Representations
2.1 Basic Notions
2.1.2 Examples
LetGbe a Lie group. A representation ofGon a finite-dimensional vector spaceV smoothly assigns to eachg ∈Gan invertible linear transformation ofV satisfying
π(g)π(g)=π(gg)
for allg,g∈G. Although surprisingly important at times, the most boring example of a representation is furnished by the mapπ : G → G L(1,C)=C\{0}given by π(g) =1. This one-dimensional representation is called thetrivial representation.
More generally, the action ofGon a vector space is calledtrivialif eachg∈Gacts as the identity operator.
2.1.2.1 Standard Representations Let G beG L(n,F), S L(n,F),U(n), SU(n), O(n), orS O(n). Thestandard representationofGis the representation onCnwhere π(g)is given by matrix multiplication on the left by the matrixg∈G. It is clear that this defines a representation.
2.1.2.2 SU(2) This example illustrates a general strategy for constructing new rep- resentations. Namely, if a groupGacts on a spaceM, thenGcan be made to act on the space of functions onM (or various generalizations of functions).
Begin with the standard two-dimensional representation ofSU(2)onC2where gηis simply left multiplication of matrices forg∈SU(2)andη∈C2. Let
Vn(C2)
be the vector space of holomorphic polynomials on C2 that are homogeneous of degreen. A basis forVn(C2)is given by{zk1zn2−k | 0 ≤ k ≤ n}, so dimVn(C2) = n+1.
Define an action ofSU(2)onVn(C2)by setting (g·P)(η)= P(g−1η)
forg ∈ SU(2),P ∈ Vn(C2), andη∈C2. To verify that this is indeed a representa- tion, calculate that
[g1·(g2·P)](η)=(g2·P)(g−11η)=P(g−21g1−1η)=P((g1g2)−1η)
=[(g1g2)·P](η)
so thatg1·(g2·P)=(g1g2)·P. Since smoothness and invertibility are clear, this action yields ann+1-dimensional representation ofSU(2)onVn(C2).
2.1 Basic Notions 29 Although these representations are fairly simple, they turn out to play an ex- tremely important role as a building blocks in representation theory. With this in mind, we write them out in all their glory. If g =
a −b b a
∈ SU(2), then g−1 =
a b
−b a
, so that g−1η = (aη1 +bη2,−bη1+aη2)whereη = (η1, η2). In particular, ifP=zk1zn2−k, then(g·P)(η)=(aη1+bη2)k(−bη1+aη2)n−k, so that
a −b b a
·(zk1zn2−k)=(az1+bz2)k(−bz1+az2)n−k. (2.3)
Let us now consider another family of representations ofSU(2). Define Vn
to be the vector space of holomorphic functions in one variable of degree less than or equal ton. As such,Vnhas a basis consisting of{zk |0 ≤ k≤ n}, so Vnis also n+1-dimensional. In this case, define an action ofSU(2)onVnby
(g·Q) (u)=(−bu+a)nQ
au+b
−bu+a (2.4)
forg=
a −b b a
∈SU(2),Q∈Vn, andu ∈C. It is easy to see that (Exercise 2.1) this yields a representation ofSU(2).
In fact, this apparently new representation is old news since it turns out that Vn ∼= Vn(C2). To see this, we need to construct a bijective intertwining operator from Vn(C2)toVn. Let T : Vn(C2) → Vn be given by(T P)(u) = P(u,1)for P∈ Vn(C2)andu ∈C. This map is clearly bijective. To see thatT is aG-map, use the definitions to calculate that
[T(g·P)](u)=(g·P) (u,1)=P(au+b,−bu+a)
=(−bu+a)nP
au+b
−bu+a,1
=(−bu+a)n(T P) (u)=[g·(T P)](u), soT(g·P)=g·(T P)as desired.
2.1.2.3 O(n)and Harmonic Polynomials Let Vm(Rn)
be the vector space of complex-valued polynomials on Rn that are homogeneous of degree m. Since Vm(Rn)has a basis consisting of {x1k1x2k2· · ·xnkn | ki ∈ Nand k1+k2+ · · · +kn =m}, dimVm(Rn)=m+n−1
m
(Exercise 2.4). Define an action of O(n)onVm(Rn)by
(g·P)(x)= P(g−1x)
forg∈ O(n),P ∈Vm(Rn), andx∈Rn. As in §2.1.2.2, this defines a representation.
As fine and natural as this representation is, it actually contains a smaller, even nicer, representation.
Write=∂x21+ · · · +∂x2nfor theLaplacianonRn. It is a well-known corollary of the chain rule and the definition of O(n)thatcommutes with this action, i.e., (g·P)=g·(P)(Exercise 2.5).
Definition 2.5.LetHm(Rn)be the subspace of allharmonic polynomialsof degree m, i.e.,Hm(Rn)= {P∈Vm(Rn)|P =0}.
IfP ∈Hm(Rn)andg ∈ O(n), then(g· P)=g·(P)=0 so thatg·P ∈ Hm(Rn). In particular, the action ofO(n)onVm(Rn)descends to a representation of O(n)(orS O(n), of course) onHm(Rn). It will turn out that these representations do not break into any smaller pieces.
2.1.2.4 Spin and Half-Spin Representations Any representation(π,V)ofS O(n) automatically yields a representation of Spinn(R)by looking at(π◦A,V)whereA is the covering map from Spinn(R)toS O(n). The set of representations of Spinn(R) constructed this way is exactly the set of representations in which−1 ∈ Spinn(R) acts as the identity operator. In this section we construct an important representation, called the spin representation, of Spinn(R)that is genuine, i.e., one that does not originate from a representation ofS O(n)in this manner.
Let(·,·)be the symmetric bilinear form onCn given by the dot product. Write n = 2mwhenn is even and writen = 2m+1 whenn is odd. Recall a subspace W ⊆Cn is calledisotropicif(·,·)vanishes onW. It is well known thatCn can be written as a direct sum
Cn =
W ⊕W neven W ⊕W⊕Ce0 nodd (2.6)
for W,W maximal isotropic subspaces (of dimensionm) and e0 a vector that is perpendicular toW⊕Wand satisfies(e0,e0)=1 . Thus, whennis even, takeW = {(z1, . . . ,zm,i z1, . . . ,i zm)| zk ∈ C}andW = {(z1, . . . ,zm,−i z1, . . . ,−i zm)| zk ∈ C}. Forn odd, take W = {(z1, . . . ,zm,i z1, . . . ,i zm,0) | zk ∈ C}, W = {(z1, . . . ,zm,−i z1, . . . ,−i zm,0)|zk∈C}, ande0=(0, . . . ,0,1).
Compared to our previous representations, the action of the spin representation is fairly complicated. We state the necessary definition below, although it will take some work to provide appropriate motivation and to show that everything is well defined. Recall (Lemma 1.39) that one realization of Spinn(R)is{x1x2· · ·x2k|xi ∈ Sn−1for 2≤2k≤2n}.
Definition 2.7. (1)The elements ofS =
W are calledspinorsand Spinn(R)has a representation onScalled thespin representation.
(2)Forn even, the action for the spin representation of Spinn(R)onSis induced by the map
2.1 Basic Notions 31 x→(w)−2ι(w),
wherex ∈ Sn−1is uniquely written asx =w+waccording to the decomposition Rn ⊆Cn =W ⊕W.
(3) Let S+ = +
W =
k
2k
W andS− = −
W =
k
2k+1
W. As vector spacesS=S+⊕S−.
(4) Forn even, the spin representation action of Spinn(R)onS preserves the sub- spaces S+ andS−. These two spaces are therefore representations of Spinn(R)in their own right and called thehalf-spin representations.
(5)Forn odd, the action for the spin representation of Spinn(R)onSis induced by the map
x→(w)−2ι(w)+(−1)degmiζ,
wherex ∈ Sn−1is uniquely written asx =w+w+ζe0according to the decom- positionRn ⊆Cn =W ⊕W⊕Ce0,(−1)degis the linear operator acting by±1 on ±
W, andmiζ is multiplication byiζ.
To start making proper sense of this definition, let Cn(C) = Cn(R)⊗R C. From the definition of Cn(R), it is easy to see thatCn(C)is simply T(Cn)mod- ulo the ideal generated by either {(z⊗z+(z,z)) | z ∈ Cn}or equivalently by {(z1⊗z2+z2⊗z1+2(z1,z2))|zi ∈Cn}(c.f. Exercise 1.30).
Since Spinn(R) ⊆ Cn(C),Cn(C)itself becomes a representation for Spinn(R) under left multiplication. Under this action, −1 ∈ Spinn(R)acts as m−1, and so this representation is genuine. However, the spin representations turn out to be much smaller thanCn(C). One way to find these smaller representations is to restrict left multiplication of Spinn(R)to certain left ideals inCn(C). While this method works (e.g., Exercise 2.12), we take an equivalent path that realizesCn(C)as a certain en- domorphism ring.
Theorem 2.8.As algebras, Cn(C)∼=
End
W n even
End
W
End W
n odd.
Proof. n even: Forz=w+w∈Cn, define:Cn →End W by (z)=(w)−2ι(w).
As an algebra map, extend to : Tn(C) → End
W. A simple calculation (Exercise 2.6) shows(z)2 = m−2(w,w) = m−(z,z) so that descends to a map :Cn(C)→End
W.
To see thatis an isomorphism, it suffices to check thatis surjective since Cn(C)and End
W both have dimension 2n. Pick a basis{w1, . . . , wm}ofW and let {w1, . . . , wm} be the dual basis for W, i.e., (wi, wj)is 0 when i = j and 1 when i = j. With respect to this basis,acts in a particularly simple fashion. If 1 ≤i1 <· · · <ik ≤m, then(wi1· · ·wikwi1· · ·wik)killsp
W for p <k, maps
k
W ontoCwi1∧ · · · ∧wik, and preservesp
W for p>k. An inductive argument onn−ktherefore shows that the image ofcontains each projection of
W onto Cwi1∧ · · · ∧wik. Successive use of the operators(wi)and(wj)can then be used to mapwi1∧· · ·∧wikto any otherwj1∧· · ·∧wjl. This implies thatis surjective (in familiar matrix notation, this shows that the image ofcontains all endomorphisms corresponding to each matrix basis elementEi,j).
n odd: Forz=w+w+ζe0∈Cn, let±:Cn →End W by ±(z)=(w)−2ι(w)±(−1)degmiζ. As an algebra map, extend± to± : Tn(C) → End
W. A simple calculation (Exercise 2.6) shows that ±(z)2 = m−(z,z) so that ± descends to a map± : Cn(C)→ End
W. Thus the map : Cn(C)→ End
W
End W
given by(v)=(+(v) ,−(v))is well defined.
To see thatis an isomorphism, it suffices to verify thatis surjective since Cn(C)and
End
W
End W
both have dimension 2n. The argument is sim- ilar to the one given for the even case and left as an exercise (Exercise 2.7).
Theorem 2.9.As algebras,
C+n(C)∼= End+
W
End− W
n even End
W
n odd.
Proof. n even: From the definition of in the proof of Theorem 2.8, it is clear that the operators in (Cn+(C)) preserve±
W. Thus restricted toCn+(C),may be viewed as a map to
End+
W
End− W
. Since this map is already known to be injective, it suffices to show that dim
End+
W
End− W
= dimCn+(C). In fact, it is a simple task (Exercise 2.9) to see that both dimensions are 2n−1. For instance, Equation 1.34 and Theorem 1.35 show dimCn+(C) =2n−1. Alternatively, use Exercise 1.3.1 to show thatCn−1(R)∼=Cn+(R).
n odd: In this case, operators in(Cn+(C))no longer have to preserve ± W. However, restriction of the map+ toCn+(C)yields a map ofCn+(C)to End
W (alternatively,−could have been used). By construction, this map is known to be surjective. Thus to see that the map is an isomorphism, it again suffices to show that dim
End W
=dimCn+(C). As before, it is simple (Exercise 2.9) to see that both
dimensions are 2n−1.
At long last, the origin of the spin representations can be untangled. Since Spinn(R) ⊆ Cn+(C), Definition 2.7 uses the homomorphismfrom Theorem 2.9 forneven and the homomorphism+fornodd and restricts the action to Spinn(R).
In the case ofn even,can be further restricted to either End±
W to construct the two half-spin representations. Finally,−1 ∈ Spinn(R)acts bym−1, so the spin representations are genuine as claimed.