A NNALES DE L ’I. H. P., SECTION A
M. D. G OULD
A trace formula for semi-simple Lie algebras
Annales de l’I. H. P., section A, tome 32, n
o3 (1980), p. 203-219
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203
A trace formula
for semi-simple Lie algebras
M. D. GOULD (*)
Department of Mathematical Physics, University of Adelaide,
G. P. O. Box 498, Adelaide, South Australia, 5001
Ann. Henri Vol. XXXII, n° 3, 1980,
Section A :
Physique theorique.
ABSTRACT. A full set of invariants for a
general semi-simple
Lie groupis constructed. The
eigenvalues
of these invariants(i.
e. theirimages
underthe Harish-Chandra
homomorphism)
are determinedby
anexplicit
formulawhich is
simple
and easy toapply.
The invariants constructed are naturalgeneralizations
of the universal Casimir element.1. INTRODUCTION
The dermination of a full set of
algebraically independent
CasimirInvariants and their
eigenvalues
is aproblem
which has along
and distin-guished history. Quadratic
operatorscommuting
with the elements of the universalenveloping algebra
were first introducedby
Casimir[7] ]
in his
purely algebraic
treatment of thecomplete reducibility
of finitedimensional
representations.
Later these operators, now nolonger
neces-sarily quadratic,
came to be known as « Casimir operators ».By
theearly
1960’s it was known, from
previous
work of Racah[2 ],
Gel’fand[3] and Chevalley [4 ],
that anysemi-simple
Liealgebra
of rank nalgebraically independent
Casimir invariants. Such invariants, for the various classical groups, have since beenexplicitly
constructed[5] ]
and theireigenvalues
on irreducible
representations
of the group determined[6 ].
(*) Present Address : School of Physical Sciences, The Flinders University of South Australia, Bedfork Park, South Australia, 5042.
Annales de l’Institut Henri Poincaré-Section A-Vol. XXXII, 0020-2339/1980/203/$ 5.00 ‘ 8
@ Gauthier-Villars
204 M. D. GOULD
It is the aim of this paper to present a unified method for the construction of a full set of
algebraically independent
invariants for anarbitrary
semi-simple
Liealgebra.
Theeigenvalues
of these invariants onrepresentations admitting
an infinitesimal character are determinedby
anexplicit
formulawhich is
simple
and easy toapply.
Let L be a
semi-simple
Liealgebra
of rank and letV(~)
be a finite dimen- sional irreduciblerepresentation
of L withhighest weight
/L Ourapproach
is
intimately
related to the nature of the tensorproduct
spaceV()w)
@ Vwhere V is a
representation (finite
or infinitedimensional) admitting
aninfinitesimal character. The case where V is a Verma module has been treated in detail
by Bernstein,
Gel’fand and Gel’fand[7] while
the moregeneral
case where V is a Harish-Chandra module has been treatedby
Kostant
[8 ].
We shall be concerned with a certain matrix A, with entries from L, which may be viewed as an operator on the spaceV(/)
0 V. Tracesof powers of A are shown to be Casimir invariants which
(at
least for thesimple
Liealgebras)
generate the centre of the universalenveloping algebra.
In
determining
theeigenvalues
of these invariants we make use ofWeyl’s
dimension formula
along
the linessuggested
in recent work of Okubo[9] ]
and Edwards
[7~] (and
also compare with Louck and Biedenharn[77]).
We remark however that our extension is nontrivial and
exploits
theClebsch-Gordan formula
originally
obtainedby Klimyk [12 ]. By
thismeans we obtain a character formula which is a
polynomial
in thehighest weight
of therepresentation
and hence extends(via
Zariskicontinuity)
to infinite dimensional
representations.
2. PRELIMINARIES
Let L be a
complex semi-simple
Liealgebra,
H a Cart ansubalgebra,
H* the dual space to H and C c H* the set of roots with respect to the
pair
(L.H).
Let 1>+ c I> be a system ofpositive
roots, A c 1>+ a base for ~, d the half sum of thepositive
roots and W theWeyl
group for thepair (L, H).
Let A denote the set ofintegral
linear functions on Hand A+ c A the set of dominantintegral
linear functions on H.Finally
let( , )
denote the inner
product
induced on H*by
theKilling
form and forset
, 03B1> = 2(03BB, a)
.a set /I.,
a >
= -20142014-.For any v E H* let Tv denote the translation map defined
by
= A + vfor
any 03BB
in H*. The translatedWeyl
group W is defined as theconjugate
of W in the group of invertible affine transformations of H*. Thus every element of W is of the form 8- = where 6 E W. W therefore
acts on H*
according
to6(~~) = 6(~,
+5) -
5 for any Ii in H*. Now let U be the universalenveloping algebra
of L and letU(H)
c U denote theenveloping algebra
of H. It is often convenient toidentify U(H)
with theAnnales de l’Institut Henri Poincaré - Section A
205
A TRACE FORMULA FOR SEMI-SIMPLE LIE ALGEBRAS
ring
ofpolynomial
functions on H*. TheWeyl
group W acts onU(H)
where if 6 E Wand hE
U(H), ~
E H*, thenSimilarly
the translatedWeyl
group acts onU(H) according
toNow let Z be the centre of U and let B be the
nilpotent subalgebra
of Lspanned by
root vectorscorresponding
to roots For any z~Z it is known[13] ]
that there is aunique
element such thatAccordingly
one obtains analgebra homomorphism
Z -~U(H)
definedby ~ -~. Following
Harish-Chandra andDynkin
one in fact has thefollowing
result.2.1. THEOREM. -- For
any z~Z
one has and the map Z -~U(H)~, ~
-~/~
is analgebra isomorphism.
By
virtue of this result one mayidentify
the centre Z of U with thering
of W-invariant
polynomial
functions on H*.If l=rank L = dim H then one knows that there exist
which are
algebraically independent
and such thatU(H)w
isgenerated
as an
algebra by
thel~;
In view of Theorem
(2 .1 )
one knows that there exist elements ... , r~e Z which may be identified with thepolynomial functions 11, ...? ~
underthe Harish-Chandra
homomorphism.
It isclear,
fromequation ( 1 ),
thatthe zi are
uniquely
determinedby fzi
=i.
Hence, as acorollary
to Theo-rem
(2 .1 ),
we obtainCOROLLARY. - Z
= C [z l’ ...,zl] where the zi
arealgebraically
inde-pendent.
By a character X we shall mean an
algebra homorphism
of Z into the scalars C. If z 1, ... , z~ E Z arealgebraically independent
then a characterX is
uniquely
determinedby
the scalars which may bearbitrary complex
numbers. Thus if c = ..., there exists a
unique
character Xc such that206 M. D. GOULD
By
this means one may set up abijection
between the characters over Z and elements of C1.We say that a module Mover U admits an infinitesimal character if the elements of the centre Z take constant values on M. Such a module determines an
algebra homomorphism
Z ~C,
z -~ wherexM(z)
is theeigenvalue
of the central element z on M. In such a case wesay that M admits the infinitesimal character XM’ If vo is a maximal
weight
vector, of
weight ~,
say, then t~o determines analgebra homomorphism
x~ : Z -~ C, where is the
eigenvalue
of z E Z on vo. In view of equa- tion(1)
we see that x~ isuniquely
determinedby
The characters XÂ
play a
fundamental role in characteranalysis
since it isa theorem of Harish-Chandra
[13, 7~] ]
that every character X over Z is of the form X = XÂ for some 03BB E H*. Thecharacter ~03BB
does not characterize theweight ~, uniquely
since it mayhappen
that XÂ EH*,
but À =1= ,u.One in fact has the
following
result due to Harish-Chandra.2 . 2 . THEOREM. 2014 ~03BB = ~ if and
only and
areW-conj ugate.
Twosuch
weights
are called linked and we write ~, ~ ,u.Remark. 2014 When z = CL is the universal Casimir element one obtains from
equations (1)
and(2)
the well known formula3 . REDUCTION OF THE PRODUCT SPACE
V(Â) (8) V(Jl)
3.1.
Regular Weights.
The reduction of a finite dimensional
representation
V into irreduciblerepresentations frequently
reduces to a determination of the infinitesimal charactersoccurring
in V. If the character EH*,
occurs in V theremust exist a finite dimensional irreducible
representation, V(v)
say, which admits xu as an infinitesimal character.By
virtue of Theorem(2)
theweights
and v must, under suchcircumstances,
beconjugate
under W.Thus a character ~ can
only
occur if theweight
isW-conjugate
to adominant
integral weight.
This leads one toinvestigate
conditions under which agiven integral weight
is linked to a dominantintegral weight.
For we define the translated
hyperplanes
Annales de Henri Poincaré - Section A
207
A TRACE FORMULA FOR SEMI-SIMPLE LIE ALGEBRAS
The translated
hyperplanes
intersect in the translatedorigin - 03B4
andpartition
H* intofinitely
manyregions (the
translatedWeyl chambers).
The
correspondence
A -~(~,
+ 5,a)
for ~, e H* determines apolynomial
function on H*.
Consequently
the translatedhyperplanes
are closed inthe Zariski
topology
on H*(see [13 ],
p.133).
Since 0 is finite it follows that the set is also Zariski closed. Itscomplement
in H*, denoted R, is therefore open in the Zariskitopology.
We call elements of Rregular.
Since all open sets
(non empty)
are dense in the Zariskitopology
it followsthat the set of
regular
elements is Zariski dense in H*.Suppose
nowthat ,u
E A and let b be the half sum of thepositive
roots.One knows
(see
forexample Humphreys [13 ]),
since that,u + b is
W-conjugate
to aunique
dominantintegral weight,
v say. Thefollowing
conditions areequivalent.
The
equivalence
ofi)
andii)
is trivial while theequivalence
ofii)
andiii)
follows from the fact that the
Weyl
group acts on the rootsystem 03A6 by permuting
the roots among themselves. Theremaining eigenvalences
follow from the definition of dominant
integral weight
and the fact that everypositive
root can be written as a linear combination of elements from A with coefficientsbeing non-negative integers.
Since
+ 03B4 isW-conjugate
to v E A + it is clearthat
isconjugate
under W to v - 5.
Suppose
nowthat
satisfies parti)
ofequation (4).
One knows
that ( 5,
= 1 for a E ~and,
in view ofv)
ofequation (4),
v 2014 ~ must
satisfy ~2014
for a E.By
definition theweight
~ 2014 5 is dominant
integral. By
virtue of theequivalences (4)
we thereforehave the
following
easy result.LEMMA.
Let
E A.Then
isconjugate
under W to a dominantintegral weight
if andonly if
E R.Clearly
then the set ofintegral weights conjugate
under W to a dominantintegral weight
isjust
the set A ofregular integral weights.
One has A+ ci A = A n R. It is a well known property of the Zariskitopology (see [13 ],
p.134)
that theweight
lattice A is Zariski dense in H*. Hence A,being
a non-empty intersection of a Zariski dense subset A and a Zariski open set R, is Zariski dense in H*.3-1980.
208 M. D. GOULD
32
Klimyk’s
Formula.In this section we consider the reduction of the
product
spacewhere
V(Â)
andV(tl)
denote irreducible modules over U withhighest weights ).. and respectively.
... ,03BBk}
be the set of distinctweights
in
V(03BB)
and suppose theweight 03BBi
occurs withmultiplicity n(i).
It isknown
[8,
13,7~] ]
that the infinitesimal characters admitted are of the form Nowlet 03BDi
be theunique
dominantintegral weight W-conjugate
If
(~ + /~ + ~ x) == 0
forsome a
(i.
e.if
+03BBi
+ 5 is fixedby
some element inW)
then section(3.1) implies
that the infinitesimal character cannot occur. On the other handif ,u + 03BBi
E Athen vt
2014 5 E A+and
+A,
isW-conjugate
to 03BDi - b.In this case the
representation b)
occurs in the Clebsch-Gordandecomposition
ofV(~)
(x) withmultiplicity
which may be obtainedby comparison
withSteinberg’s
formula[15 ]. By
this means we obtainthe ansatz
originally
due toKlimyk [12 ].
The reduction of the spaceV()w)
0V(/l)
isprobably
bestrepresented
in terms of formal characters(see [13 ],
p.124).
Letch , ~+,
denote the formal character of and let denote the formal character ofV()..)
@V(~). Application
ofthe above considerations
yields
the formulawhere the
multiplicites
aregiven by
where sn
(6~) _
:t 1 is thesign
of theWeyl
group element fi,.Remark. Formula ( 5) may be written in the more fam i l i a r >
where
t(,u)
= 0 A and if ,u E A,t(,u)
= sn(o-)
where cr is theunique
element of W such that A+ . We note further that the
highest weights occurring
inV(r~)
0V(J1)
arenecessarily
of the form /1 +~,~ (see [13,
14]).
Hence the
representations occurring
inV(Â)
0V(/1)
are of the form +/.,).
for
+ withmultiplicity
~~~here 6~
is theunique
element of W such that +i,j
+ 5) = ,u +/~
+ (BAnnales de l’Institut Henri Poincaré - Section A
209
A TRACE FORMULA FOR SEMI-SIMPLE LIE ALGEBRAS
4. CASIMIR INVARIANTS
Following
the notation of section(3.2)
we shall henceforth let be a finite dimensional irreducible module over U withhighest weight
and ..., denote the set of distinct
weights
inV(À)
occurring
withmultiplicities n( 1),
...,n(k) respectively.
Now U may be embedded in the
algebra
U 0 Uby
thediagonal
homo-morphism
defined for x E L
by
dx == 1 (x) x + x (x) 1. Ingeneral
du forarbitrary
u E U is a more
complicated expression
which may be writtenwhere u~, Now let Y be the
algebra
Y =[End V(/t)] 0
U and let7T; be the
representation
affordedby V(~). Following
Kostant[8 ]
weconsider the map defined for xE L
by
which we extend to an
algebra homomorphism
to all of U. Moregenerally
if u E U with du as in
equation (6)
thenWhen z is an element of the centre Z Kostant shows that the operator
9(z)
satisfies a certainpolynomial identity
over the centre ofð(U).
We1
consider here the operator A
= - 2 [~(z) 2014 7T;(z)
0 1 - 1 (x)z ].
For ourpurposes no loss of
generality
is incurredby choosing
z to be the universal Casimir element cL. We henceforth let A denote the operatorNow let
be the map defined
by
where 03C1i
E EndV{),,},
ui E U andwhere tr [pj
denotes the trace of the endo-Vol.
210 M. G. GOULD
morphism
pi E EndV(~).
Inparticular
for u E U,a(u)
as inequation (7),
we have
In view of the
properties
of trace it is immediate that L is well defined and linear. In fact T is a U-modulehomomorphism
as one can see fromfor all y E Y, u E U. Also for u E U we have (x)
u)
=d(~)M,
whereJ(~)
= dimV(~),
which shows that L issurjective.
Moreover, from theproperties
of trace, one obtainsNow let C denote the centralizer of
3(U)
in Y. If w E C we have, for arbi- traryxeL,which may be
rearranged
togive
Applying T
to both sides of thisequation, using
the fact that l’ is a U-homo-morphism,
weobtain,
in view ofequation (10),
It follows then that
T(C) £
2. In view of thesurjectivity
andlinearity
of Twe in fact obtain
T(C)
= Z.For y E Y let us call
z(y)
the trace of y. Now the operator A definedby equation (8)
and all of its powersbelong
to C. Hence traces ofarbitrary polynomials
in Abelong
to the centre Z. Ofparticular importance
to usare the fundamental invariants
Remark. -
Let {x1,
...,(n
= dimL)
be a basis for L and let~,
..., xn}
be thecorresponding
dual basis with respect to theKilling
form of L. Now fix a basis for the reference
representation V(À)
and letE End
V(/~
x E L, denote the matrixrepresenting x
in this basis.With respect to this basis the entries of the matrix A are
given by
Annales de 1’Institut Henri Poincare - Section A
A TRACE FORMULA FOR SEMI-SIMPLE LIE ALGEBRAS 211
In
particular
A is a matrix with entries from L.Polynomials
in A may bedefined
recursively according
tofor
positive integral
m. The invariants definedby equation (11)
may then be writtenFrom this
expression
it can be seen that these invariants aregeneralizations
of the well known Gel’fand invariants of and
U(n).
Note that theactual basis chosen for
V(Å)
is immaterial as far as the invariantsare concerned.
One may define an
adjoint
A of Aby
where
( )T
denotestransposition.
In agiven
basis for the matrix elements of A and A are relatedby
As for the matrix A onemay define
polynomials
in the matrix A. One also obtains a set of invariants for the Liealgebra by taking
traces ofpolynomials
in the matrixA.
Inparticular
one may consider theadjoint
invariantsIt is clear that the
adjoint
invariant is theimage
of the invariant under theprincipal antiautomorphism
of U(see [14 ],
p.73).
5. EIGENVALUES OF THE
Throughout
this section we let denoteWeyl’s
dimension function which is thepolynomial
function on H* definedby
212 M. D. GOULD
The translated
Weyl
group acts on daccording
towhere we have used the fact that
(, )
is invariant under theWeyl
group.One
knows,
moreover, that 6 permutes the roots and if(7)
denotesthe number of factors in the reduced
expression
for cr((cr)
is called thelength
ofr)
then the number ofpositive
roots sent tonegative
roots is1«(1).
It follows then ad = sn
(6)d,
where sn(6) _ ( -
is thesign
of theWeyl
group element cr. We say that d is skew under W.It shall be convenient to consider the
polynomial
functionsd~,
~ = 1, ..., ~ obtained from dby
translation with respect to the distinctweights 03BB1,
...,03BBk of V(Ii);
The translated
Weyl
group acts on thefunctions di according
to= sn
(~)d(,u
+ Recall however that 7just
permutes the distinctweights 03BBi; 03C3(03BBi)
=03BB03C0(i)
where 03C0 is apermutation
of the numbers 1, ..., k.Thus
di
= snAccordingly
it follows that anysymmetric
combi-nation of the
polynomial functions di
will be skew under the translatedWeyl
group.Suppose
now that is a finite dimensional irreducible module over U withhighest weight
E A + and let be therepresentation
affordedby V(J1).
One may extend 7~ to an
algebra homomorphism
defined
by
where pz E End
V()B’) and i
E U. Inparticular
if A is the operator definedby equation (8)
thenis
clearly
an operator on theproduct
spaceV()
@V(J1.)
whoseentries are
endomorphisms
ofV(Jl).
In thefollowing
we shallidentify
Aand A basis may be chosen for
V()
(x)V(,u)
for which A isrepresented by
adiagonal
matrix. Theeigenvalues
of A areuniquely
determinedby
the characters
occurring
in the reduction ofV(À)
0V(J1.).
IfV(v), v E A + ,
is an irreducible module
occurring
in thedecomposition
ofV()
@V(Jl)
one knows, in view of section
(3.2),
thatV(v)
admits an infinitesimal cha-racter for some i = 1, ..., k. On
V(v)
the operator A therefore takesI’Institut Henri Poincaré - Section A
213
A TRACE FORMULA FOR SEMI-SIMPLE LIE ALGEBRAS
the constant value - 1 2[~ +i(cL)-~(cL)-~ (cL)
which, by
virtue ofequation (3), equals
2The
correspondence -~ - (i~, ~,
+2~) 2014 - (~ 2(~
+~)
+~i)
determinesa linear
polynomial
function on H*. We henceforth denote thispolynomial
function
by
ai ;Remark. 2014 From the above considerations it is an easy matter to deduce that in the
representation
affordedby V(J1)
the operator A satisfies thepolynomial identity
This result in fact follows from the
easily
established fact that adiagonal
matrix D with distinct
eigenvalues d1,
...,dk
satisfies thepolynomial identity
Applying
the same considerations to theadjoint
matrix A ones arrives atthe
adjoint identity
where
The translated
Weyl
group acts on thepolynomial functions at according
towhere we have used the fact that
(, )
is invariant under theWeyl
group.Moreover, since 6 E W
just
permutes the distinctweights
of we have(7(/.J
=03BB03C0(i)
where 03C0 is apermutation
of the numbers 1, ..., k. Hence we obtain214 M. D. GOULD
and it follows that any
symmetric
combination ofthe ai
determines aW-invariant
polynomial
function.Following
the notation of section(3.2) let 03BDi
be theunique
dominantintegral weight
which is Wconjugate
to + 03BBi + ð and let 03C3i be the cor-responding Weyl
groupelement ;
_ ,u +~,i
+ ~ We may then write the Clebsch-Gordan reduction of theproduct
spaceV(~,)
0V(J1) formally
aswhere
Now consider the linear map defined
by
where
03C1i ~ End V(Ii), ui ~ U
and where trpi,]
denotes the trace of theendomorphisms
pi andrespectively.
It was shown earlier thaton each of the spaces
03B4) occurring
in(14)
the operator A takes the constant value= - 2 l
where we haveused the fact that
5)
admits the infinitesimal character x~+ ~,~. Moregenerally
the operator Am takes the constant value on the space~).
It followsimmediately
thatOn the other hand
let t
denote the linear mapdefined
by
Also let
denote the map defined
by equation (9).
Then it iseasily
shown thattr~
is the
composite
maptr
= From this we obtainAnnales de l’Institut Henri Poincare - Section A
A TRACE FORMULA FOR SEMI-SIMPLE LIE ALGEBRAS 215
where are the Casimir invariants defined
by equation (11) and ~
is the infinitesimal character of
V(t). Comparing equations (15)
and(16)
one obtains
Note that this
expression
is well defined sinced(,u)
~ 0for
E A+ c A.Now 03BDi
- 5 isW-conjugate,
via i, to+ 03BBi
+ 03B4. Hence oneobtains,
in view of the fact that d is skew under W,
Thus we
obtain,
aftersubstituting
forweyl’s
dimensionformula,
theexplicit
resultHowever the
multiplicities m(i)
aregiven by m(i)
= sn or elsem(i)
= 0which
only
occurs when+ ~, a)
= 0 for some a E 1>+. We thereforefinally
obtain the character formulaUsing
the fact that W acts onthe 03BBi by permuting
them among them- selves and the fact thatW-conjugate weights
occur with the same multi-plicity
one may provedirectly
thatequation (17)
in fact determines aW-invariant
polynomial
function asrequired by
Harish-Chandra’s Theorem. One may make formula( 17) fully explicit using
the Kostantmultiplicity
formula[7d].
Note that we have
strictly only proved
this formula forweights
However this is sufficient to prove the formula for
arbitrary
E H*(see
for
example
Kostant[8 ]). Alternatively
one may extend formula(17)
to A
using
the fact that elements of A areW-conjugate
to dominantintegral weights.
Then,using
the fact that A is Zariski dense in H*, extend viacontinuity
to all of H*. To be moreexplicit
let gm be thepolynomial
functionon H* defined
by
3-1980.
216 M. D. GOULD
One may prove
directly
that grn is skew under W. In terms of thepolynomial
functions gm and d we may rewrite
equation ( 17)
asSuppose
nowthat
E A. Thenaccording
to lemma(3.1)
isW-conjugate
to a
unique
dominantintegral weight
v E A +. Write ,u = Since ,u and v areW-conjugate
we may writeBut
where the last
equality
follows from the fact that both gm and d are skew under W. This shows that formula( 17)
extends toarbitrary
Notethat
( 17)
is well defined forsuch
sinced(,u)
cannot vanishfor
E A.Now let
~
be theunique
W-invariantpolynomial
function determinedby
the Casimir under the Harish-Chandrahomomorphism ;
We have
proved
then thatTo avoid
singularities
we express thisby writing
Thus the
polynomial
functiondfm - gm
vanishes on a Zariski-dense subset of H*. Sincepolynomial
functions are continuous in the Zariskitopology
on H* we may write == In
particular
thepolynomial
function dnecessarily
divides g,~ and it makes formal sense to write = This proves the character formula(17)
forarbitrary
Note that formula
( 17)
isstrictly only
correct for allregular
elementsof H*. However for
weights lying
on a translatedhyperplane
care must betaken in order to avoid
singularities.
In such a case formula(17)
mustfirst be
expanded
into apolynomial.
6. In the notation of section
(4)
let A denote theadjoint
matrix of Aand let denote the
corresponding adjoint
invariants.Proceeding
as before one may determine the
eigenvalues
of the invariantsI,n(i~).
How-ever in this case it suffices to note that there exists a
unique
element T of Wwhich sends 03A6 into - 03A6
(see
e. g.[13] and [14 ]).
If z ~ Z one may deduce(by [14 ],
p.246)
forarbitrary
Annales de l’Institut Henri Poincaré - Section A
217
A TRACE FORMULA FOR SEMI-SIMPLE LIE ALGEBRAS
where z is the
image
of z under theprincipal anti-automorphism
of U.Hence In
particular
One may deduce from this the formula
where denotes the
polynomial
function definedby equation (13).
7. GENERALIZATIONS
In
previous
sections we worked, forsimplicity,
with the universal Casimir element cL of U. One mayequally
well work with any(non-trivial)
elementof Z. If z E Z is any central element one may show,
by
the sametechniques,
that the operator
satisfies a
polynomial identity
where
the ai
are thepolynomial
functions over H*given by
By taking
traces of powers of the matrixA(z)
we obtain a set of invariantsIm(À) == T[A(z)y".
Theeigenvalues
of these invariants aregiven
stillby
formula
( 17)
except now thepolynomial functions ai
aregiven by equation ( 18)
instead of( 12).
Applications.
The character formula ( 17) in many ways reflects the nature of the tensor
product
spaceV(},,)
(x)V(Il).
In fact one may obtainKlimyk’s multiplicity
formula
by examining equation
( 17). It ishoped
therefore that this formulamay be of use in
obtaining
useful informationconcerning
the moregeneral
tensor
product
spaceV()B.)
0 V where V is an infinite dimensional moduleadmitting
an infinitesimal character Kostant[8 ]
has shown218 M. D. GOULD
that when V is a Harish-Chandra module that
V(À)
0 V admits a compo- sition series whose factors admit infinitesimal characters xv+ ~~. However it is clear that not all infinitesimal characters xv + ~,i need be admitted. It would therefore be of interest to determineprecisely
which characters do occur(the
infinite dimensionalanalogue
of the Clebsch-Gordanseries).
It is
suggestive
that such considerations areclosely
related to the structureof maximal ideals in the universal
enveloping algebra [14 ].
From the
point
of view ofapplications
tophysics
one sees that thepoly-
nomial identities satisfied
by
the matrix A(see
remarksfollowing
equa- tion(12))
are useful for the construction ofprojection
operators :The matrix elements of such
projection
operators, inunitary representations
of the group, are bilinear combinations of Clebsch-Gordan coefficients.
This opens up the
possibility
of acomplete
determination of the multi-plicity
freeWigner
coefficients of asemi-simple
Lie groupby
a close exami- nation of theprojection
operatorsP [i ].
Forapplications along
theselines see
[77] ] (and
referencesquoted therein).
ACKNOWLEDGMENTS
The author
gratefully acknowledges
the assistance extendedby
Prof. H. S. Green and the financial support of a Commonwealth Post-
graduate
Research Award. The author also thanks Dr. A. L.Carey
forseveral useful discussions
concerning
thepresentation
of thismanuscript.
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Annales de l’Institut Henri Poincare - Section A
219 A TRACE FORMULA FOR SEMI-SIMPLE LIE ALGEBRAS
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Phys.
( M anuscrit reçu Ie 5 novembre 1979)
Vol. XXXII,