We now apply the results of the previous sections to give a “symplectic” proof of Schur’s Lemma.
Recall first the result of Section 5.8.4: for admissable a∈ MG, the natural G-representation on thee
space of sections ΓFRa(LaR) is irreducible. Choosing qa ∈ L˙aR over R(µ,a) ∈ π−1Ra(a), T a maximal torus inGµ, and a choice ∆+ of positive roots determined by i
~µ, the irreducible G-representatione ΓFRa(LaR) is the dual of that with highest weight i
~µ, ΓFa
R(LaR)'qa(H~iµ)∗.
We also apply the geometric quantization procedure to the symplectic manifold (π−1Rb(b),−ωRb), where b is an admissible element of MG. The momentum map corresponding to the reduced G- actions is now−JbR. The polarizationFRb is a totally complex and strictly positive, and determines a complex structureJ
FRb =−JFb
Ron π−1Rb(b). This complex structure is the one induced by the series of diffeomorphisms
π−1(b) Rb 'R(ν,b)
G
G−ν ' GC P−ν− = GC
Pν+
,
(recall, the momentum of R(ν,b) is now −ν), where Pν+ is the positive parabolic subgroup of GC, which has Lie algebra p+ν = gCν ⊕n+ν. There exists a U(1)-bundle over π−1Rb(b) with connection of curvature −εh0ωRa. The associated line bundle is (LbR)∗, the dual to the usual one; this can be seen by the Chern-Weil correspondence, since the extra minus in the symplectic form corresponds to transition functions which are inverses of those on LbR. Taking q∗b a point in this bundle over R(ν,b)∈ π−1Rb(b) and following through the Borel-Weil argument1, we obtain
ΓFRb (LbR)∗ 'q∗
b H~iν. We are interested in calculating Hom
Ge
ΓFb
R(LbR),ΓFRa(LaR)
, the set ofG-intertwiners betweene the irreducible representation spaces ΓFb
R(LbR) and ΓFRa(LaR). Under the canonical isomorphism Hom(V1, V2)'V1∗⊗V2, this is the same as
ΓFb
R
(LbR)∗ ΓFa
R(LaR)Ge
= ΓFb
R⊕FRa (LbR)∗LaRGe ,
the space ofG-invariant sections of (Le bR)∗LaR which are covariantly constant with respect to the polarizationFRb ⊕FRa. The space π−1Rb(b)×π−1Ra(a) is compact, and so using Proposition D.3.1 we can straight away express a necessary condition for the existence of nonzeroG-invariant sections.e Corollary D.5.1. The space ofG-invariant sectionse Γ
FRb⊕FRa (LbR)∗LaRGe
is nonzero only if the G-orbitsaandb both belong to J−1G(O) for some coadjoint orbitO.
Proof. The momentum map on π−1Rb(b)×π−1Ra(a)isJRb ◦pπ−1 (b) Rb
−JRa◦pπ−1 (a) Ra
. For nonzeroG-invariante
1Here it is convenient to take the Peter-Weyl equivalence as “α⊗v ∈ (Hλ)∗⊗ Hλ corresponds to the map eg7→α(eg−1·v) inC∞(G,e C)⊂L2(G).” The convention from Section 5.8.3 yields Γe
FRb (LbR)∗ 'q∗
0
Hw0(−~iν)∗
, wherew0is the longest element of the Weyl group, though the latter space is isomorphic toH~iν.
sections to exist, the image of this map must contain 0, implying that π−1Rb(b) ×π−1R(a)a contains a point of the form (R(µ,a), R(µ,b)) for some µ∈ g∗. The G-orbits aand b both intersect J−1(µ) in the unreduced spaceM, and therefore are subsets of J−1(O), whereOis the coadjoint orbit through µ.
Now restrict to the case a, b∈ J−1G(O). So take pointsR(µ,a)∈ π−1R(a)a and R(µ,b)∈ π−1Rb(b) (note these are labeled by the sameµ∈g∗), and combine the previously discussed diffeomorphisms,
π−1(b)
Rb ×π−1(a)
Ra '(R(µ,b),R(µ,a))
GC Pµ+
×GC Pµ−
.
We are interested in the orbits of the diagonal GC-action in this space. A generalGCorbit can be written as
GC·(ePµ+, gPµ−)
for some g∈GC. To find explicitly what these orbits look like, we employ the generalized Bruhat decomposition(see for example [BL00, Chapter 1]). This characterizes the orbits of the spaceGC/Pµ− under the natural leftPµ+-action, the so-called Schubert cells. Choosing a maximal torus T ⊂Gµ (implyingTC⊂Pµ±), the decomposition says that within each orbitPµ+-orbit, there exists a point nPµ−, where n ∈ GC is an element of the normalizer of TC. In particular, Pµ+gPµ− contains a pointngPµ−, where ng normalizesTC, and so gives a corresponding element wg of the Weyl group W =NGC(TC)/TC=NG(T)/T. In general, for the case whenGµ is larger than the maximal torus T, there are severalng’s and severalwg’s in thePµ+-orbit throughgPµ−—see [BL00] for a discussion.
The orbits
GC·(ePµ+, gPµ−) and GC·(ePµ+, ngPµ−)
agree, since Pµ+ acts trivially on the ePµ+. The stabilizer group of the diagonal GC-action at the point (ePµ+, ngPµ−) is
Pµ+∩(ngPµ−n−1g ) =Pµ+∩PAd−∗ n−1
g
µ=Pµ+∩Pw−
g·µ.
This stabilizer group is of smallest dimension whenwg = id, corresponding tog=e. In that case, we getPµ+∩Pµ− =GCµ, and the corresponding orbitGC·(ePµ+, ePµ−)⊂GC/Pµ+×GC/Pµ− is largest.
The real dimension of the orbit is
dimRGC
GCµ = 2 dimR G Gµ
,
which is the dimension of π−1Rb(b)× π−1R(a)a '(R(µ,b),R(µ,a)) GC Pµ+ × GC
Pµ− itself. Hence the GC-orbit is through (R(µ,b),R(µ,a)) is open in π−1Rb(b)×π−1Ra(a).
It can be shown that the otherGC-orbits have complex codimension≥1 in π−1Rb(b)×π−1Ra(a). This is particularly clear for the case when Gµ is a torus T, since then any nontrivial element w of the Weyl group will cause the initially completely disjoint root spaces ofp+µ =tC⊕n+µ andp−µ =tC⊕n−µ to overlap between p+µ and p+w·µ in at least one root space, which has complex dimension 1. Since this represents the stabilizer of theGC-orbit, the statement follows.
AGeC-invariant sectionsover the orbitGC·(R(µ,b),R(µ,a))⊂ π−1Ra(a)×π−1Ra(a)is determined by its value at (R(µ,b),R(µ,a)). Proposition D.3.1 demonstrates thatsis bounded onGC·(R(µ,b),R(µ,a)), and so it can be extended uniquely to the rest of theπ−1Rb(b)×π−1R(a)a , since the union of the remaining orbits has complex codimension ≥ 1. Hence the space of sections Γ
FRb⊕FRa (LbR)∗LaRGe is one complex dimensional.
Tracing through the use of the Peter-Weyl and Borel-Weil theorems, we can say that a general section in ΓFb
R⊕FRa (LbR)∗LaR
, expressed in terms of the corresponding U(1)-equivariant function, is a complex linear combination of sections of the form
k(˙ ega·q∗a egb·qb) =α↓(ega−1·v)α(egb·v↑)
(extended by U(1)-equivariance), whereα↓is a lowest weight vector in (H~iµ)∗(of weight−i
~µ),v↑ is a highest weight vector inH~iµ, andα∈(H~iµ)∗, v∈ Hi~µ are arbitrary.
k(˙ ega·q∗a geb·qb) =X
i
α↓(ega−1·ei)ei(egb·v↑) =α↓(ega−1·egb·v↑),
whereei is a basis forH~iµ, and ei is the corresponding dual basis in (H~iµ)∗. Returning to the form Hom
Ge
ΓFb
R(LbR),ΓFRa(LaR)
, we get the space of all complex multi- ples of the following map: given qb ∈ ( ˙τRb)−1(R(µ,b)), qa ∈ ( ˙τRa)−1(R(µ,a)), and a section ˙tbR ∈ Cid∞
U(1)( ˙LbR,C), the image section ˙taR∈Cid∞
U(1)( ˙LaR,C) is defined by t˙aR(eg·qa·w) := ˙tbR(eg·qb·w)
for all eg ∈ G, ande w ∈ U(1). In other words, it is the map induced by the bundle-connection isomorphism ˙LbR'L˙aR, dependent onqb andqa.
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