(1) The setBehis a closed smooth submanifold ofB,Bsis open inB,Msis open inM, the action
ofKonBehis principal, andpeh:Beh→Msis a smooth principalK-bundle.
(2) The action ofGonBsis proper,Msis a Hausdorff open subspace ofM, the action ofGonBs
is principal, andps:Bs→Msis a holomorphic principalG-bundle.
(3) There exists a unique K¨ahler metrichsonMs, such thatpeh∗ (Θs) = Ωeh, whereΘsis the K¨ahler
form ofhs,Ωeh=i∗eh(Ωs),ieh:Beh→ Bis the inclusion map, andΩis the K¨ahler form onB.
PROOF : The stabiliser Gρ of any pointρ ofB equals H, and, by Theorem 6.8, the induced
action ofGonBis principal. LetΨθ:B →Lie(K)∗be the restriction ofΦθ. By Lemma 7.7,Φθis a
moment map for the action ofKonA, henceΨθis a moment map for the action ofKonB. Moreover, Ψθ(B) ⊂ Φθ(A) ⊂ Ann(Lie(H∩K)), andBm := Ψ−1θ (0) = Φ−1θ (0)∩ B = Aeh∩ B = Beh.
Finally, by Lemma 7.5, we haveBms :=BmG=BehG=Bs, andPG(Bm,Bm) =PG(Beh,Beh) ⊂
8. THELINEBUNDLE ON THEMODULI OF STABLE REPRESENTATIONS
8.A. Line bundles on quotients of vector spaces
Let V be a finite-dimensional C-vector space, h·,·i a Hermitian inner product on V, and Ω = −2=(h·,·i)its fundamental2-form. We will considerV to be a K¨ahler manifold in the usual way. Let Gbe a complex Lie group, andKa real Lie subgroup ofG. Suppose that we are given a holomorphic linear right action ofG, and that the induced action ofKonV preserves the Hermitian inner product h·,·ionV.
Letχ : G → C× be a character ofG, and suppose thatχ(K) ⊂ U(1). Then,Te(χ)(Lie(K))
is contained in theR-subspaceLie(U(1)) =√−1RofLie(C×) =C. Fix a non-zero real number λ. Letα be the element ofLie(K)∗ defined byα(ξ) = −
√ −1
λ Te(χ)(ξ)for allξ ∈ Lie(K). Since
χ(gag−1) =χ(a)for alla, g∈G,αisK-invariant. Therefore, by Lemma 7.1, the mapΦ
α :V → Lie(K)∗, which is defined by
Φα(x)(ξ) = 12Ω(ξ](x), x) +α(ξ)
for allx∈V andξ∈Lie(K), is a moment map for the action ofKonV.
LetEdenote the trivial holomorphic line bundleV×ConV. Define a right action ofGonEby setting(x, a)g = (xg, χ(g)−1a)for all(x, a)∈E andg ∈G. LetΓ(E)denote theC-vector space of smooth sections ofEonV. For eachξ ∈Lie(G)ands∈Γ(E), define another sectionξs∈Γ(E)
by (ξs)(x) = d dt ¯ ¯ ¯ t=0 ¡ s(xexp(tξ)) exp(−tξ)¢
for allx∈V. For everyx∈V, define a Hermitian inner producth(x) :E(x)×E(x)→Cby h(x)((x, a),(x, b)) = exp(λkxk2)ab
for alla, b∈C. This gives a smooth Hermitian metrichonE.
Lemma 8.1 — Let∇be the canonical connection of the Hermitian holomorphic line bundle(E, h) onV. Then:
(1) For allξ∈Lie(K)ands∈Γ(E), we have∇ξ](s) =ξs−λ
√
−1Φξαs.
(2) The first Chern formc1(E, h)of∇equals−2λπΩ.
PROOF: Letξ ∈ Lie(K). Define a maps0 :V → E bys0(x) = (x,1)for all x ∈ V. It is a
holomorphic frame ofE onV. We have
for allx∈V andv∈Tx(V) =V. Therefore, ∇ξ](s0)(x) =λhξ](x), xis0(x) =− λ√−1 2 Ω(ξ ](x), x)s 0(x).
On the other hand,
(ξs0)(x) = Te(χ)(ξ)s0(x) =λ √ −1α(ξ)s0(x). Thus, (ξs0)(x)− ∇ξ](s0)(x) =λ √ −1¡α(ξ) +1 2Ω(ξ ](x), x)¢s 0(x) =λ √ −1Φα(x)(ξ)s0(x). It follows that ξs0− ∇ξ](s0) =λ √ −1Φξαs0.
Now, letsbe an arbitrary element ofΓ(E). Then, there exists a smooth complex functionf on V, such thats=f s0. It is easy to see that
ξ(f s0) =ξ](f)s0+f(ξs0). Therefore, ξs− ∇ξ](s) = ¡ ξ](f)s0+f(ξs0) ¢ −¡ξ](f)s0+f∇ξ](s0) ¢ =f(ξs0− ∇ξ](s0)) =f λ √ −1Φξαs0 =λ √ −1Φξαs. This proves (1).
Letω be the connection form of∇with respect to the holomorphic frames0 ofE onV, andR
the curvature form of∇. Then,ω =λ∂N, andR=∂ω, whereN :V →Ris the smooth function x7→ kxk2. Therefore, c1(E, h) = √ −1 2π R=− λ√−1 2π ∂∂N.
It is easy to see that√−1∂∂N = Ω. Thus,c1(E, h) =−2λπΩ, as stated in (2). 2
LetHbe a normal complex Lie subgroup ofG,Gthe complex Lie groupH\G, andπ :G→G the canonical projection. Let K be the compact subgroup π(K) of G, and πK : K → K the
homomorphism of real Lie groups induced byπ.
LetX be aG-invariant open subset ofV,Xmthe closed subsetΦ−1α (0)∩XofX, andXms =
projection. LetYms=p(Xms),pms:Xms→Ymsthe map induced byp, andpm=pms|Xm :Xm→
Yms.
The subsetEX =X×Cis aG-invariant open subset ofE. LetFdenote the quotient topological
spaceEX/G, andq :EX →F the canonical projection. There is a canonical continuous surjection
fromF toY, and every fibre of this map has a canonical structure of a1-dimensionalC-vector space. Thus,F is a family of1-dimensionalC-vector spaces onY. LetFm (respectively,Fms) denote the
restriction of this family to the subspaceYm (respectively, Yms) ofY. For everyx ∈ X, the map
q:EX →F restricts to aC-isomorphismq(x) :E(x)→F(p(x)).
Note that ifHis contained in the kernel of the characterχ:G→C×, then we have an induced action of G on E, and hence on EX. If, moreover, the action ofG on X is principal, then so is
its action on EX. Thus, in that case, there is a unique structure of a complex premanifold on F,
such thatq is a holomorphic submersion. With this structure, the family F of 1-dimensional C- vector spaces is a holomorphic line bundle onY(It is the holomorphic line bundle associated with the holomorphic principalG-bundlep:X →Y, and the character ofGinduced byχ:G→C×.). For every holomorphic (respectively, smooth) sectiontofF on any open subsetV ofY, there exists a unique holomorphic (respectively, smooth) sectionsofEX onp−1(V), which isG-invariant (that is,
s(xa) =s(x)afor allx∈p−1(V)anda∈G), such thatq(s(x)) =t(p(x))for allx∈p−1(V). Proposition 8.2 — Consider the context of Corollary 7.4. Suppose thatGx =H for allx∈ X,
the induced action ofGonXis principal,H ⊂Ker(χ), and
Φα(X)⊂Ann(Lie(H∩K)), PG(Xm, Xm)⊂HK.
Then, there exists a unique smooth Hermitian metrickms on the holomorphic line bundleFms
on Yms, such that c1(Fms, kms) = −2λπΘms, where Θms is the K¨ahler form on the open complex
submanifoldYmsofY.
PROOF : For every point y ∈ Yms, define kms(y) : F(y) ×F(y) → Cby kms(y)(a, b) =
h(x)(a0, b0), wherexis any point ofp−1
m (y)anda0, b0 ∈E(x)are such thatq(a0) =aandq(b0) =b.
Then, sincepm : Xm → Ymsis a smooth principalK-bundle, and the metric hisK-invariant, the
above rule gives a well-defined smooth Hermitian metrickmsonFms.
Suppose tis a smooth section ofFmson an open subsetV ofYms, y ∈ Yms, andw ∈ Ty(Y).
We will define an element ∇0w(t)ofF(y) as follows. Letx ∈ p−1m (y), and choosev ∈ Tx(Xm),
such thatTx(pm)(v) = w. Letsbe the uniqueK-invariant section ofE onp−1m (V)which projects
Tx0(pm)(v0) =w, then there exists a uniqueg∈K, such thatx0 =xg. Now,v0−Tx(ρg)(v)belongs
toKer(Tx0(pm)), and is hence of the formξ](x0)for someξ ∈Lie(K). Thus, by Lemma 8.1,
∇v0(s) =∇ξ](x0)(s) +∇Tx(ρg)(v)(s) =
¡
(ξs)(x0)−λ√−1Φξα(x0)s(x0)¢+¡∇v(s)
¢ g, since the action ofK preserves the metrichonE, and hence its canonical connection∇also. Now, sinces is K-invariant, we have ξs = 0, and since x0 ∈ Xm, we have Φξα(x0) = 0. Therefore,
∇v0(s) = ¡ ∇v(s) ¢ g, henceq(∇v0(s)) =q( ¡ ∇v(s) ¢
). It follows that∇0w(t) is well-defined. Since qmis a smooth principalK-bundle, this rule defines a smooth connection∇0onFms.
We claim that∇0is the canonical connection of the Hermitian holomorphic line bundle(Fms, kms)
onYms. As∇is compatible with the metrichonE, andK preservesh,∇0 is compatible with the
metrickms onFms. Therefore, we only need to check that∇0 is compatible with the holomorphic
structure onFms. Lettbe a holomorphic section ofFmson an open subsetV ofYms,y ∈ V, and
w∈Ty(Y). We have to check that∇0√−1w(t) =
√ −1∇0
w(t). Letsbe theG-invariant holomorphic
section ofE on p−1ms(V)corresponding tot. Letx ∈ p−1m (y), and choosev ∈ Tx(Xm), such that Tx(pm)(v) = w. Then, by Lemma 7.2,
√
−1v = v0 +√−1ξ](x), wherev0 ∈ T
x(Xm) andξ ∈ Lie(K). By definition,∇0w(t) =∇v(s). Similarly, sinceTx(pm)(v0) = Tx(p)(
√
−1(v−ξ](x))) =
√
−1w, we have∇0√
−1w(t) = ∇v0(s). Now, since ∇is compatible with the holomorphic structure
onE, we get
∇v0(s) =∇√−1(v−ξ](x))(s) =
√
−1(∇v(s)− ∇ξ](x)(s))
But, as we saw above,∇ξ](x)(s) = 0. It follows that∇0√
−1w(t) =
√ −1∇0
w(t). This proves the
above claim.
Thus, the canonical connection∇0 on(Fms, kms)is the descent of∇throughpm :Xm → Yms.
Therefore,
p∗mc1(Fms, kms) =i∗mc1(E, h),
whereim:Xm→Xis the inclusion. But, by Lemma 8.1,c1(E, h) =−2λπΩ, hence
p∗mc1(Fms, kms) =− λ 2πi ∗ mc1(E, h) =− λ 2πp ∗ mΘms.
Aspmis a smooth submersion, it follows thatc1(Fms, kms) =−2λπΘms. 2
8.B. The line bundle on the moduli space
>0, such thatn(θa−µθ(d))∈Zfor alla∈Q0. Letλ=−n. Letχ:G→C×be the character
χ(g) = Y a∈Q0
det(ga)n(µθ(d)−θa).
Then,χ(K) ⊂ U(1), andH ⊂Ker(χ), sincePa∈Q0(µθ(d)−θa)da = 0. Letα =− √ −1 λ Te(χ). Then, α(ξ) = √ −1 n Te(χ)(ξ) = √ −1 n X a∈Q0 n(µθ(d)−θa)Tr(ξa) =hξ, ηi, whereη=¡√−1(θa−µθ(d))1Va ¢ a∈Q0. Thus, Φα(ρ)(ξ) = 12Ω(ξ](ρ), ρ) +α(ξ) = 12Ω(ξ](ρ), ρ) +hξ, ηi= Φθ(ρ)(ξ)
for allρ∈ Aandξ∈Lie(K).
LetE be the trivial line bundle onAwith the action ofGdefined byχas above. LetFs be its
quotient byGonMs. As above,Fsis a holomorphic line bundle onMs. Now, the following result is
an immediate consequence of Proposition 8.2.