A construction by deformation of unitary irreducible representations of $SU(n+1)$

0.16 MB·

by  Benjamin Cahen

A CONSTRUCTION BY DEFORMATION OF UNITARY
IRREDUCIBLE REPRESENTATIONS OF SU(n+1)
BENJAMIN CAHEN
7
1
0
2 To the memory of my father, Alfred Cahen (1932-2016)

n
a
Abstract. We derive unitary irreducible representationsof SU(n+1) from a minimal
J
realization of sl(n+1,C) by using various techniques from deformation theory.
7
1

]
A
Q 1. Introduction
.
h The deformations of Lie algebras were intensively studied in the years 1960-70[10], [21],
t
a [22], [18] and still remain objects of active research, see for instance [7], [8] and [6]. On
m
the other hand, the deformations of Lie algebra representations have not been studied as
[
systematically, with some notable exeptions, see [23], [13], [19] and also [20].

1 Constructing (formal) deformations of Lie algebra representations is a way to derive a
v
family of representations from a given one and then to get many representations from a
6
5 few ones. However, the existence and classification problems for deformations depend on
7
some Lie algebra cohomology modules which are not easy to compute in general, see for
4
0 instance [19] and [3].
. The aim of the present note is to use deformation theory in order to recover some
1
0 unitary irreducible representations of SU(n+1). More precisely, we construct the unitary
7
irreducible representations of SU(n + 1) considered in [4] by deformation of a so-called
1
: minimal realization of sl(n + 1,C) [15]. It is known that such a minimal realization is
v
i connected to the minimal (non trivial) nilpotent coadjoint orbit of SL(n+1,C) [16], [1],
X
[17]. By taking a parametrization of this orbit, we can simplify the computation of the
r
a deformations of the minimal realization by using the Moyal star product and the Weyl
correspondence as in [1] and [3].
Although the derivation presented here can be considered as a simple exercice in defor-
mation theory, we have not found it explicitly done in details in the literature. Moreover,
we could hope for applications of this method to the description of representations of Lie
algebras (in particular of unitary dual of Lie groups) in more general situations (some
examples can already be found in [20] and [3]).
Thisnoteisorganizedasfollows. InSection2, wedescribe theunitaryirreducible repre-
sentations of SU(n+1) introduced in [4] as an analogue to the holomorphic discrete series
representations of SU(1,n) and we compute their differentials which can be extended to
representations of sl(n + 1,C). In the notation of [11], p. 143, these representations
2000 Mathematics Subject Classification. 17B10;17B20; 17B56;22E46; 53D55.
Key words and phrases. Deformation of representation; Lie algebra; unitary group; Chevalley-
Eilenbergcohomology;Moyalstarproduct;Weylcorrespondence;minimalrealization;minimalcoadjoint
orbit.
1
2 BENJAMIN CAHEN
have highest weight mǫ with m integer ≥ 1. Section 3 is devoted to some generalities
1
on (formal) deformations of Lie algebra homomorphisms and, in Section 4, we recall the
Moyal star product and the Weyl correspondence [9], [24]. In Section 5, we show how a
symplectic chart of the minimal nilpotent coadjoint orbit of sl(n+1,C) naturally leads
to a minimal realization of sl(n+1,C) and we expose the method which will be used to
recover the representations of SU(n+1). In Section 6, we compute the first cohomology
module corresponding to the deformation of the minimal realization and then we derive
the desired representations of SU(n+1) in Section 7. In particular, by this way we can
recover all the irreducible unitary representations of SU(2).
2. Representations of SU(n+1)
Here we consider a family of representations of SU(n+1) indexed by an integer m ≥ 1.
In [4], we showed that this family can be contracted to the unitary irreducible represen-
tations of the Heisenberg group of dimension 2n + 1 as the holomorphic discrete series
representations of SU(1,n).
ThegroupSU(n+1) consists ofallcomplex (n+1)×(n+1) matricesg withdeterminant
1 such that g∗g = I . Here we write the elements of the group SU(n + 1) as block
n+1
matrices
a b
g =
(cid:18)c d(cid:19)
with matrices a(1×1), b(1×n), c(n×1) and d(n×n).
The Lie algebra su(n+1) of SU(n+1) consists of all matrices of the form
iα b
(cid:18)−b∗ A(cid:19)
where α ∈ R, b ∈ Cn and A is an anti-Hermitian n×n matrix (that is, A∗ = −A) such
that iα+Tr(A) = 0.
The group SU(n+1) acts naturally on the projective space P (C) and this action in-
n
duces anholomorphic action(defined almost everywhere) of SU(n+1)onCn by fractional
linear transformations
a b
g ·z = (a+bzt)−1(c+dzt)t, g =
(cid:18)c d(cid:19)
where the subscript t denotes transposition.
For each integer m ≥ 1, let P be the space of all complex polynomial functions on Cn
m
of degree ≤ m. We endow P with the Hilbert product
m
hf ,f i := f (z)f (z)dµ (z)
1 2 m 1 2 m
Z
Cn
where the measure µ on Cn is defined by
m
(m+1)...(m+n)
dµ (z) := (1+kzk2)−m−n−1dx dy ...dx dy .
m πn 1 1 n n
Here we use the notation z = (x + iy ,x + iy ,...,x + iy ) where x ,y ∈ R for
1 1 2 2 n n k k
k = 1,2,...,n.
Now, let π be the representation of SU(n+1) on P defined by
m m
a b
(π (g)f)(z) = (bzt +a)mf(g−1·z), g−1 = .
m (cid:18)c d(cid:19)
A CONSTRUCTION BY DEFORMATION... 3
We can easily verify that π is unitary. Moreover, the differential dπ of π can be
m m m
extended to a representation of sl(n+1,C) = su(n+1)c also denoted by dπ . We have
m
(dπ (X)f)(z) = −m(βzt +α)f(z)+df(z)((α+βzt)z −(γ +δzt)t)
m
where
α β
X =
(cid:18)γ δ(cid:19)
with matrices α(1×1), β(1×n), γ(n×1) and δ(n×n).
In order to give more explicit formulas for dπ , let us introduce the following basis of
m
sl(n + 1,C). For 1 ≤ i, j ≤ n + 1, write E for the matrix whose ij-th entry is 1 and
ij
all of the other entries are 0. Then the matrices H = E −E (1 ≤ k ≤ n) form
k k+1k+1 11
a basis for the Cartan subalgebra h of sl(n+1,C) consisting of all diagonal matrices of
sl(n+1,C) and, obviously, the matrices H (1 ≤ i ≤ n) and E (1 ≤ i 6= j ≤ n+1)
k ij
form a basis for sl(n+1,C). Then we have
n
∂f ∂f
(dπ (H )f)(z) =mf(z)−z − z
m k k j
∂z ∂z
k Xj=1 j
n
∂f
(dπ (E )f)(z) =−mz f(z)+z z
m 1k+1 k k j
∂z
Xj=1 j
∂f
(dπ (E )f)(z) =−
m k+11
∂z
k
∂f
(dπ (E )f)(z) =−z
m i+1j+1 j
∂z
i
for 1 ≤ k ≤ n and 1 ≤ i 6= j ≤ n.
Wecaneasilyseethatdπ -henceπ -isirreducible. Indeed, letV beanonzero subspace
m m
of P which is invariant under dπ (X) for each X ∈ sl(n+1,C). Then there exists at
m m
least one nonzero element f in V. Thus, by applying the operators dπ (E ) to f, we
m k+11
get 1 ∈ V and by applying the operators dπ (E ) and dπ (E ) to 1 we see that
m 1k+1 m i+1j+1
V = P .
m
Let us denote by ǫ , 1 ≤ k ≤ n, the linear form on h defined by
k
ǫ : Diag(a ,a ,...,a ) → a .
k 1 2 n+1 k
It is well-known that the root system of sl(n+1,C) relative to h is
∆ = {ǫ −ǫ : 1 ≤ i,j ≤ n+1},
i j
see for instance [11]. The ordering on ∆ is usually taken so that the positive roots are
ǫ −ǫ (1 ≤ i < j ≤ n+ 1). In this context, we can verify that dπ has highest weight
i j m
mǫ and highest weight vector f = zm.
1 n
3. Generalities on deformations
In this section, we recall some definitions and results of deformation theory. The ma-
terial of this section is essentially taken from [23], [13], [19], see also [12] and [3].
4 BENJAMIN CAHEN
Let g be a Lie algebra over C and let A be an associative algebra over C with unit
element 1. Then A is also a Lie algebra for the commutator [a,b] := ab − ba. Let
ϕ : g → A be a Lie algebra homomorphism.
Definition 3.1. (1) A formal deformation of ϕ is a formal series Φ = tkΦ
k≥0 k
where Φ0 = ϕ and Φk is a linear map from g to A for each k ≥ 1, suchPthat
(3.1) Φ([X,Y]) = [Φ(X),Φ(Y)]
for each X and Y in g. Here we have extended the bracket of A to formal series
by bilinearity.
(2) Two formal deformations Φ and Ψ of ϕ are said to be equivalent if there exists a
series a = 1+ta +t2a +... ∈ A[[t]] such that for each X ∈ g, we have
1 2
(3.2) a−1Φ(X)a = Ψ(X).
The study of the formal deformations of ϕ naturally leads us to consider the structure
of g-module on A defined by X ·a = [ϕ(X),a] for X ∈ g and a ∈ A and the Chevalley-
Eilenberg cohomology of g with values in the g-module A. Indeed, denoting by ∂ the
corresponding cobord operator, we immediately see that Eq. 3.1 is equivalent to the fact
that for each n ≥ 0 and each X,Y ∈ g, we have
(∂Φ )[X,Y] :=[ϕ(X),Φ (Y)]+[Φ (X),ϕ(Y)]−Φ ([X,Y])
n n n n
n−1
=− [Φ (X),Φ (Y)].
k n−k
Xk=1
In particular, we see that if such a deformation Φ exists then Φ is a 1-cocycle.
1
We have the following result, see for instance [13], Section III and [19], Section I.
Proposition 3.2. (1) If we have H2(g,A) = (0) then, for each 1-cocycle α : g → A,
there exists a formal deformation Φ such that Φ = α.
1
(2) If we have H1(g,A) = (0) then each formal deformation Φ of ϕ is equivalent to ϕ.
In [3], we proved the following result.
Proposition 3.3. Assume that H1(g,A) is one-dimensionaland that there exists a formal
deformation Φ of ϕ such the class of Φ generates H1(g,A). For each sequence c =
1
(c ) of complex numbers, consider the formal series S (t) := c tk and the formal
k k≥1 c k≥1 k
deformation Φc of ϕ defined by Φc(X) = S (t)rΦ (X) for Peach X ∈ g.
r≥0 c r
Then the map c → Φc is a bijection froPm the set of all sequences c = (ck)k≥1 of C onto
the set of all equivalence classes of formal deformations of ϕ.
Note that the preceding definitions and results can be applied to the particular case
of a representation ϕ of g in a complex vector space V, since ϕ is also a Lie algebra
homomorphism from g to End(V), or, more generally, to a subalgebra A of End(V).
4. Weyl correspondence and Moyal star product
Here we first recall the Moyal star product, see for instance [2]. Take coordinates (p,q)
on R2n ∼= Rn ×Rn and let x = (p,q). Then one has x = p for 1 ≤ i ≤ n and x = q
i i i i−n
A CONSTRUCTION BY DEFORMATION... 5
for n+1 ≤ i ≤ 2n. For u,v ∈ C∞(R2n), define P0(u,v) := uv,
n
∂u ∂v ∂u ∂v
P1(u,v) := − = Λij∂ u∂ v
(cid:18)∂p ∂q ∂q ∂p (cid:19) xi xj
Xk=1 k k k k 1≤Xi,j≤n
(the Poisson brackets) and, more generally, for l ≥ 2,
Pl(u,v) := Λi1j1Λi2j2···Λiljl∂l u∂l v.
xi1...xil xj1...xjl
1≤i1,...,Xil,j1,...,jl≤n
Then the Moyal product ∗ is the following formal deformation of the pointwise mul-
M
tiplication of C∞(R2n)
tl
u∗ v := Pl(u,v)
M
l!
Xl≥0
where t is a formal parameter. Moreover, the corresponding Moyal brackets are given by
1 t2l
[u,v] := (u∗ v−v ∗ u) = P2l+1(u,v).
∗M 2t M M (2l+1)!
Xl≥0
Now, we restrict ∗ to polynomials on R2n and take t = −i/2. Then we get an
M
associative product ∗ on polynomials which we denote by ∗. This product corresponds to
the composition of operators in the usual Weyl quantization procedure as we will explain
below.
The Weyl correspondence on R2n is defined as follows, see [5], [9], [14]. For each f
in the Schwartz space S(R2n), we define the operator W(f) acting on the Hilbert space
L2(Rn) by
W(f)ϕ(p) = (2π)−n eisqf(p+(1/2)s,q)ϕ(p+s)dsdq.
ZR2n
As it is well-known, that the Weyl calculus can be extended to much larger classes of
symbols (see for instance [14]). In particular, if f(p,q) = u(p)qα where u ∈ C∞(Rn) then
we have
α
∂
(4.1) W(f)ϕ(p) = i (u(p+(1/2)s)ϕ(p+s)) ,
(cid:18) ∂s(cid:19) (cid:12)s=0
(cid:12)
(cid:12)
see [24]. For instance, if f(p,q) = u(p) then W(f)ϕ(p) = u(p)ϕ(p) and if f(p,q) = u(p)q
k
then
(4.2) W(f)ϕ(p) = i (1/2)∂ u(p)ϕ(p)+u(p)∂ ϕ(p) .
k k
(cid:0) (cid:1)
Moreover, we have W(f ∗ f ) = W(f )W(f ) for each functions f ,f on R2n of the
1 2 1 2 1 2
form u(p)qα, in particular for polynomials, see [9], p. 103.
Note also that, since the map W and the product ∗ on polynomials can be defined in a
purely algebraic way, see Eq. 4.1, we can extended them to the polynomials in complex
variables p,q without any modification.
6 BENJAMIN CAHEN
5. Minimal realization
In [1], a general method for constructing minimal realizations of semisimple complex
Lie algebras from minimal coadjoint orbits was introduced. In the particular case of the
Lie algebra g := sl(n+1,C) of G := SL(n+1,C), this method goes as follows.
First, we can identify the dual g∗ of g with g by means of the bilinear form on g defined
by hX,Yi := Tr(XY). In this identification, the coadjoint action of G corresponds to the
adjoint action of G and the coadjoint orbits to the adjoint orbits.
This is a simple exercice to show that the minimal (non trivial) nilpotent (co)adjoint
orbit O of G consists of all rank one matrices of g.
Now, let us consider the map Ψ from C2n to O′ := O∪(0) defined by
n
− p q q ... q
j=1 j j 1 n
n
Ψ(p,q) := −p1P j=1pjqj p1q1 ... p1qn.
. . .P. . . . . . . . . . . . . . .
 
n
−p p q p q ... p q 
 n j=1 j j n 1 n n
P
Then the image of Ψ is a dense open subset of O′.
For each X ∈ g, let us denote by X˜ the corresponding coordinate function on C2n:
˜
X(p,q) := hΨ(p,q),Xi.
Proposition 5.1. (1) For each X,Y ∈ g, we have
[X˜,Y˜] = {X˜,Y˜} = [X˜,Y].
∗
(2) The map ρ : X → W(iX˜) is a representation of g in C[q] := C[q ,q ,...,q ].
0 1 2 n
Proof. (1) Let X and Y in g. The equation {X˜,Y˜} = [X˜,Y] can be verified by a direct
˜ ˜
computation. On the other hand, since X and Y are polynomials of degree ≤ 1 in the
variables q ,q ,...,q , we have Pk(X˜,Y˜) = 0 for each k ≥ 3, hence we get [X˜,Y˜] =
1 2 n ∗
{X˜,Y˜}.
(2) Let X and Y in g. By (1), we have
(iX˜)∗(iY˜)−(iY˜)∗(iX˜) = i[X˜,Y].
Then, by the remark at the end of Section 4, we get [W(iX˜),W(iY˜)] = W(i[X˜,Y]) hence
(cid:3)
the result.
The representation ρ is a minimal realization of g, that is, a realization of g as Lie
0
algebra of differential operators acting on functions of n variables with n minimal, see
[15].
Note that
Span{E ,...,E ,E ,...,E }
n+12 n+1n 21 n+11
is a Heisenberg Lie algebra of dimension 2n−1 with central element E , the only non
n+11
trivial brackets being [E ,E ] = E for k = 2,...,n. Then Ψ was chosen so that
n+1k k1 n+11
˜ ˜ ˜
E = p q ,E = q (for k = 2,...,n) and E = q . In fact, these conditions
n+1k k−1 n k1 k−1 n+11 n
determine Ψ uniquely.
Now, we aim to study the deformations of ρ . By using the map f → W(if) this is
0
equivalent to studying thedeformations of theLiealgebra homomorphism X → Φ (X) :=
0
X˜ from g to M := C[p,q] endowed with [·,·] .
∗
A CONSTRUCTION BY DEFORMATION... 7
As explained in Section 3, we endow M with the g-module structure defined by X·f :=
˜
[X,f] and then consider the corresponding Chevalley-Eilenberg cohomology.
∗
6. Determination of H1(g,M)
Recall that H1(g,M) is the quotient space Z1(g,M)/B1(g,M) where Z1(g,M) consists
of all linear maps ϕ : g → M satisfying
(6.1) ∂ϕ(X,Y) := [X˜,ϕ(Y)] +[ϕ(X),Y˜] −ϕ[X,Y] = 0
∗ ∗
(the 1-cocycles) and B1(g,M) consists of all maps from g to M of the form X → [X˜,f]
∗
for f ∈ M (the 1-coboundaries).
The aim of this section is to compute H1(g,M). We begin with the following ’Poincar´e
lemma’.
Lemma 6.1. Let F (q), i = 1,2,...,n be a family of polynomials in the variable q =
i
(q ,q ,...,q ) such that, for each i,j = 1,2,...,n, one has ∂Fi = ∂Fj. Then there exists
1 2 n ∂qj ∂qi
a polynomial F(q) such that ∂F = F for each i = 1,2,...,n.
∂qi i
Proof. By the usual Poincar´e lemma, the result is true for polynomials in real variables
q which implies that it is also true for polynomials in complex variables q . (cid:3)
i i
Proposition 6.2. The space H1(g,M) is one dimensional, generated by the class of
the cocycle ϕ defined by ϕ (E − E ) = 1, ϕ (E − E ) = 0 for k = 2,...,n,
1 1 11 22 1 kk k+1k+1
ϕ (E ) = p for k = 1,2,...,n and ϕ (E ) = 0 for i ≥ 2.
1 1k+1 k 1 ij
Proof. We have divided the proof into several steps. The method of the proof is quite
elementary and consists in transforming progressively a given 1-cocycle to an equivalent
one which is more simple by adding suitable 1-coboundaries.
Let us consider a 1-cocycle ϕ : g → M.
1) First we apply Eq. 6.1 to X = E and Y = E for k,l = 1,2,...,n. Writing
k+11 l+11
ϕ = ϕ(E ) for simplicity, we get ∂ϕk = ∂ϕl for each k,l = 1,2,...,n. Then, by
k k+11 ∂pl ∂pk
decomposing each ϕ as ϕ = ϕα(p)qα with the usual multi-index notation, we have
k k α k
∂ϕα ∂ϕα
k = l for each k,l,α. P
∂pl ∂pk
Thus, by Lemma 6.1, for each α there exists a polynomial ϕα(p) such that ∂ϕα = ϕα
∂pk k
for each k = 1,2,...,n.
Now, let φ := ϕα(p)qα. For each k = 1,2,...,n, we have
α
P ∂φ
[φ,q ] = = ϕ .
k ∗ k
∂p
k
Hence, replacing ϕ by the equivalent 1-cocycle ϕ−[φ,·] , we can always assume that
∗
ϕ(E ) = 0 for each k = 1,2,...,n.
k+11
2) We apply Eq. 6.1 to X = E , k,l ≥ 2, k 6= l and Y = E , j = 1,2,...,n. Taking
kl j+11
1) into account, we can immediately see that ϕ(E ) is a polynomial in the variables
kl
q ,q ,...,q .
1 2 n
3) Similarly, applying Eq. 6.1 to X ∈ h and Y = E , we verify that ϕ(X) is a
j+11
polynomial in the variables q ,q ,...,q .
1 2 n
4) Now, we fix k = 1,2,...,n − 1 and we apply Eq. 6.1 to X = E and Y =
n+1k+1
n−1(E − E ). Write ϕ := ϕ(E ) for simplicity and recall that ϕ is
j=1 n+1n+1 j+1j+1 k n+1k+1 k
P
8 BENJAMIN CAHEN
a polynomial in q ,q ,...,q by 2). Then we see that there exists a polynomial u (q) in
1 2 n k
q ,q ,...,q such that
1 2 n
n
∂ϕ
k
(6.2) −nϕ = q +q u (q).
k j n k
∂q
Xj=1 j
Let ϕ = ϕm and u = um be the decompositions of ϕ and u into homogeneous
k m k k m k k k
polynomiaPls of degree m inPq1,q2,...,qn . Then Eq. 6.2 implies that
−n ϕm = mϕm +q u (q)
k k n k−1
Xm Xm
and we conclude that, for each k = 1,2,...,n − 1, there exists a polynomial ψ in
k
q ,q ,...,q such that ϕ = q ψ .
1 2 n k n k
Taking X = E and Y = E in Eq. 6.1 for k,l = 1,2,...,n − 1, we get
n+1k+1 n+1l+1
∂ψk = ∂ψl for each k,l. This implies the existence of a polynomial ψ in q ,q ,...,q such
∂ql ∂qk 1 2 n
that ψ = ∂ψ for each k = 1,2,...,n−1.
k ∂qk
Thus, by replacing ϕ by ϕ − [·,ψ] , we are led to the case where ϕ(E ) = 0 for
∗ n+1k+1
each k = 1,2,...,n − 1 and the condition ϕ(E ) = 0 for each k = 1,2,...,n is still
k+11
satisfied.
5)Letk = 1,2,...,n,l = 1,2,...,n−1withk 6= l. ByapplyingEq. 6.1toX = E
k+1l+1
and Y = E for j = 1,2,...,n−1, we see that ϕ(E ) only depends on q . Thus,
n+1j+1 k+1l+1 n
taking into account the equality
[E ,E ] = E −E
k+1l+1 l+1k+1 k+1k+1 l+1l+1
we get ϕ(E −E ) = 0. Hence, applying Eq. 6.1 to X = E −E and
k+1k+1 l+1l+1 k+1k+1 l+1l+1
Y = E we obtain ϕ(E ) = 0.
k+1l+1 k+1l+1
Finally, we apply Eq. 6.1 to X = E and Y = E and we also obtain
j+1n+1 k+1j+1
ϕ(E ) = 0.
k+1n+1
6) Now, take X ∈ h and Y = E in Eq. 6.1. Then we see that ϕ(X) only depends
k+1j+1
on q .
n
Let H = E − E ∈ h. Then we can replace ϕ by ϕ + [·,F(q )] for a suitable
0 11 22 n ∗
polynomial F(q ) so that ϕ(H ) is a constant which we denote by a.
n 0
7) Taking X = E and successively Y = E , (k = 1,2,...,n) and Y = E ,
12 k+11 n+1k
(k = 2,...,n−1) in Eq. 6.1, we see that
ϕ(E ) = ap +f(q )
12 1 n
where f(q ) is a polynomial. Moreover, taking also X = E and Y = H , we get
n 12 0
−2f(q ) = q ∂f hence f = 0 and ϕ(E ) = ap .
n n∂qn 12 1
8) Finally, we apply Eq. 6.1 to X = E and Y = E where k = 2,...,n we obtain
12 2k+1
ϕ(E ) = ap .
1k+1 k
(cid:3)
7. Derivation of the representations π
m
In this section, we retain the notation of the previous sections. Proposition 6.2 leads
us to consider the formal deformations Φ of Φ : X → X˜ such that Φ = aϕ for a ∈ C.
0 1 1
We have the following result.
A CONSTRUCTION BY DEFORMATION... 9
Proposition 7.1. For each a ∈ C, the map Φ : g → M[[t]] defined by Φ (X) = X˜ +
a a
taϕ (X) is a formal deformation of Φ in M.
1 0
Proof. Taking into account that ϕ is a 1-cocycle (see Section 6), the result follows im-
1
mediately from the equality [ϕ (X),ϕ (Y)] = 0 for X,Y ∈ g. (cid:3)
1 1 ∗
By using the properties of W (see Section 4), we get the following proposition.
Proposition 7.2. For each a ∈ C, let m(a) := −1/2(a+n+1). Then the map ρ defined
a
by
i
ρ (X) = W X˜ − aϕ (X)
a 1
(cid:18) 2 (cid:19)
for X ∈ g is a representation of g in C[p] and we have
n
∂f ∂f
(ρ (H )f)(z) =m(a)f(z)−z − z
a k k j
∂z ∂z
k Xj=1 j
n
∂f
(ρ (E )f)(z) =−m(a)z f(z)+z z
m 1k+1 k k j
∂z
Xj=1 j
∂f
(ρ (E )f)(z) =−
a k+11
∂z
k
∂f
(ρ (E )f)(z) =−z
a i+1j+1 j
∂z
i
for 1 ≤ k ≤ n and 1 ≤ i 6= j ≤ n.
Proof. The fact that ρ is a representation g follows fromProposition 7.2 andthe formulas
a
for ρ can be easily verified by Eq. 4.2. (cid:3)
a
In other words, the formulas for ρ are the same as the formulas for dπ , see Section
a m
2, but note that these two representations don’t act on the same spaces since ρ acts on
a
C[p] and dπ on the finite dimensional space P .
m m
In order to recover the representations π of Section 2, we select now the values of a
m
(or, equivalently, of m(a)) for which there exists a non trivial finite dimensional subspace
of C[p] that is invariant under ρ .
a
Proposition 7.3. Let a ∈ C. Assume that P is a non trivial finite dimensional subspace
of C[p] that is invariantunder ρ . Then m(a) is a non negative integer, we have P = P
a m(a)
and the restriction of ρ to P coincides with dπ .
a m(a)
Proof. Let a ∈ C. Let P 6= (0) a finite dimensional subspace of C[p] which is invariant
under ρ . Define m := max{deg(f) : f ∈ P\(0)}. Let f be an element of P of degree m.
a
m
Let us decompose f as f = f where, for each k, f is an homogeneous polynomial
k=0 k k
of degree k. Then we have fPm 6= 0 and
m
ρ (E )f = p (k −m(a))f .
a 1l+1 l k
Xk=0
10 BENJAMIN CAHEN
We see that if m 6= m(a), we get a contradiction. Thus we have m(a) = m hence m(a)
is a non negative integer and P ⊂ P . Since P is irreducible under the action of
m(a) m(a)
dπ , see Section 2, we can conclude that P = P . (cid:3)
m(a) m(a)
Then we have recovered the representations dπ of Section 2, hence the representations
m
π by integration. Notethat by taking n = 1 we see that this method gives all the unitary
m
irreducible representations of SU(2).
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