C OMPOSITIO M ATHEMATICA
J OHAN A. C. K OLK V. S. V ARADARAJAN
Lorentz invariant distributions supported on the forward light cone
Compositio Mathematica, tome 81, n
o1 (1992), p. 61-106
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Lorentz invariant distributions supported
onthe forward light
cone
JOHAN A.C.
KOLK1 and
V.S.VARADARAJAN2,*
(Ç)
1 Mathematisch Instituut, Rijksuniversiteit Utrecht, The Netherlands;
2Department
of Mathematics, University of California . Los Angeles, U.S.A.Received 12 June 1990; accepted 11 March 1991
1. Introduction
From the
physical point
ofview,
for instance in thequantum theory
of theelectromagnetic field,
it is of interest togive,
for any finite-dimensional module U for theconnected
Lorentz groupG,
adescription
of the spaceJ( U)
of all theU-valued distributions on
(the
dual ofthe)
Minkowskispace-time
that areinvariant under G and
supported
on the closed forwardlight
cone, see[2].
Lorentz invariant distributions have of course been studied in
depth,
notonly
on
physical space-time
but on the moregeneral
spaces Rm’" with aquadratic
form of
signature (m, n),
see[10], [12], [13],
the workof Gârding
and J.E. Roos in[3], [11], [16].
However the case of the vector-valued distributions as well asthe situation when their
supports
arerequired
to be in theforward (as opposed
to the
full) light
cone have not received theemphasis they
deserve in the mathematical literature. Our aim here is tosupplement
these papers with a consideration of these twoaspects.
We restrict ourselves to the case ofsignature ( 1, n).
We shall now
briefly explain
some of the main ideas of the paper. For this purpose it isenough
to consider scalar distributions. In his seminal paper[14]
M. Riesz studies the wave
operator
D and itscomplex
powers. ForRiesz,
as wellas for Gelfand and Shilov who took this up
later,
thestudy
of invariant distributions associated to thequadratic
form co wasessentially
aquestion
of theanalytic
continuation of the powers WS(s E C).
Thedescription
of all invariantdistributions with or without
support conditions,
did not emerge as anobjective
until the works of Methée and others referred to above focussed attention on
this
goal. Finally,
it was Harish-Chandra who realized(in
a differentcontext)
thefruitfulness of
regarding
the space of invariant distributions as a module for thealgebra
ofpolynomial
differentialoperators, especially
for the Liealgebra
a*Supported in part by NSF Grant DMS 88-03289. This author would also like to express his
gratitude to the Mathematical Institute of the State University at Utrecht, where he spent a part of the Fall of 1988, and where a substantial part of this work was carried out.
isomorphic
tosl(2, C),
with basis 0, cv, and their commutator[D, (0],
which isessentially
the Euler vector field. This idea is also ourstarting point
and one ofthe main results is a
complete description
of this module structure. Rallis and Schiffmann in[11]
have studied relatedmodules,
but as modules for 0only.
The closed forward
light
conesupports
two invariant measures:b,
the Diracmeasure at the
origin,
and(Xri,
the invariant measure on the open forwardlight
cone. One may
naively expect
that all invariant distributionssupported by
theclosed forward
light
cone may be obtained as linear combinations of then’ô
and~k03B1+0(k 0).
This is in fact true ifd,
the dimension of theunderlying
space, is odd. But when d is even this is nolonger
the case. Then the distribution~d-2/203B1+0
becomes amultiple
of b - a circumstancecompatible
with theHuygens principle -
and so one cannotgain
access to those invariant distributions which are nonzero away from theorigin
and have a transversal order at thepoints
of the forward cone that isd - 2 2. Thus,
togenerate
allinvariant distributions one has to start with an invariant distribution 03C4 of transversal
order 2 ;
the ones withhigher
transversal order are then obtained from theEk T,
and the ones with lower transversal order are obtained from the(Okr.
One also sees from thisdescription
that the module of invariant distribu- tions will have a much morecomplicated
structure when d is even, and that inparticular
it will not becyclic
for C alone.The construction of i is therefore one of the central concerns of this paper. For
use in
applications
inphysics
it is alsoimportant
to obtain anexplicit space-time expression
for i. This is done in Section 7.2 where the formula for i on the spacego
of Schwartz functionsvanishing
at theorigin
is calculated. The extension of i to the full Schwartzspace Y is
notunique; however,
all extensions are invariant andthey
are determinedby
the value at one element of g outsidego.
The method we use for
constructing i
is not theonly
onepossible.
Thetheory
of Riesz distributions can be used for this purpose, as we discuss in
greater
detail in[8].
Forinstance,
if(RJ
is the Rieszfamily (R,
is the restriction to the open solid forwardlight
cone of thepower cos
with a suitablenormalization),
it turnsout
that,
up to a constantdepending
ond,
the distribution r isequal
toAlso the
principal
results onJ(C)
can now beobtained, although
at variousstages
in thearguments,
one has to make use of ageneralization
to the one-sidedcontext of the
theory
of Methée. Therefore we havepreferred
todevelop
in thispaper the entire
theory
in a direct andelementary
manner,independent
of therather
sophisticated
Methée calculus.This work was
inspired by questions posed
to VSVby
a groupof physicists
atthe Istituto Nazionale di Fisica Nucleare in
Genova, consisting
of Professors G.Cassinelli,
G. Olivieri and P. Truini. We aregrateful
to Professor J.J. Duister- maat forstimulating
discussions on the results of this paper.2. Notations and statement of main results We set
We work in
R’,",
for n3,
with coordinates(p0, p1,.
.. ,Pn)
and fundamentalquadratic
formwhere p03BC
=+ p, according
as y is 0 or > 0. We write G forSO(1, n)°,
theconnected
component containing
theidentity
of thesubgroup
ofGL(R1°")
ofelements
fixing
03C9. We have:g = (aij) ~ G
if andonly if g
fixes w,det(g) = 1,
andaoo > 0
(
1actually).
MoreoverLie(G)
= g is its Liealgebra, acting
onR1,n
viathe vector fields
Here ,u, v =
1,..., n, for M
~ v,and a,,
=êlêp,.
The forwardlight
cone isand its closure
Cl(Xô )
=X+0 ~ {0}.
G
operates naturally
on the usual space of test functionsC~c(R1,n) (resp.
Y(R1,n))
and hence on the dual space of distributions(resp. tempered
distribu-tions).
Moregenerally
let U be a finite-dimensional G-module. A U-valued distribution(resp. tempered distribution)
is a continuous map T:C~c(R1,n) ~
U(resp. Y(R1,n) ~ U).
G acts on theseby
T is invariant
if
g - T =T,
for all g E G. For anyT, supp(T)
denotes itssupport.
Our main concern is with
J(U) = {T| T
an invariant distribution withsupp(T)
cCl(X+0)}.
Many computations involving
the distributions inJ(U) only exploit
theirbehavior under
multiplication by
thequadratic
form cv andapplication
of thewave
operator
Clearly
it is also natural to consideralong
with these the commutator of (O and D, cf.[5];
up to constants it isequal
to the radial or Euler vector fieldwhich detects the
homogeneity properties
of the distributions. These threeoperators generate
a 3-dimensionalsimple subalgebra
a of thealgebra
ofpolynomial
differentialoperators.
Infact,
for anypolynomial h
letM(h)
be theoperator
ofmultiplication by
h.Then,
if we writewe have the commutation rules
so that
is a three-dimensional
simple subalgebra
of thealgebra
ofpolynomial
dif-ferential
operators
onR1,n.
It is clear that aoperates
onJ(U)
so thatJ(U)
is ana-module;
and we shallgive
acomplete description
ofJ(U)
as a module for a.First we come to the modules U that
actually
do occur in thisset-up.
Let Y be thealgebra
ofpolynomial
functions onR 1 ,no
We have a natural action of G on Y.An element u ~ Y is said to be harmonic if ~u = 0. We write Jf for the
(graded)
G-submodule of Y of harmonic
polynomials
andHj
for itssubspace
ofhomogeneous
elements ofdegree j.
These are stable and irreducible under G. If U is an irreducible finite-dimensionalG-module,
thenJ(U)
is nonzero if andonly
if U ~Jfj.
The closed forward cone is stratified
by
two orbits: the vertex and the open forward cone.Accordingly
we have aninjection
of the a-moduleK(U)
of theLorentz invariant distributions
supported by
the vertex intoJ(U).
LetJ( U)
bethe space of germs of Lorentz invariant distributions defined on invariant open
neighborhoods
ofX ô
withsupports
contained inX t.
Then restriction of distributionssupported
onCl(X+0)
to open invariantneighborhoods of xt gives
a
mapping J(U) ~ J(U).
These maps lead to the sequenceIt is
important
to establish at the outset that(*)
is exact.Although K( U)
andJ(U)
arerelatively simple
to describe asa-modules,
the exact sequence(*)
doesnot
split
when d is even(it does,
if d isodd).
Theanalysis
of(*)
is thus a basicissue when d is even. We shall now
proceed
to a more detailed discussion of thispoint.
We recall the Verma modules
V(À) (À
EC), cf. §
3 infra. If 03BB, = i is aninteger 0,
then
F(i)
denotes the irreducible finite-dimensional module of dimension i +1,
and we have the exact sequenceNext we consider the module
M(i)
which may be characterized as themodule, unique
up toisomorphism,
for which there is anonsplitting
exact sequenceA
precise
formulation of the results above is now thatalways
and that
When d is even the submodule
F(d 2 - 2 corresponds of course to the linear
span of the elements
~k03B1+0 0 k d 2- 2 .
In order to
study (*),
weconsider,
for anyinteger
i0,
modules W for which there is anonsplitting
exact sequenceThe moduli space for all such modules W is
P 1 (C),
and for every value of the modulus we construct W as aquotient
of the universalenveloping algebra
of aby
anexplicitly given
ideal.It therefore remains to
verify
the exactness of(*)
as well as to determine theexact
parameter
thatcorresponds
to the module under consideration. For bothquestions
the essential case is that of U ~ C. Note that03C9:R1,nB{0} ~ R
is asurjective submersion,
therefore thepullback 03C9*(03B4(d-2/2)0)
ofthe d 2-derivative
of the Dirac measure
ô.
at 0 in Rgives
a distribution onR1,nB{0} homogeneous
of
degree - d.
Thesurjectivity
ofJ(C) ~ J(C)
now comes down toproving
that03C9*((03B4(d-2/2)0)
can be extended to a Lorentz invarianttempered
distributionsupported
onCl(X+0).
Suchtempered
extensions exist andthey
are auto-matically
Lorentz invariant. The determination of the modulus for thegeneral
case
J(U)
is a delicate calculation.Our main
theorem,
which summarizes the resultsabove,
is as follows.2.1. THEOREM. As usual d = n + 1.
(a)
For anyinteger
i 0 the setof isomorphism
classesof
a-modules Wadmitting
anonsplitting
exact sequence(W, i)
is in naturalbijection
withP1(C) = C ~ {~}.
LetW(i : y)
be the modulecorresponding
to y EP1(C).
(b) If U is
anirreducible finite-dimensional G-module,
thenJ(U)
is nonzeroif
and
only if
U ~Hj.
I n this case,The invariant y is a subtle one. However there is a
qualitative
differencebetween the module
W(i : oo)
and the modulesW(i : y)
with yfinite, namely, W(i : oo)
is aweight
module while theW(i : y)
with y finite are not so; the action of H on these involves atwo-step nilpotent corresponding
to eacheigenvalue
ofmultiplicity
2.In the
description
of theJ(U)
there is a noticeable contrast between the case of odd or even dimensiond ;
this isdirectly
related to theHuygens principle.
If d isodd there exist no invariant distributions T
supported by
the forwardlight
conethat are fundamental solutions of the wave
operator,
i.e. with UT = ô. Infact,
inthis case the Dirac measure at 0 is a
highest weight
vector in the Verma moduleoccurring
as the secondsummand;
and therefore it cannot be in theimage of ~.
It would be
interesting
to know whether there existgeometric
realizations for the modulesW(i : y)
for finite values of y different from1 2(i+1),
as spaces ofvector-valued distributions or variations thereof. And
furthermore,
can oneexplain
theisomorphisms
between the moduleshaving
the same value of3. Construction of the modules
W( j : y)
In this section we shall
give
the construction of the modulesW( j : y).
Weconsider modules for g =
sI(2, C)
with the commutation rulesFor any module V and any
c E C, V[c]
denotes thegeneralized eigenspace
of Hfor the
eigenvalue
c,namely
the space of all v E v such that(H - c)’v
=0,
forsome k.
Let us write
V(03BB)
for the Verma module with basis(v03BB,
VÀ-2,...)
such thatWe next introduce the modules
M( j), with j
aninteger
0. Let us considermodules M for which there is a
nonsplitting
exact sequenceThen H acts
semisimply
on M withsimple spectrum {j, j - 2,....}.
It is obvious thatFor any
integer j
0 one can construct such a moduleM(j)
as follows. Themodule
M(j)
has a basis{m(k)}k=j,j-2,...
such thatConversely
any Msatisfying (E)
andnonsplit
isisomorphic
toM( j);
to see thiswe observe that X is
surjective
on M and so we can find0 ~ m(k)
EM[k],
X m(k)
=m(k
+2) (m(j
+2)
=0);
the commutation rules then lead to the above action of Y3.1. We shall next consider modules W
admitting
an exact sequencePROPOSITION. Let w ~
W[- j - 2]
map to a nonzero elementof M( j).
ThenYX w and
(H + j
+2)w
areindependent of
the choiceof w;
andProof.
The first assertion is clearsince,
for any u EV(-j - 2)[ - j - 2],
X u = 0and
(H + j
+2)u
= 0. Also theimplication =>
is obvious.Suppose conversely
that YXw = 0 and
(H + j
+2)w
= 0. Let Wk =Xkw, 0 k j
+ 1. Since wj+1 maps toXj+1 m ~
0 where m is theimage
of w, it must be nonzero. Letwhere
... >
denotes linear span. We leave it to the reader toverify
that L is asubmodule. It is obvious that L ~
M(j)
and W =V(-j - 2)
~L.
3.2. DEFINITION.
Suppose
that(3.1)
does notsplit.
Since(H + j
+2)w
andYX w map to 0 in
M( j), they
both lie inV(-j - 2)[ -j - 2]
and sosatisfy
anontrivial relation
03B1 YXw - 03B2(H + j
+2)w
= 0. Hence we have aunique point
03B3 = 03B2 03B1 ~P1(C) = C ~ {~}.
It is obviousthat 03B3 = 03B3(W)
is an invariant of W. Our aim is to show that allpoints of P1(C)
arise in this manner and that y determinesW
uniquely.
We writeW(j : 03B3)
for any W for which03B3(W)
= y.It is clear that the
spectrum
of H in Wis {j,j - 2,...};
theeigenvalues j, j - 2,..., - j
aresimple
and the others are double.LEMMA. Let y =
y(W). If y is finite, W is
not aweight module; and for
any r1,
H
+ j
+ 2r is a nonzeronilpotent
onW[- j - 2r]. If
y = oc, then W is aweight
module. In either case, w generates W
Finally, if
y isfinite,
and 1r j
+1,
Proof.
If y isfinite,
we have weW[- j - 2],
and(H + j
+2)w ~ 0, proving
that
H + j + 2
is a nonzeronilpotent
onW[- j - 2]. Furthermore,
for anyk 1,
X is abijection
ofW[- j - 2(k
+1)]
withW[- j - 2k] taking H -
2 to H.This
implies, by
inductionon k,
that H+ j
+ 2k is nonzeronilpotent
onW[- j - 2k]
for all k. In either case let W’ =U(g) ·
w, whereU(g)
is the universalenveloping algebra
of g. It is clear that W’ maps ontoM( j).
So to prove that W’ = W we need toverify
that W’ containsker( W ~ M(j)).
If y is finite andv =
(H + j
+2)w,
then v ~0, (H + j
+2)v
=0,
and v maps to zero inM( j),
sothat v
generates ker(W - M(j)).
If y = oo, we take v =YX w;
thenagain
v ~ 0and maps to 0 in
M(j)
so that itgenerates ker( W ~ M(j)).
We prove the last assertion of the lemmaby
induction on r. This is clear for r = 1. Let r > 1 andassume it for lower values of r. Now
therefore we
get,
if we remember that(H + j
+2)2W
=0,
3.3. We write
« ... »
for the idealgenerated by - - .
LEMMA. Let
U(g)
be the universalenveloping algebra of
g. Thenwhere
Proof
If0 ~ u ~ M(j)[- j - 2],
the map a ~ a·u(a
EU(g)) gives
an exactsequence
U(g)/K(j) ~ M(j) ~
0. The elementsY"X’,
for r0, s j
+ 1 spanU(g)
modK( j).
On the otherhand,
if r 1and s 1,
So
Yr,
for r0,
andX S,
for 1s j
+1,
spanU(g)
modK( j).
Theweights
ofY" and X S are
respectively - j -
2 - 2rand - j -
2 + 2s. This shows that thespectrum
of H onU(g)/K(j)
issimple
and is containedin {j,j - 2...}, and, hence,
that
U(g)/K(j) ~ M(j).
3.4. We now consider the
W( j : y).
We treat first the case of finite y. The vector wis as in Section 3.1. For any
integer j
0 andy E C let
Let
LEMMA.
I( j : y)w
= 0 and03932j
EI( j : y).
Inparticular F?
= 0 onW( j : y).
Proof
The first relation is obvious. Since0393j ~ (H - j + 403B3)(H + j + 2)
mod
1( j : y)
weget rJ ==
0 mod1( j : y), proving
the secondrelation.
3.5. LEMMA.
dim(U(g)/I(j: y))[ - j - 2]
= 2.Proof.
Let us write 7 for1( j : y),
W forU(g)/I,
and defineNow
YrXs(H + j
+2)t,
for r, s, t0,
form a basis forU(g)
and it is clear that thesubfamily
with r 0 and(s, t) satisfying
one of:(a) s j
+2; (b)
t2; (c)
s1,
t
1,
span l’. HenceYrXs,
for r0, 0 s j
+1,
andyr(H + j + 2),
forr
0,
form a basis forU(g)
mod l’. Theweights
of these are,respectively, - j - 2 + 2s - 2r and - j - 2 - 2r,
so that if W’ =U(g)/l’,
and a ~ a’ is thenatural map
U(g) ~ W’,
thenW’[- j - 2]
has the basisu’, vs,
for0 s j
+1,
where u = H
+ j
+ 2and vs
= YSX S. Now I’ c I and I = l’ +U(g)(YX - 03B3(H + j
+2)).
Hence W =W’/W"
where W" is the submodule of W’generated by (YX - y(H + j
+2))’
which lies inW’[- j - 2].
ButW"[- j - 2]
isspanned by
the elements a ·(YX - y(H + j
+2))’ where a
is of theweight
0 for theadjoint representation, i.e., a
commutes with H. Now the centralizer of H inU(g)
is the
algebra generated by H
and YX and is therefore abelian.Hence, W"[- j - 2]
isL(W’[- j - 2])
where L is theendomorphism
of W’ which isinduced
by
leftmultiplication by YX - y(H + j
+2)
onU(g). Hence,
But a
simple
calculation shows thatSo for the kernel ic and the range p of L on
W’[- j - 2]
we havewhere (...)
denotes linear span. The vectorsdetermining
p arelinearly independent,
so that K =~u’, v’j+1~
and the dimension of pis j
+ 1. This provesthat
W[- j - 2] = W’[- j - 2]/03C1 ~ C2.
3.6. Since
I(j: y)
cK( j),
we now may introduce the module V’ =K( j)/I( j : y).
LEMMA. We have V’ ~
V(-j - 2)
and the sequenceis exact and
nonsplitting.
Inparticular, for
W =U(g)/I(j : y)
we havey(W)
= y.Proof.
We know YX’(r 0,
sj
+1), Y’(H + j
+2) (r
,0)
spanU(g) mod I(j:03B3).
But if r1,
s1,
Hence,
These have
weights - j - 2 + 2s, - j - 2 - 2r, respectively.
So thespectrum
of H on W is containedin {j, j
-2,...}
with theeigenvalues
ofmultiplicity
1 or 2according
asthey are - j or - j - 2. Hence,
thespectrum
of H on V’ issimple
and containedin {- j - 2, - j - 4,...}. By
3.5 we knowthat - j - 2 is
aneigenvalue. Hence, - j - 2
is thehighest weight
ofV’, showing
thatV’ ~
V( - j - 2).
On the otherhand,
H+ j
+2 ft I( j : y)
as otherwise we wouldhave
K(j)
cI( j : y)
cK( j), giving V’
= 0. We may therefore take w to be theimage
of H+ j
+ 2 in W and find thaty( W)
= y. · 3.7. We now take up the case y = oo. We setLEMMA.
dim(U(g)/I( j : oo))[- j - 2]
= 2.Proof.
Writeand a’ for the
image
in W’ ofa E U(g).
Let W" =U(g)·(03932j)’.
Thenyrxs(H + j
+2Y,
for r, s, t0,
form a basis forU(g)
and thesubfamily
witheither ,s
j
+ 2 or t 1 span l’.Hence,
YrXs(r 0, 0 s j + 1)
form a basis ofU(g)
mod l’. Inparticular,
Vs =(Y’X’)’ (0 s j
+1)
form a basis ofW[- j - 2J.
Since1 j
is m the center ofU(g)
we have W"’ =03932j.
W’ so thatA
simple
calculation shows thatfrom which it follows
easily
that(restricted
toW’[ - j - 2]) ker(03932j)
hasdimension 2 and that
W’[ - j - 2]
=ker(03932j)
~im(03932j).
3.8. As
I(j:~) ~ K(j)
we are allowed to introduce the module V’ =is a
nonsplitting
exact sequence. Inparticular, y(W)
= oo.Proof.
Write1 = l(j: (0).
Theimages
Vrs of YrXs inU(g)/I
for r 0 and0
sj
+ 1 span W. An easy calculation showsthat,
with yj =rj/4,
and, hence,
that for suitable constants ars,brs,
It follows that for 1
s j
and constants crs we haveThus vro and vr(j+1)
span m
and theirweights
arerespectively - j -
2 - 2r andj - 2r. Consequently,
thespectrum
of H in V’ issimple
and contained in{- j - 2, - j - 4, ...}. As - j - 2
is aneigenvalue by 3.7,
it follows as before that V’ ~V(- j - 2).
If w is theimage
of 1 inW,
we must haveYX w ~ 0; for, otherwise,
YX E I which wouldimply
thatK(j)
~ I ~K(j)
so that V’ would be 0. Since(H + j
+2)w
=0,
we have03B3(W)
= ~.3.9. THEOREM. For each
y E P1(C)
there isexactly
oneW( j : y)
up to isomor-phism. It is a weight module for
y ocwhile for finite
y the actionof H + j
+ 2k onW(j:03B3)[-j-2k] for
eachk
1 is a nonzeronilpotent.
ln either case,if w ~ W(j:03B3)[-j-2]
maps to a nonzero element inM(j),
there is aunique isomorphism of W( j : y)
withU(g)/I(j: y)
that takes w to theimage of
1.Proof.
Let W be such that(3.1 )
is exact andnonsplitting
and lety( W) = y. If J
is the annihilator of w in
U(g),
it is direct that1( j : y)
c J. Hence there is aunique surjective homomorphism U(g)/I(j: y)
~ W that takes theimage
of 1 to w. Thismust be an
isomorphism
since themultiplicities
of theeigenvalues
of H are thesame for both modules. The
remaining
statements are clear. Note that thisproves Theorem
2.1(a)
3.10. The modules
W(j:03B3)
do not seem to have been encountered before inrepresentation theory; however,
the modulesW( j : oo)
are the same as themodules
T
discussed in[18,
p.185].
3.11. We shall now
give
anotherdescription
ofW( j : y).
Let W be thecategory
of modules V such that:(1)
v= ~c~C V[c],
where theV[c]
are thegeneralized weight
spaces forH;
(2) V[c] ~ 0, only
for c ~Z,
and finite-dimensional for all c.As before let
and for any V ~ B and
integer
c0,
letVc
be the maximalsubspace
of V onwhich 0393 -
c(c
+2) is nilpotent. The n
are submodules and V= ~ Vc.
We writeE,
for theprojections V ~ Vc.
The functor03B2j,
forintegers j 0,
is then definedby
It is clear that this is a covariant functor which is exact, that
is,
takes exactsequences to exact sequences. Our aim is to prove the
following
theorem whoseproof requires
somepreparation.
THEOREM. We have
3.12. We
write {fj, fj-2,..., f- il
for a basis ofF(j) with fk
ofweight k
andYfk = fk-2 (Yf - k
=0).
Thefollowing proposition
is thespecial
case, forg =
sI(2, C),
of[1,
Lemma5].
PROPOSITION. Let c E C. Then
V(c) 0 F(j)
hasa flag of
submodules whose successivequotients
areMore
precisely,
letv(c)
be a nonzerohighest weight
vectorof V(c)
and letwhere D
=C [ Y]
cU(g).
Then :(1)
theLr
are submodules and 0 cLo
c... cLi
=V(c)
0F( j);
(2) Lr/Lr-1 ~ V(c + j - 2r);
(3)
Theimage of v(c) O fj- 2r ~ Lr
lies inV(c+j-2r)[c+j-2r]
and is not zero.3.13. DEFINITION.
Suppose
W ~ B has aflag
of submanifoldsAssume that for all
m = 0,..., p,
we have that(Wm/Wm-1)c(m)
=Wm/Wm-1,
for somec(m) E C;
thatis,
r has asingle eigenvalue c(m)(c(m)
+2)
onWm/Wm-l’
For afixed r, 0
r p, the numberc(r)
orWr/Wr-1
is said to beisolated if
c(m)(c(m)+2) ~ c(r)(c(r) + 2)
for m ~ r.3.14. Let the context be as above and let 03C0 be the natural map
LEMMA
(1) Suppose
x ~Wm
and 03C0x ~0,
thenEc(m)x ~ 0;
infact,
(2) Suppose further
thatWr/Wr-1 is
isolated. Then W’ :=Ec(r)W
is contained inW
andWr = W’ ~ Wr-1.
Proof
The assertion(1)
follows from thecommutativity
of thediagram
For
(2)
we write W" =(I - Ec(r))(W).
Then W =W’ ~
W" so thatW"c(r)
= 0. Forany m,
Taking
m = r we see thatW
~ W" =Wr-1
n W".Hence,
On the other
hand,
if m ~ r, theimbedding
shows that
Wm-l 1 ~ W’ = Wm
n W’.Hence,
This proves that
Wr = W’ ~ Wr-1.
3.15. Let c ~
C, Lr
as inProposition 3.12,
andc(r)
= c+ j -
2r.LEMMA. We have
and its
image
inL, /4 -
1 generatesL, /4 -
1.Proof.
Follows fromProposition 3.12(2)
and Lemma3.14(1).
3.16. We continue in the above context but with
V(-2)
inplace
ofV(c).
LEMMA. Let E =
Ej,
then:Proof.
Assertion(1)
is immediate from Lemma 3.15. For(2),
notice thatV(-2) ~ Fi
has theflag
of submodulesL,
withLr/Lr-1 ~ V(-2+j-2r)
withr = 0, 1,..., j.
ThenLj/Lj-l
i is isolated and soby
Lemma3.14(2)
we find that03B2j(V(-2)) ~ V(-j - 2).
We now prove(3).
From(2)
and the exact sequencewe
get
the exact sequenceIt is thus
enough
to prove that this does notsplit.
If itdoes,
thenV( - j - 2)
is adirect summand of
03B2j(M(0)) and, hence,
ofM(O) Q F( j).
So there exists v ~ 0 inM(O)
(8)F(j)
ofweight - j -
2 with X v = 0. Writewhere , and
is of weight
ThenThis means that v-j+2r is of
weight
0 so that r= - 1,
acontradiction.
Let us write
W(y)
forW(0 : y).
From the above lemma we obtain the exact sequenceWe are thus reduced to
verifying
that this does notsplit
andcomputing
the y- invariant of03B2j(W(03B3)).
Considerand let we
W(y)[ -.2]
be as in 3.1. Let3.17. LEMMA. With notation as above we have
the following :
Proof
A standard51(2)
calculation shows thatXjf-j = j!2fj.
Also we knowthat
Xj+1f-j
=0, X2w
= 0.Hence,
But YX w = u E
V(-2)[-2]
andyj(u 0 fj) ~ L0 ~ V(j - 2), according
toPropo-
sition 3.12.
Hence, ELO
=0,
so thatAgain Yj-1(u ~ fj-2) ~ L1,
andEL1
=0,
so thatThis
argument
can be continued until weget
For
(2)
wesimply
note3.18. We recall the exact sequence
LEMMA. Let w’ =
E(w
Qf-j).
Then w’ E03B2jW(03B3)[-j- 2]
and maps to a nonzeroelement
of M( j)[ - j - 2].
Moreover the above exact sequence does notsplit,
andso,
03B2jW(03B3)
=W( j : y’) for
somey’ E P’(C);
andy’
isfinite if
andonly if
y is.Proof.
Theimage
of w’ inM(j)[-j-2] is E(m( - 2) (8) f - )
wherem( - 2) E M(o)
is theimage
of w. UnderM(0) ~ V(-2)
we havem(-2) ~ v’ ~ V(-2)[-2]
withv’ ~ 0,
so that underM(j) ~ V(-j-2),
theelement
E(m(-2) ~ f-j)
goes toE(v’ ~ f-j)
which is~ 0 by
Lemma3.16(1).
Hence, E(m( - 2) (D f- j)
~ 0.We now have
Suppose
now that y E C. Then v ~ 0 and therefore(H + j
+2)w’ ~
0. Soy’ E
C.Suppose
next that y = oo . Then v = 0 andu ~ 0.
So(H + j + 2)w’ = 0;
moreover
This
implies
thatYXw’ ~
0.Indeed,
if YX w’ =0,
then as(H + j
+2)w’ = 0, Proposition
3.1 willapply
togive
the conclusion that03B2j(W(03B3)) = V(-j- 2)
~M( j),
whichimplies
thatYi’ 1Xj+1
= 0 on03B2j(W(03B3)).
3.19. We can
complete
theproof
of Theorem 3.11by proving
thefollowing
lemma.
LEMMA.
03B2j(W(03B3)) = W(j:(j
+1)y).
Proof.
We needonly
consider the case when y is finite. ThenBut u = YX w =
y(H
+2)w,
so thatLet
y’
be such that YX w’ =y’(H + j
+2)w’.
Thenby
Lemma3.2,
Hence
equating
theright-hand sides,
we find4. The modules
li
We now return to the framework of Section 2. The
properties
of the modulesli
are
quite
classicaland
have beenstudied,
in avastly
moregeneral
context,by
Kostant
[9, §§0-2].
For our purposes we need a few refinementswhich, although readily
deducible from Kostant’stheory,
are hard to find in the literature. Let X be acomplex
vector space of finite dimension on which aconnected
complex
reductivealgebraic
group Loperates.
Let Y be thealgebra
of
polynomial
functions on X. For any x in X we writeL(x)
for the orbit of xunder L. If x E
X,
letF(L(x))
denote the space of functions onL(x)
that arerestrictions of the
polynomial
functions toL(x);
moreover, letR(L(x))
denote thespace of L-finite functions on
L(x), namely,
the linear span of all finite- dimensional spaces of functions onL(x)
that are stable under L and carry aholomorphic (=rational) representation
of L. It is obvious thatF(L(x)) c R(L(x)),
and it isimportant
to know when there isequality
here. It is known(see [9])
that if thevariety Cl(L(x))
is normal andCl(L(x))BL(x)
is ofcodimension at least 2 in
Cl(L(x)),
thenF(L(x))
=R(L(x)).
Let Y be thealgebra
ofconstant coefficient differential
operators
on X. We shallnaturally identify
itwith the
symmetric algebra
over X. We writeI(Y) (resp. I(Y))
for thesubalgebra
of L-invariant elements of Y
(resp. Y).
An elementf
of Y is called harmonic ifa(u)f
= 0 for all u EI(Y)
whose constant terms are zero. We denoteby
H thegraded
space of harmonic elements andby li
itshomogeneous components.
Let us now introduce the
variety
fi of zeros of the idealI(Y)+·
EP whereI(Y)+
is the set of all elementsof I(Y)
that vanish at theorigin
of X.Following
Kostant let us make the
following assumptions:
(1) I(Y)+·
Y is aprime ideal;
(2)
there exists an orbit 0 dense inN;
(3)
there exists asymmetric nondegenerate
bilinear form( . , . )
for X invariantunder L.
Under these
conditions,
which we shall call(K),
we have thefollowing
results(cf. [9]):
(1)
the natural mapI(&» ~ H ~
P is anisomorphism;
inparticular Y
is afree
I(P)-module;
(2)
H is the linear span of the monomials1(x)",
for n =o, l, 2, ... ,
where1(x)
isthe linear function
y ~ (x, y)
onX;
(3)
let x e X be calledquasiregular
ifthen if x is
quasiregular,
the(surjective)
restrictionmap P ~ F(L(x))
is anisomorphism
onto when restricted to e.The structure of Yf
(hence
of9)
as an L-module may be obtained under suitable additionalassumptions.
Let x E X be anyquasiregular element;
forinstance,
we may take x to be such thatL(x)
is dense in X. Let Lx be the stabilizer of x in L and letR(L/Lx)
be the left L-module of allregular
functions onL that are
right
Lx-invariant. If in additionCl(L(x))BL(x)
is ofcodimension
2 inCl(L(x)),
we have the L-moduleisomorphism
e -
R(L/Lx),
which, together
with Frobeniusreciprocity
shows that e containsprecisely
those irreducible
representations
of L whosecontragredients
contain nonzerovectors invariant under
Lx,
and that themultiplicity
of such arepresentation
isthe dimension of the space of Lx-invariant vectors in the
contragredient representation. Finally,
we have anisomorphism
of Y with Y under which the linear function
1(x):
y H(x, y)
goes over to the directional derivationê(x)
in the direction of x. The bilinear form for EP definedby
is then