdu
tion
1
0 Adjoint orbits in
omplex semi-simple Lie algebras are known to
0
arry a
ompatible hyperKähler metri
invariant under the
ompa
t
0
groupa
tion(see[18,17,16,2℄). Nilpotentorbitsareparti
ularlyinter-
/
h
estingas they admita hyperKähler stru
ture whi
h is
loselyrelated to
t
a
twistor spa
es and quaternion-Kähler geometries [20℄ and whi
h
omes
m
equipped with a hyperKähler potential. If one only asks for a Kähler
:
v potential
ompatible with the hyperKähler stru
ture, then several ex-
i
X amples are known. Hit
hin [8℄ gave an expression for a global Kähler
ar potsel(nnt,iaCl)for a hyperKähler stru
ture on the regular semi-simple orbit
of in terms of theta fun
tions. Biquard and Gaudu
hon [3℄ de-
termineda simple formulafor the Kähler potential for the hyperKähler
metri
on semi-simple orbits of symmetri
type. These orbits
ome in

ontinuous familiesand by taking a limit Biquard and Gaudu
hon also
obtain Kähler potentials for
ertain nilpotent orbits.
In [15, 13℄, Kähler and hyperKähler potentials were obtained for
orbits of
ohomogeneity one and two by
onsidering the invariants pre-
served by the
ompa
t group a
tion. The
ohomogeneity of a
omplex
1991 Mathemati
s Subje
t Classi(cid:28)
ation. (2000 version) Primary 53C26; Se
-
ondary 53D20, 14L35.
Key words and phrases. HyperKähler manifold, hyperKähler potential, hyper-
Kähler quotient,
lassi
al Lie algebras,nilpotent orbit.
1
2 PIOTR KOBAK AND ANDREW SWANN
gC
orbit O ⊂ is de(cid:28)ned as the
odimension of the generi
orbits of the
G

ompa
t group on O. As the
ohomogeneity in
reases, we move fur-
ther away from homogeneous manifolds and the geometry of the orbits
be
omes more
ompli
ated.
Butthereareotherways ofratingthelevelof
ompglCexityofnilpotent
orbits. sIun(nth,eC
)asseo(wnh,eCn)ea
hsspi(mnp,Cle)
omponent of is
lassi
al (i.e.,
equals gC , , or ) it
an be shown that nilpotent
orbits iHnN arise as hyperKähler redu
tions of the (cid:29)at hyperKähler
spa
es (see [11℄). This gives a more expli
it des
ription of the hy-
perKähler metri
and the
orrespro2ndinHgNpotential, as thHeNlatter
omes
simply from the radial fun
tion on . The spa
e in the
on-
stru
tion arises from a diagram of unitary ve
tor spa
es; the longer the
diagram, the more
ompli
atedthe geometry of the orbit. But even or-
2
bits that arise from the simplest diagrams (i.e., those of length ) may
have arbitrary high
ohomogeneity, whi
h puts them beyond the s
ope
of the (cid:16)low
ohomogeneity approa
h(cid:17) mentioned above. In [10℄, we su
-

essfully applied this te
hnique tos
l(o3n,sCtr)u
t the hyperKähler potentia4l
for the regular nilpotent orbit in , whi
h has
ohomogeneity .
The aim of this paper is to apply the same
onstru
tion to
al
ulate
hyperKähler potentialsfornilpotentorbitswithdiagramsof lengthtwo
orthree. Thisin
ludes
lassi
alorbitsof
ohomogeneityoneortwo and
also all orbits obtainable as limits of semi-simplsel(onr,bCit)s of symmetri
type. In parti
ular, we are able to prove (in the
ase) that the
Kähler potentials obtained by Biquard and Gaudu
hon on nilpotent
orbits are in fa
t hyperKähler potentials. This is not apparent from
their work, parti
ularly be
ause we found in [13℄ that several of these
orbits admit families of invariant hyperKähler metri
s wsioth(nK,Cäh)ler po-
tentials. We also determine the potential for orbits in whi
h
(3,22k,1ℓ)
have lengththree diagramsand Jordan type . In the simplest

ases there is a striking resemblan
e to the formulæ we have for the
k > 2

ohomogeneity two
ase, but for matters
ompli
ate rapidly.
In the
al
ulations we use (cid:28)nite
overing maps between nilpotent
orbits and the Beauville bundle
onstru
tion. It is worth pointing out
that these te
hniques
ombined with knowledge of the invariants of the

ompa
t group a
tion
an be used to (cid:28)nd the potential in several other
g
aCses, for example for nilpotent orbits in the ex
eptional Lie algebra
2
(see [14℄).
Expli
it knowledge of hyperKähler potentials is of interest in the
study of real nilpotent orbits,
f. [5℄, and we expe
t to pursue this in
future work.
The paper is organised as follows. Se
tion 2 re
alls the hyperKäh-
ler quotient
onstru
tion of
lassi
al nilpotent orbits and gives some
HYPERKÄHLER POTENTIALS 3
general results on hyperKähler potentials. In se
tion 3 we derive for-
2
mulæ for the potential for orbits with diagrams of length and then,
in se
tion 4, apply the result to the low
ohomogeneity
ase. Finally,
in se
tion 5 we work out the potential for the simplest orbits with
3
diagrams of length .
A
knowledgements. We are grateful for (cid:28)nan
ial support from the Ep-
sr
of Great Britain and Kbn in Poland.
2. Ba
kground and General Results
We beginby reviewingthe generaltheory ofthe relationshipbetween
hyperKähler quotients, hyperKähler potentials and nilpotent orbits.
(N,g) I J K
A Riemannian manifold with
omplex stru
tures , and
IJ = K = JI
satisfyingthequaternionidentities − , et
., ishyperKähler
g
if is Hermitian with respe
t to ea
h of the
omplex stru
tures and
ω (X,Y) := g(X,IY) ω ω
I J K
the two-forms , and are
losed. Su
h a
manifoldis thus symple
ti
in three di(cid:27)erent ways. If one distinguishes
I N
the
omplex stru
ture , then be
omes a Kähler manifold with a
ω := ω +iω
c J K
holomorphi
symple
ti
two-form .
An interesting general problem is to (cid:28)nd hyperKähler stru
tures
I

ompatible with a given
omplex stru
ture and a holomorphi
sym-
ω
c
ple
ti
form . One natural sour
e of su
hC manifolds is adjoint orbits
G
O of a
omplex seImi-simple Lie group . Su
h an orbit inheritgsCa

omplex stru
ture as a submanifold of the
omplex ve
tor spa
e .
X
The
omplex symple
ti
form on O is given at ∈ O by
ω ([A,X],[B,X]) = X,[A,B] ,
cO h i
, gC G
where h· ·i is thCe negative of the Killing form on . If is a
ompa
t
G G
real form of , then O admits a -invariant hyperKähler stru
ture
I ω
cO

ompatible with and [18, 17, 16, 2℄.
TheMarsden-Weinsteinquotient
onstru
tionwasadaptedtohyper-
H
Kähler manifolds in [9℄. Suppose a Lie group a
ts on a hyperKähler
N g I J K
manifold preserving , , and . Suppose also that there exist
µ µ µ N h
I J K ∗
symple
ti
moment maps , and from to for the a
tion
H ω ω ω I
I J K
of with respe
t to the symple
ti
forms , and . For , this
V h µV := µ ,V
means that for ea
h ∈ , the fun
tion I h I i satis(cid:28)es
dµV = ξ yω ,
I V I
(2.1)
ξ V
V
where is the ve
tor (cid:28)eld generated by the a
tion of . We then
de(cid:28)ne a hyperKähler moment map by
µ: N h ImH, µ = µ i+µ j +µ k.
∗ I J K
→ ⊗
4 PIOTR KOBAK AND ANDREW SWANN
N H
The hyperKähler quotient of by is de(cid:28)ned to be
N///H := µ 1(0)/H.
−
H N N///H
If a
ts freely on , then is a hyperKähler manifold of dimen-
dimN 4dimH H
sion − . Even if the a
tion of is not free, there is a
N///H
naturalwaytowrite asaunionofhyperKählermanifolds[6℄. We
I µ = (µC,µR)
will often distinguish the
omplex stru
ture and write ,
µC = µJ + iµK µR = µI µC
where and . The map is thenCa
omplex
H N
symple
ti
moment map for the (in(cid:28)nitesimal) a
tion of on .
G
Fornilpotentorbitsinthe
lassi
alLiealgebras,a -invarianthyper-
Kähler metri
may be
onstru
ted by (cid:28)nite-dimensional hyperKähler
quotients [11℄. The only other orbits foslr(nw,hCi
)h su
h a
onstru
tion
is known are the semi-simple orbits in [19℄ together with (cid:28)-
nite quotients of a
ouple of orbits in ex
eptional algebras [12℄. Let us
brie(cid:29)y re
all the
onstru
tion for nilpotent orbits.
2.1. NilpoteAnt Osrlb(nit,sC)for Spe
ialALkin1e=ar0GrouApks.=G0iven a nilpo-
−
tent element ∈ su
h t0ha=t V ⇄6 V ⇄anVd ⇄ ⇄onVe d=e(cid:28)Cnens
0 1 2 k
theasso
iatedimage(cid:29)ag tobe{ } ··· ,
V = ImAk i
i −
where . We
onsider the
omplex ve
tor spa
e
k 1
−
W = Hom(V ,V ) Hom(V ,V )
i i+1 i+1 i
⊕ (2.2)
i=0
M(cid:0) (cid:1)
(...,α ,β ,...) W
i i
and represent elements of by diagrams
0 = V ⇄α0 V ⇄α1 V ⇄α2 α⇄k−1V = Cn.
0 1 2 k
{ } ···
β0 β1 β2 βk−1
Cn
Taking to be equipped with a Hermitian two-form, indu
es Her-
V i = 0,1,2,...,k
i
mitian inner produ
ts on ea
h , , and we get a norm
W
on given by
k 1
−
r2 = (...,α ,β ,...) 2 = Tr(α α +β β ).
k i i k i∗ i i i∗ (2.3)
i=1
X
α
i∗
The inner produ
ts enables us to make sense of Hermitian adjoints
β W
and i∗ and to endow the ve
tor spa
e with a quaternioni
stru
ture
j(...,α ,β ,...) = (..., β ,α ,...)
by de(cid:28)ning i i − i∗ i∗ .
H = U(V ) U(V )
1 k 1
The produ
t ×···× − of unitary groups a
ts in a
W
natural way on :
(a ,...,a )(...,α ,β ,...,α ,β )
1 k 1 i i k 1 k 1
− − −
= (...,a α a 1,a β a 1 ,...,α a 1 ,a β ).
i+1 i −i i i −i+1 k 1 −k 1 k 1 k 1
− − − −
HYPERKÄHLER POTENTIALS 5
W
This a
tion preserves the quaternioni
stru
ture on , and the hyper-
µ = (µC,µR)
Kähler moment map is given by
µC = (...,αiβi βi+1αi+1,...),
−
µR = (...,αiαi∗ −βi∗βi +βi+1βi∗+1 −αi∗+1αi+1,...). (2.4)
W///H
The hyperKähler quotient= SL(n,Cis)Ahomeomorphi
to the
losure O
of the nilpotent orbit O , whi
h is a ψsin:gWular alggle(nbr,aCi
)
variety. The identi(cid:28)
ation is indu
ed by the map →
given by
ψ(...,α ,β ,...) = α β .
i i k 1 k 1
− − (2.5)
W W α
0 i
If ⊂ denotes the open set where ea
h is inje
tive and ea
h
β ψ: W ///H
i 0
ψ is surje
tive, then → O is a di(cid:27)eomorphism. GILn(nfa,C
t),
is the
omplex symple
ti
moment map for the a
tion of
W ///H
0
on and so the general theory of moment maps implies that
ψ ω W ///H
∗ cO agrees with the
omplex symple
ti
stru
ture on 0 . Note
j W α β β α
that on a
tsXon O bXy k−1slk(−n1,C7→) − k∗−1 k∗−1 whi
h agrees with
∗
the real stru
ture 7→ − on de(cid:28)ning the Lie algebra of the
SU(n)

ompa
t group .
2.2. Nilpotent Orbits in Orthogonal and Symple
ti
Algebras.
The abovseo
(non,Cst)ru
tionspm(na,yCb)e adapted to the remaining
lassi
al LAie
algebras angdC Ak. W=e0start wAikth1a=nil0potent eδlemen0t
−
ignC t=hesoL(nie,Cal)gebra1 gwCi=thsp(n,C) and 6 . Let be , if
, or , if . We
onsider the image (cid:29)ag
0 ⇄ (V ,ω ) ⇄ (V ,ω ) ⇄ ⇄ (V ,ω ) = (Cn,ω ),
1 1 2 2 k k k
{ } ··· (2.6)
ω : V V C
i i i
where × → are non-degenerate bilinear forms satisfying
ω (X,Y) = ( 1)k i+δω (Y,X).
i − i
−
dimV k i+δ
i †
(This implies that is even if − is odd). We denote by ·
ω
i
the adjoint with respe
t to the forms and de(cid:28)ne Lie groups
H = A U(V ) : A A = Id .
i { ∈ i † Vi}
H Sp(V ) k i+δ O(V ) k i+δ
i i i
Then is , if − is odd, or , if − is even.
H = H H W
1 k 1
Take ×···× − andletW+betWhequaternioni
ve
torspa
e
as in formula (2.2). The subspa
e ⊂ de(cid:28)ned by the equations
βi = αi†, i = 1,...,k −1,
is a quaternioni
ve
tor spa
e. The equations (2.4) de(cid:28)ne a hyperKäh-
H W+ ψ
ler moment map for the a
tion of on . Using the map of (2.5),
W+///H
the hyperKähler quotient may be identi(cid:28)ed with the
losure of
6 PIOTR KOBAK AND ANDREW SWANN
HCA hC
the nilpotent orbit k ⊂ k. Again, this identi(cid:28)
ation is
ompatible
ω
cO
with the
omplex-symple
ti
form and the real stru
ture.
ρ: N R
2.3. HyperKähler Potentials. A real-valued fun
tion →
N ρ
on a hyperKähler manifold is
alled a hyperKähler potential if
is simultaneously a Kähler potential for ea
h of the Kähler stru
tures
(ω ,I) (ω ,J) (ω ,K) I ω = i∂ ∂ ρ
I J K I I I
, and . For , this means that , or
equivalently
ω = 1dIdρ.
I −2
N
In general, will not admit a hyperKähler potential even lo
ally.
ρ ζ = 1 gradρ
Inζd,eIeζd,,Jζth,Ke ζexisten
e of implies that if we set H =2 R Stph(e1n)
∗ ∼
{ } generates an in(cid:28)nitesimal a
tion of ×
su
h that
L g = 0, L I = 0, L J = 2K,
Iζ Iζ Iζ
and
Jζ Kζ
with similar expressions for the a
tion of and , obtained by
(I,J,K)
permuting
y
li
ally (see [20, 4℄).
We need to know how hyperKähler potentialsbehave with respe
t to
hyperKähler quotients. An indire
t proof of a slightly weaker form of
the following result may be found in [20℄. Beware that the hypotheses
given in [4℄ are not quite strong enough.
(N,g,I,J,K)
Theorem 2.1. Let be a hyperKähler manifold admit-
ρ H
ting a hyperKähler potential . Suppose a Lie group a
ts freely and
N g I J K ρ
properly on preserving , , , and . Suppose also that there
µ H N
is a hyperKähler moment map for the a
tion of on and that
µ Sp(1)
is equivariant with respe
t to the in(cid:28)nitesimal a
tion of de(cid:28)ned
ρ
by , meaning
L µ = 0, L µ = 2µ ,
Iζ I Iζ J K
− et
. (2.7)
ρ
Thenthe fun
tion indu
es ahyperKähler potential on the hyperKähler
N///H
quotient .
i: µ 1(0) ֒ N π: µ 1(0)
− −
Proof. Let → be the in
lusion and write →
Q := N///H
for the proje
tion. The hyperKähler stru
ture on the
π ωQ = i ω
quotient is de(cid:28)ned by the relations ∗ I ∗ I, et
. In parti
ular, at
x µ 1(0)
−
ea
h ∈ the tangent spa
e to the (cid:28)bre is spanned by the ve
tor
ξV V h (Txµ−1(0))⊥ = IξV,JξV,KξV : V h
(cid:28)elds , for ∈ and { ∈ }. Thus
Y T µ 1(0) ξ IY JY KY
x − V
if ∈ is orthogonal to ea
h , then , and lie in
T µ 1(0)
x −
too.
ρ H
As iρs i:nvQarianRt under theπaρ
tio=n iofρ , it des
endπs dtoρ d=e(cid:28)inedρa
Q ∗ Q ∗ ∗ Q ∗
fun
tion → satisfying . This implies .
HYPERKÄHLER POTENTIALS 7
dρ 2ζ ζ H
Now is metri
dual to , so
ommutes with the a
tion of , and
ζ µ 1(0)
−
we
laim that is tangent to .
The equivarian
e
ondition (2.7) gives,
2µV = L µV = Iζ y(ξ yω ) = ω (ξ ,ζ),
K − Iζ J − V J K V
J
using the version of (2.1). But now
L µV = ζ ydµV = ζ y(ξ yω ) = ω (ξ ,ζ) = 2µV .
ζ K K V K K V K
L µ = 2µ ζ µ 1(0)
ζ −
Thus and preserves .
V h
For ∈ , we have
g(ζ,ξ ) = 1dρ(ξ ) = 1L ρ = 0,
V 2 V 2 ξV
ρ H Iζ µ 1(0)
−
as is -invariant. So is also tangent to . In parti
ular,
i Idρ = Ii dρ
∗ ∗
and we have
π ( 1dIdρ ) = i ( 1dIdρ) = i ω = π ωQ,
∗ −2 Q ∗ −2 ∗ I ∗ I
ρ ωQ
so Q is a Kähler potential for I . Similar
omputations apply for
J K ρ Q =
Q
and and we have that is a hyperKähler potential on
N///H
.
W W+
For the (cid:29)at hyperKähler spa
es and introdu
ed above, the
r2
hyperKähler potential is given=bWy t/h//eHfun
tsilo(nn,C)of equa=tioWn (+2//./3H). A
shoy(pne,rCK)ählerspp(ont,eCnt)ial on O 0 ⊂ or Or2 0 ⊂
or is then given by the restri
tion of to the zero
set of the hyperKähler moment map.
One
an now ask whether this hyperKähler potential is any sense
unique. In fa
t, one
an answer su
h a question for nilpotent orbits in
general. The following is an extension of an argument in [5℄.
G
Pσroposition 2.2. Let be a
ompa
t sgeCmi-simple LgiCe group and let
be the
orresponding real stru
ture on . Let O ⊂ be a nilpotent
orbit with the Kirillov-Kostant-Souriau
omplex symple
ti
stru
ture
(I,ω ) (g,I,J,K)
cO . Suppose is a hyperKähler stru
ture on O su
h that
ω + iω = ω g G
(a) J K cO, (b) is invariant under the
ompa
t group
and (
)Hthe stru
tujre aHdmits a hyσperKähler potential su
h that for the
∗ ∗
indu
ed -a
tion ∈ a
ts as |O. Then the hyperKähler stru
ture
is unique.
G
Proof. By averaging with the -a
tion we may assume that there is a
G ρ ζ = 1 gradρ
-invariant hyperKähler potential on O. Let 2 , as above.
L ω = 2ω L ω = 2ω
ζ I I ζ cO cO
Then and , so
ω = 1d(ζ yω ).
cO 2 cO
ω (2,0) ζ yω (1,0)
cO cO
Note that as is a -form, is of type .
8 PIOTR KOBAK AND ANDREW SWANN
ω
However, as O is nilpotent, the form cO is exa
t in Dolbeault
oho-
ω = dθ θ ([X,A]) = X,A
mGCology: cO , with θX 1ζyω h i, whi
hHi1s(ho,loCm)o=rp0hi
and
-invariant. Therefore −2 cO is
losed. But O , as for
θ 1ζ yω = df
nilpotent orbits hafv:e (cid:28)niteCfundamental groups. So − 2 cO ,
for some fun
tion O → .
df (1,0) G ζ
Now is of type and holomorphi
. It is also -invariant, as
G f G

ommGutes with . Therefore we may avef˜rage overdthf˜e=a
θtion1oζfyωto
get a -invariant holomorphiC
fun
tion satisCfying − 2 cO.
G G
Hofw˜ever, su
h afun
tζioynωis = 2-iθnvariantand a(
1t,s0t)ransitiveζlyonO,
so is
onstant and cO . Therefore, the -part of agrees
(1,0)
with the part of the Euler ve
tor (cid:28)eld on O. As both these ve
tor
I ζ
(cid:28)elds preserve , we have that equals the EulCer ve
tor (cid:28)eld.
∗
ζ We nIoζw have that the quotient oPf(O)by theθ -a
tion generated by
and is the proje
tivised orbitσ O withP(as)its
omplex-
onta
t
stru
ture and with real stru
ture . By [21℄, O is the twistor spa
e
M
of a unique quaternion-Kähler manifold of positive s
alar
urva-
(M)
ture and O is the asso
iated hyperKähler manifold U . Thus the
hyperKähler stru
ture is uniquely determined.
3. Nilpotent Orbits with Diagrams of Length Two
gC gC
Assume that iska
lassi
al
omplexXsimple Lsiel(anl,gCeb)raand O ⊂
isanorbitofarank nilpotentmatrix ∈ O ⊂ whi
hsatis(cid:28)es
X2 = 0 X (2k,1n 2k)
−
. Then has Jordan type . Su
h orbits are pre
isely
those that arise from diagrams of length two:
α
0 ⇄ Ck⇄Cn.
{ }
β
α: C2 Cn β: Cn C2
It follows from Ÿ2.1 that there exist → and → ,
X = αβ
su
h that , with
βα = 0 ββ = α α.
∗ ∗
and (3.1)
g = su(n) g
When this is the full set of equations for O. If is either
o(n) sp(n)
or , then we have additionally
β = α .
†
(3.2)
rankα = rankβ = rankX = k α β
In all
ases , so is inje
tive and is
surje
tive.
We shall use the above equations to
al
ulate the hyperKähler po-
ρ ρ
tential on O. From Theorem 2.1 we know that is the restri
tion of
r2
the radial fun
tion . By (2.3) we have
ρ = Tr(α α+ββ ) = 2Trα α = 2TrΛ,
∗ ∗ ∗
(3.3)
HYPERKÄHLER POTENTIALS 9
Λ = α α = ββ Λ
∗ ∗
where e ,...,e . SinC
ke is self-aΛdjoint,there exists an orthonor-
1 k
mal basis { } for in whi
h is diagonal,
Λ = diag(λ ,λ ,...,λ ).
1 2 k
ρ = 2(λ + +λ )
1 k
Thus ··· .
Note that
β e ,β e = ββ e ,e = Λe ,e = λ δ .
∗ i ∗ j ∗ i j i j i ij
h i h i h i
β e 2 = λ β λ > 0
∗ i i ∗ i
In parti
ular, k k . But is inje
tive, so and
β e ,...,β e Imβ
∗ 1 ∗ k ∗
{ } is an orthogonal basis for .
X X Imβ X X = Λ2
∗ ∗ ∗
Now
onsider the matrix . On , we have , sin
e
X Xβ e = β α αββ e = β Λ2e = λ 2β e .
∗ ∗ i ∗ ∗ ∗ i ∗ i i ∗ i
(Imβ ) = kerβ X = αβ X X
∗ ⊥ ∗
On the other hand, and , so vanishes
(Imβ ) X X λ 2,...,λ 2
∗ ⊥ ∗ 1 k
on . As a result has eigenvalues . Writing
SpecX X = µ ,...,µ µ k
∗ 1 r i i
{ } with distin
t and of multipli
ity we get
Theorem 3.1. Let O be the adjoint orbit of a non-zero nilpotent ma-
X X2 = 0
trix in a
omplex
lassi
al Lie algebra, and assume that .
Then the hyperKähler potential for the
anoni
al hyperKähler metri
on O is given by the formula
ρ(X) = 2 k µ 1/2
i i
(3.4)
µi∈SpXec(X∗X)
Remark 3.2. The above formula
an be obtained from (3.3) by expli
-
X
itlysolving(3.1)ansdl(n(3,.C2))foragivXenniUlp(ont)entelement . Forexample

onsider orbits in . Then is -
onjugate to
0 A
M = ,
0 0 (3.5)
(cid:18) (cid:19)
A = diag(a ,...,a ) a
1 k i
where with real and positive. To see this
X X e ,...,e
∗ 1 k
note that determines a set of orthonormal eigenve
tors
µ ,...,µ Xe ,Xe = µ δ
1 k i j i ij
with positive eigenvalues . Moreover, h i , so
µ 1/2Xe i = 1,...,k X2 = 0
i− i
, are also orthonormal. Sin
e it follows
that
0 = X2e ,Xe = Xe ,X Xe = µ Xe ,e .
i j i ∗ j j i j
h i h i
(cid:10) (cid:11)
In e(cid:27)e
t the ve
tors
e ,...e ,µ 1/2Xe ,...,µ 1/2Xe
1 k 1− 1 k− k
fConrm an orthonorXmal set. Complete this to anaor=thoµn1o/r2mal basis in
i i
. In this basis has the required form, with .
10 PIOTR KOBAK AND ANDREW SWANN
sl(n,C) so(n,C) sp(n,C)
Type
1 (21n 2) (221n 4) (212n 2)
− − −
Cohomogeneity
2 (221n 4) (31n 3) (241n 8) (2212n 4)
− − − −
Cohomogeneity ,
Table 1. Nilpotent orbits of low
ohomogeneity in
las-
si
al Lie algebras
X SU(n) λM λ
It follows that is -
onjugate to for some satisfying
λλ = 1
. The moment map equations (3.1) are now solved by
A1/2
α = λ β = λ 0 A1/2 ,
0 and
(cid:18) (cid:19)
(cid:0) (cid:1)
A1/2 = diag(a1/2,...a1/2) A = α α = ββ
where 1 k . In parti
ular ∗ ∗. We
Spec(XX ) = Spec(A2) = a 2,...,a 2
∗ 1 k
have { } and, by (3.4)
k
ρ(X) = 2 a .
i
| |
i=1
X
This agrees with the formula obtained in [3℄. There Biquard & Gaudu-

hon showed that this formula gives a Kähler potential for a hyper-
Kähler stru
stlu(rne,Con) the nilpotent orbit. This was done by
onsidering
the orbit in as a limit of semi-simple orbits. However, we have
now shown that the Biquard-Gaudu
hon Kähler potential is in fa
t a
hyperKähler potential.
4. HyperKähler Potentials for Low Cohomogeneity
Orbits
In the simplest
ase O is a minimal nilpotent orbit in a
lassi
al Lie
algebra. Su
h orbit arises from a length two diagram. Its Jordan type
is given in Table 1. Minimal orbits are
ohomogeneity one so any two
X,X X = X
′ ′
elements ∈ O are
onjugate ifandonlyifk k k k. It follows
X X X
∗
that for all ∈ O the matrix has only one non-zero eigenvalue,
λ κ ρ = 2κλ1/2 ρ2 = 4κ2λ
say , with multipli
ity . Then, by (3.4) , so .
TrX X = κλ
∗
But , so
1 sl(n,C) sp(n,C)
ρ2 = 4κTrX∗X, where κ = 2 for so(n,C), , (4.1)
(
for .