and then by taking quotients to obtain irreducible representations. In the case ofSL(2,C), the story is very simple. The only nontrivial parabolic subgroup is the Borel subgroup B of upper-triangular matrices. It has a decompositionB =M AN corresponding to the decomposition b=a⊕m⊕nwith
a=hHi, A={exptH} (t∈R),
m=hiHi, M ={exptiH} 'U(1), (7.1.14) n=hE, iEi, N ={exp(tE+t0iE)}.
Then, lettingν ∈(a∗)C and κ∈m∗ be (real) linear functionals defined by
ν(H) =−iw , κ(iH) =k ; ν(iH) =κ(H) = 0, (7.1.15) and letting ρ be the half-sum of positive restricted roots satisfying
ρ(H) = 2, ρ(iH) = 0, (7.1.16)
the principal series representation Pk,w is induced from the representation exp(iν+ρ)⊗ expiκ⊗idof B =M AN:
Pk,w= indGBC(eiν+iκ+ρ). (7.1.17) Observe that the defining data here is a pair of linear functionalsκandν, corresponding to the parametersk andw, that act on the maximal torusTC'GL(1) of the groupGC. The functional acting on the compact part ofTCis quantized. This is true in general for maximal parabolic subgroups: principal series representations are induced from representations of the maximal torus that act trivially on the off-diagonal part of the parabolic subgroup. The representationTC(hereeiν+iκ+ρ) is called the quasi-character of the induced representation.
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[125] (see also [127] and [48] in the physics literature). Here, we will give a brief overview of the general procedure, stating necessary results, and then specializing to the example of SU(2) andSL(2,C).
The adjoint action of G on g can be written as g : X → gXg−1. Likewise, there is a coadjoint action of G on the dual space g∗. Since g∗ and g can be identified for reductive groups by a non-degenerate trace form, we can write weights λ∈g∗ as matrices; then the coadjoint action isg:X →gλg−1. The coadjointorbit
Ωλ ={gλg−1}g∈G (7.2.1)
through a point λhas the geometry of the coset G/Hλ (or sometimes a quotient thereof), whereHλ is the stabilizer ofλ:
Hλ={h∈G|hλh−1 =λ}. (7.2.2)
In the case of compact or complexG,Hλ is conjugate to the maximal torus ofGfor generic λ. The left-regular G-action onG/Hλ is equivalent to the coadjoint action on Ωλ, and is the action used for representations.
The coadjoint orbit Ωλ ' Hλ has a natural G-invariant symplectic structure. The symplectic form can be written as
ω=−Tr(λ g−1dg∧g−1dg) (7.2.3) for g ∈ G/Hλ. To geometrically quantize Ωλ one must form a line bundle L → Ωλ with curvatureω, choose apolarization that effectively cuts out half the degrees of freedom (half the coordinates) on Ωλ, and construct a Hilbert space V as the space of square-integrable polarization-invariant sections of L. Considering λ again to be a linear functional λ∈g∗, the line bundle L can is obtained as a quotient of C×G by the representation eiλ of Hλ ∈ G acting onC; in other words, it is essentially the space C∞(G, eiλ) of the induced representation indGH
λ(eiλ) described in Section 7.1. The line bundle exists if and only if the representationeiλ is integrable.
In the case of compact group G, where Hλ ' T is a maximal torus, one can induce a complex structure onG/Hλ via the equivalence
G/Hλ 'GC/B , (7.2.4)
where B is the upper-triangular Borel subgroup of GC containing Hλ. Then, to obtain the unitary finite-dimensional representations, one must choose aholomorphicpolarization.
Specifically, if −λ is the dominant weight of a sought representation R, then the space H∂0¯(G/Hλ;eiλ) of holomorphic sections of the bundleL is finite dimensional andGacts on it in the left-regular representation to furnishR.
In the case of complex groupGC, principal series representations are obtained by consid- eringreal polarizations on the line bundle overGC/Hλ, whereλis now a non-complex-linear element of the dualg∗
C. The set of all sections of this line bundle is the representation space of a representation induced from Hλ by eiλ. To get the desired representation induced from the Borel subgroup B =HλN by eiλ⊗id, the polarization is chosen precisely such that sections are independent of coordinates in N. (This process can also be extended to describe induction from generic non-maximal parabolic subgroups,e.g. by choosingλsuch that its isotropy group Hλ is non-minimal.)
In both compact and complex cases, the representations of the groupG(C)end up being described by choices of coadjoint orbits. Since a coadjoint orbit is defined by an element λ ∈ g∗(
C) ' g(C) up to conjugacy, this establishes an equivalence between representations and conjugacy classes of g(C).
Example: SU(2)
In the simplest case of SU(2), let us write a matrixg∈SU(2) as g=
w −¯z z w¯
, w, z ∈ C. (7.2.5)
with |w|2+|z|2 = 1. The right action of a diagonal matrix diag(eiθ, e−iθ) in the maximal torus H = T acts by sending (w, z) 7→ (eiθw, eiθz). From this, we immediately see that SU(2) has the geometry ofS3, and theH action is just rotation in theS1 fiber of the Hopf bundle S1→S3 →S2. Therefore, a generic coadjoint orbit looks like
Ω'SU(2)/T=S2'P1. (7.2.6)
We already know what all holomorphic bundles on P1 look like: they are tensor powers of the canonical line bundle, O(˜λ) for ˜−λ∈Z. These have curvature
ω=−Tr(λg−1dg∧g−1dg), (7.2.7)
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whereλis the matrix
λ= λ˜ 2i
1 0
0 −1
, (7.2.8)
or equivalently the corresponding element of (g∗)C acting as Tr[λ, ·]. Moreover, for ˜λ <0, the bundle O(−˜λ) has an (|˜λ|+ 1)-dimensional space of square-integrable holomorphic sections. A basis of this space is given by the polynomials{1, z, ..., z−˜λ}in local coordinates on SU(2)/H where w 6= 0; this is precisely the (|λ|˜ + 1)-dimensional representation of SU(2). It is easy to see that the generators of the Lie algebrasu(2) act on local functions f(z) exactly as in (7.1.4).
By using the symplectic form (7.2.6), and working in projective coordinates (w : z) on P1, it is also possible to show that the variables ¯wand ¯zbecome identified after quantization with the conjugate momenta
λ˜w¯∼ −∂w, λ¯˜z∼ −∂z. (7.2.9)
Example: SL(2,C)
Let us write an element ofSL(2,C) in complex coordinates as g=
w −x z y
, wy+zx= 1, z, w, x, y∈C. (7.2.10) We can also putw=b+ic, y=b−ic, z =d+ie, x=d−ie(four new complex coordinates);
thenSL(2,C) ={b2+c2+d2+e2 = 1} 'T∗S3 'T∗(SU(2)).
There is a single Cartan subgroup (or maximal torus) H = TC, and almost all of SL(2,C) is conjugate to it. In the coadjoint orbitSL(2,C)/TC, there is a rescaling symmetry (w, z, x, y) 7→ (aw, az, a−1x, a−1y) for a ∈ C∗. A slightly nontrivial argument shows that SL(2,C)/TC is isomorphic to{b2+c2+f2= 1} ∈C3 (we have essentially set f2=zxand used the scaling to set x = 1 or z = 1, or both, except at a point). Therefore, a genetic coadjoint orbit looks like
Ω'SL(2,C)/TC 'T∗S2. (7.2.11) Unlike the case ofSU(2), this is a noncompact manifold, and will lead to infinite-dimensional Hilbert spaces (representations) upon geometric quantization.
In the case of SU(2), we implicitly extended the isomorphismg∗ 'gto an isomorphism (g∗)C'gby complex linearity. We need do the same for the Lie algebrasl(2,C), viewed as
a real Lie algebra in order to access principal series representations. To simplify notation, letg=sl(2,C) (rather than usinggC). Then, as in Section 7.1, the complexification of this real Lie algebra satisfiesgC'g×g. An appropriate complexification of the nondegenerate trace form ReTr is given byh,i : gC×g'(g×g)×g→C, such that
h(XL, XR), Yi= 1
2Tr(XLY) +1
2Tr(XRY ,¯ (7.2.12) where ¯Y denotes the usual complex conjugation in the (“broken”) complex structure on sl(2,C). This leads to an identification of linear functionals and matrices
ν∈(a∗)C ↔ ν =−iw 2
1 0
0 −1
, κ∈(m∗)C ↔ κ=−ik 2
1 0
0 −1
. (7.2.13)
The pairν⊕κ∈(m∗⊕a∗)C⊂(g∗)C is identified with
ν⊕κ ↔ (XL, XR) = (ν+κ, ν−κ) ∈sl(2,C)C'sl(2,C)×sl(2,C). (7.2.14) The natural symplectic form resulting from the above isomorphism is
ων,κ = 1
2Tr((ν+κ)g−1dg∧g−1dg) +1
2Tr((ν−κ)g−1dg∧g−1dg). (7.2.15) Due to the compact cycleS2 in the coadjoint orbit Ω'T∗S2, this form can is the curvature of a line bundle L if and only if k ∈ Z. Moreover, when the line bundle L exists, it will be unique (since T∗S2 is simply-connected). Using a polarization in which sections of L depend only on a single complex projective coordinate (w : z), we see that the sections induced from eiν+iκ+ρ will transform as
f(aw, az,¯aw,¯ ¯a¯z) =a−12(w+k)−1¯a−12(w−k)−1f(w, z,w,¯ z),¯ a∈C∗. (7.2.16) This, of course, is precisely the right transformation for the principal series representation Pk,w.
The conjugate momenta to the complex projective coordinates (w : z) and ( ¯w : ¯z) are x∼ −2
w+k∂z, y∼ −2
w+k∂w, x¯∼ −2 w−k
∂¯z, y¯∼ −2 w−k
∂¯w. (7.2.17)
On sections ofL, one has that z∂z+w∂w=−1
2(w+k)−1, z¯∂¯z+ ¯w∂¯w =−1
2(w−k)−1. (7.2.18)
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