Monodromy of the trigonometric Casimir connection for sl_2

0.43 MB·English

by  Sachin Gautam

MONODROMY OF THE TRIGONOMETRIC CASIMIR
CONNECTION FOR sl
2
1
1
SACHIN GAUTAMAND VALERIOTOLEDANO LAREDO
0
2
Abstract. Weshow thatthemonodromy of thetrigonometric Casimir connec-
p
e tiononthetensorproductofevaluationmodulesoftheYangianYhsl2isdescribed
S bythequantumWeylgroupoperatorsofthequantumloopalgebraU~(Lsl2). The
proof is patterned on thesecond author’s computation of themonodromy of the
2
1 rational Casimir connection for sln via the dual pair (glk,gln), and rests ulti-
mately on the Etingof–Geer–Schiffmann computation of the monodromy of the

] trigonometric KZ connection. It relies on two new ingredients: an affine exten-
A sion of the duality between theR–matrix of U~slk and thequantum Weylgroup
Q elementofU~sl2,andaformulaexpressingthequantumWeylgroupactionofthe
. coroot lattice of SL2 in terms of the commuting generators of U~(Lsl2). Using
h thisformula,wedefinequantumWeylgroupoperatorsforthequantumloopalge-
at bra U~(Lgl2), and show that they describe the monodromy of the trigonometric
m Casimir connection on a tensor product of evaluation modules of the Yangian
[ Yhgl2.

1
v
7
6
3
Contents
2
9. 1. Introduction 2
0 2. The trigonometric Casimir connection of sl 5
2
1
3. The trigonometric Casimir connection of gl 7
1 2
4. Affine braid groups 10
:
v
5. The trigonometric KZ equations 12
i
X 6. The dual pair (gl ,gl ) and trigonometric connections 13
k 2
r 7. Monodromy of the trigonometric KZ equations 16
a
8. Quantum loop algebras 19
9. Quantum Weyl groups 22
10. The dual pair (U~glk,U~gln) 29
11. Affine braid group actions on quantum matrix space 30
12. Monodromy theorems 33
Appendix A. Monodromy of the trigonometric KZ equations (after
Etingof–Geer–Schiffmann) 34
Appendix B. Proof of Proposition 9.10 39
References 42
Date: September2011.
Both authorsare supported byNSFgrants DMS–0707212 and DMS–0854792.
1
2 S.GAUTAMANDV.TOLEDANOLAREDO
1. Introduction
1.1. Let g be a complex, semisimple Lie algebra, G the corresponding connected
and simply–connected Lie group, H ⊂ G a maximal torus and W the corresponding
Weyl group. In [28], a flat W–equivariant connection ∇ was constructed on H
C
whichhaslogarithmic singularities on therootsubtoriofH andvalues inany finite–
dimensional representation of the Yangian Yhg. By anablogy with the description
of the monodromy of the rational Casimir connection obtained in [26, 27], it was
conjectured in [28] that the monodromy of the trigonometric Casimir connection
∇ is described by the action of the affine braid group B of G arising from the
C G
quantum Weyl group operators of the quantum loop algebra U~(Lg).
b
1.2. Theaim of the present paperis to prove this conjecture wheng = sl and V is
2
a tensor product of evaluation modules. Note that, by a theorem of Chari–Pressley
[5], such representations include all irreducible Yhsl2–modules. To state our main
result, let V ,...,V be finite–dimensional sl –modules, z ,...,z points in C, and
1 k 2 1 k
V(z)= V (z )⊗···⊗V (z )
1 1 k k
the tensor product of the corresponding evaluation representations of Yhsl2. The
monodromy of the trigonometric Casimir connection yields an action of the affine
braid group B on V(z).
SL2
Let V be a quantum deformation of V , that is a module over the quantum group
i i
U~sl2 such that Vi/~Vi ∼= Vi. Set ~ = 4πıh and ζi = exp(−~za), and consider the
tensor product of evaluation representations of the quantum loop algebra U~(Lsl2)
given by
V(ζ)= V (ζ )⊗···⊗V (ζ )
1 1 n k
ThequantumWeylgroupoperatorsS0,S1 ofU~(Lsl2)yieldarepresentation ofBSL2
on V(ζ) [19, 20, 24]. The main result of this paper is the following
Theorem. The monodromy action of the affine braid group B on V(z) is equiv-
SL2
alent to its quantum Weyl group action on V(ζ).
1.3. TheproofoftheabovetheoremreliesontwodualitiesbetweentheLiealgebras
sl and sl discovered in [26]1. The first duality arises from their joint action on the
k n
space C[M ] of functions on k ×n matrices, and identifies the rational Casimir
k,n
connection of sl with the rational KZ connection on n points for sl . The second
k k
duality arises from the action of the correspondingquantum groups U~slk and U~sln
on a noncommutative deformation of C[M ], and identifies the quantum Weyl
k,n
group elements of U~sln with the R–matrices of U~slk.
These dualities were used in [26] together with the Kohno–Drinfeld theorem for
sl , to show that the monodromy of the rational Casimir connection of sl is de-
k n
scribed by the quantum Weyl group operators of U~sln.
1in thecase relevant to thepresent paper, n=2.
MONODROMY OF THE TRIGONOMETRIC CASIMIR CONNECTION FOR sl2 3
1.4. In this paper, we apply a similar strategy to compute the monodromy of the
trigonometric Casimir connection of sl and, in fact, gl . The latter connection is
2 2
an extension of the former to the maximal torus of GL constructed in [28], and
2
takes values in the Yangian Yhgl2. Its evaluation on a tensor product of evaluation
modulescoincides,uptoabelianterms,withthetrigonometricdynamicaldifferential
equationsconsideredin[25]. Inparticular, wealsocomputethemonodromyofthese
equations.
The duality between the Casimir and KZ connections identifies the trigonomet-
ric Casimir connection of gl with the trigonometric KZ connection of gl (see,
2 k
e.g., [25]). In turn, the monodromy of the latter was computed by Etingof–Geer–
Schiffmann in terms of data coming from the quantum group U~glk [12]. This
reduces the original problem to interpreting this data in terms of the quantum loop
algebra U~(Lgl2).
Part of this interpretation, namely the one pertaining to the data describing the
monodromy of the finite braid group Z ∼= B ⊂ B , is provided by the duality
sl2 SL2
between U~glk and U~gl2 of [26] alluded to in 1.3. What remains is the description
of the operators giving the action of the coroot lattice Z2 ∼= Q∨ ⊂ B of GL , in
GL2 2
terms of appropriate, commuting quantum Weyl group operators of U~(Lgl2).
1.5. To thebestof ourknowledge, quantum Weyl group operators giving an action
of the coroot lattice of GL on finite–dimensional representations of the quantum
2
loop algebra U~(Lgl2) have not been defined. Moreover, for U~(Lsl2), no compact,
explicit formula appears to be known for the element S S giving the action of the
0 1
generator of the coroot lattice of SL . In this paper, we give the following solution
2
to both of these problems.
Let t ⊂ gl and h ⊂ sl be the Cartan subalgebras of diagonal and traceless
2 2
diagonal matrices respectively, and U0 ⊂ U~(Lgl2), U0′ ⊂ U~(Lsl2) the commutative
subalgebras deforming U(t[z,z−1]) and U(h[z,z−1]). Then, we prove the following.
Theorem.
(1) There exist elements L ,L in a completion of U such that {S = S ,L ,L }
1 2 0 1 1 2
satisfy the defining relations of the affine braid group B .
GL2
(2) The element L = L L−1 lies in a completion of U′, and coincides with the
1 2 0
quantum Weyl group element S S giving the action of the generator of the
0 1
coroot lattice of SL .
2
The elements L ,L are given by explicit formulae in terms of the generators of
1 2
U0. For L = L1L−21, these are as follows. Let {Hk}k∈Z be the generators of U0′
with classical limit {h⊗zk}, where h is the standard generator of h (see Section 8).
Define, for any r ∈ N,
r
r s
H = H + (−1)s H
r 0 s [s] s
s=1 (cid:18) (cid:19)
X
e
4 S.GAUTAMANDV.TOLEDANOLAREDO
and note that H =h⊗(1−z)r mod ~. Then, we show that
r
e L= exp Hr
 r 
r≥1
X e
thus extending to the q–setting the factthat the classical limit of L is the loop
z−1 0
z 7→ = exp(−hlogz)
0 z
(cid:18) (cid:19)
The operators L ,L are given by similar formulae. These generalise in fact to any
1 2
complex semisimple Lie algebra and to gl [16].
n
1.6. Once the operators L ,L are explicitly defined, a direct computation shows
1 2
that their action on quantum k ×2 matrix space coincides with that of the U~glk
operatorswhich,by[12]describethemonodromyofthetrigonometricKZconnection
of gl , thus providing an extension of the q–duality of [26] to the affine setting.
k
Theorem 1.2, and its analogue for gl follow as a direct consequence.
2
1.7. The results of the present paper extend without essential modification to the
case of g = sl and gl , and give a computation of the monodromy of the trigono-
n n
metric Casimir connection of g with values in a tensor product of arbitrary finite–
dimensional evaluation representations of the Yangian Yhg, in terms of the quantum
Weyl group operators of the quantum loop algebra U~(Lg).
1.8. Outline of the paper. Sections 2 and 3 review the definition of the Yangian
and trigonometric Casimir connections of the Lie algebras sl and gl respectively.
2 2
Section4givespresentationsoftheaffinebraidgroupsB andB ,anddescribes
SL2 GL2
theembeddingB ⊂ B resultingfromtheinclusionofthemaximaltoriof SL
SL2 GL2 2
and GL in terms of the corresponding generators.
2
In Section 5, we review the definition of the trigonometric KZ connection for
the Lie algebra gl and, in Section 6 the fact that, under (gl ,gl )–duality, the
k k 2
trigonometric Casimir connection for gl is identified with the trigonometric KZ
2
connection for gl . In Section 7 we describe, following [12], the monodromy of the
k
latter connection in terms of the quantum group U~glk.
In Section 8, we review the definition of the quantum loop algebras U~(Lgl2) and
U~(Lsl2). Section 9 contains the main construction of this paper. We first extend
thequantum Weyl group action of theaffinebraid groupBSL2 on U~(Lsl2)to one of
BGL2 on U~(Lgl2). We then show that this action is essentially inner, by exhibiting
elements in an appropriate completion of the maximal commutative subalgebra of
U~(Lgl2), whose adjoint action coincides with the quantum Weyl group action of
the coroot lattice of gl .
2
Section 10 describes the joint action of U~glk and U~gl2 on the space C~[Mk,2]
of quantum k × 2 matrices. In Section 11, we prove the equality of two actions
of the affine braid group BGL2 on C~[Mk,2]. The first arises from its structure as
U~glk–module,anddescribesthemonodromyofthetrigonometricKZequations; the
second from its structure as a tensor product of k evaluation modules of U~(Lgl2).
MONODROMY OF THE TRIGONOMETRIC CASIMIR CONNECTION FOR sl2 5
In Section 12, we prove that the monodromy of the trigonometric Casimir con-
nection for g = sl (resp. g = gl ) on a tensor product of evaluation modules is
2 2
described by the quantum Weyl group operators of U~(Lg).
AppendixA outlines the computation of themonodromy of the trigonometric KZ
connection given in[12]. AppendixBcontains theproofof atechnical resultbearing
upon the completions of the quantum loop algebras U~(Lsl2) and U~(Lgl2) required
to handle quantum Weyl group elements.
Acknowledgments. The present paper was completed while both authors visited
the Kavli Institute for Theoretical Physics at the University of California, Santa
Barbara. We are grateful to the organisers of the program Nonperturbative Ef-
fects and Dualities in QFT and Integrable Systems for their invitation, and Edward
Frenkel and KITP for supporting our stay through their DARPA and NSF grants
HR0011-09-1-0015 and PHY05-51164 respectively.
2. The trigonometric Casimir connection of sl
2
2.1. The Yangian Yhsl2 [10]. The Yangian Yhsl2 is the unital, associative algebra
over C[h] generated by elements {ξr,er,fr}r∈N, subject to the relations
(Y1) For each r,s ∈ N,
[ξ ,ξ ] =0
r s
(Y2) For each r ∈ N,
[ξ ,e ] = 2e and [ξ ,f ]= −2f
0 r r 0 r r
(Y3) For each r,s ∈ N,
[e ,f ]= ξ
r s r+s
(Y4) For each r,s ∈ N,
[ξ ,e ]−[ξ ,e ] = h(ξ e +e ξ )
r+1 s r s+1 r s s r
[ξ ,f ]−[ξ ,f ] = −h(ξ f +f ξ )
r+1 s r s+1 r s s r
(Y5) For each r,s ∈ N,
[e ,e ]−[e ,e ] = h(e e +e e )
r+1 s r s+1 r s s r
[f ,f ]−[f ,f ] = −h(f f +f f )
r+1 s r s+1 r s s r
Yhsl2 is an N–graded algebra with deg(xr) = r and degh = 1. Moreover, it is a
Hopf algebra with coproduct determined by
∆(x ) = x ⊗1+1⊗x
0 0 0
for x = e,f,ξ, and
∆(ξ ) = ξ ⊗1+1⊗ξ +h(ξ ⊗ξ −2f ⊗e ) (2.1)
1 1 1 0 0 0 0
Let {e,f,h} be the standard basis of the Lie algebra sl . Then, the map
2
e → e f → f h → ξ
0 0 0
defines an embedding of sl2 into Yhsl2. In particular, Yhsl2 is acted upon by sl2 via
the adjoint action. This action is integrable since the graded components of Yhsl2
are finite–dimensional.
6 S.GAUTAMANDV.TOLEDANOLAREDO
2.2. The trigonometric Casimir connection [28]. Let G = SL (C), H ⊂ G the
2
maximal torus consisting of diagonal matrices and h ⊂ sl its Liealgebra. TheWeyl
2
group W ∼= Z2 of G acts on H and on the centraliser of h in Yhsl2.
Thetrigonometric Casimirconnection ofsl istheflat, W–equivariantconnection
2
on H with values in Yhsl2 given by
hκ
∇sl2 = d− −t dα
C eα−1 1
(cid:18) (cid:19)
where κ = e0f0+f0e0 is thebtruncated Casimir element of sl2, α ∈ h∗ is defined by
α(h) = 2, dα is the corresponding translation–invariant one–form on H, and
h
t = ξ − ξ2
1 1 2 0
2.3. Evalution homomorphism. For any s∈ C[h], there is an algebra homomor-
phism evs : Yhsl2 → Usl2[h] which is equal to the identity on sl2 ⊂ Yhsl2 and is
otherwise determined by [5, Prop. 2.5]
h
t 7→ sh− κ
1
2
Note that if s ∈ hC[h], evs maps elements of positive degree in Yhsl2 to hUsl2[h]
and therefore extends to a homomorphism Y[hsl2 → Usl2[[h]], where Y[hsl2 is the
completion of Yhsl2 with respect to its grading.
2.4. Let k ∈ N∗, s = (s ,...,s )∈ C[h]k and consider the homomorphism
1 k
evs = evs1⊗···⊗evsk◦∆(k) :Yhsl2 → Usl⊗2k[h]
where ∆(k) : Yhsl2 → Yhsl⊗2k is the iterated coproduct.
Proposition. The image of ∇sl2 under the homomorphism ev is the Usl⊗k[h]–
C s 2
valued connection on H given by
b
∆(k)(κ)
∇sl2 = d− h −A dα
C,s eα−1
!
where b
k h k
A= s h(a) − κ(a) −2h f(a)e(b)
a
2
a=1 a=1 1≤a<b≤k
X X X
and, for any x ∈ Usl , x(a) = 1⊗(a−1) ⊗x⊗1⊗(k−a).
2
Proof. The element t defined in §2.2 satisfies ∆(t ) = t ⊗1+1⊗t −2hf ⊗e ,
1 1 1 1 0 0
so that
∆(k)(t )= t(a)−2h f(a)e(b)
1 1 0 0
a a<b
X X
The result follows since ev (t ) = sh− hκ. (cid:3)
s 1 2
MONODROMY OF THE TRIGONOMETRIC CASIMIR CONNECTION FOR sl2 7
3. The trigonometric Casimir connection of gl
2
3.1. The Yangian Yhgl2 [9]. Yhgl2 is the unital, associative algebra over C[h] gen-
erated by elements {t(r)} , subject to the relations2
ij 1≤i,j≤2,r≥1
[t(r+1),t(s)]−[t(r),t(s+1)]= h t(r)t(s)−t(s)t(r)
ij kl ij kl kj il kj il
(cid:16) (cid:17)
foranyr,s ≥ 0,wheret(0) = h−1δ . TheseimplythatE 7→ t(1) givesanembedding
ij ij ij ij
of gl2 into Yhgl2, and that Yhgl2 is an N–graded algebra with
deg(t(r)) = r−1 and deg(h) = 1
ij
Moreover, Yhgl2 is a Hopf algebra with coproduct given by
∆(t (u)) = t (u)⊗t (u)
ij ik kj
k
X
where t (u) = h t(r)u−r.
ij r≥0 ij
3.2. The embedPding Yhsl2 ⊂ Yhgl2[4, 22]. Let e(u),f(u),ξ(u) ∈ Yhsl2[[u−1]] be
the generating series
e(u) = h e u−r−1 f(u)= h f u−r−1 ξ(u) = 1+h ξ u−r−1
r r r
r≥0 r≥0 r≥0
X X X
Then, the following defines an embedding of graded Hopf algebras ı :Yhsl2 → Yhgl2
[22, Rem. 3.1.8]
e(u) 7→ t (u)t (u)−1 f(u)7→ t (u)−1t (u)
21 11 11 12
ξ(u) 7→ t (u)t (u)−1−t (u)t (u)−1t (u)t (u)−1
22 11 21 11 12 11
In particular,
(1) (1) (1) (1)
ı(e )= t ı(f )= t ı(ξ )= t −t
0 21 0 12 0 22 11
ı(ξ ) = t(2)−t(2)+h (t(1))2−t(1)t(1)−t(1)t(1)
1 22 11 11 22 11 21 12
which implies that the element t = ξ (cid:16)−hξ2/2 is mapped to (cid:17)
1 1 0
h h
(2) (2) (1) (1)
ı(t ) = t −t + (t −t )(I +1)− κ (3.1)
1 22 11 2 11 22 2
(1) (1) (1) (1) (1) (1)
where I = t +t and κ= t t +t t .
11 22 12 21 21 12
Remark. The restriction of ı to sl2 ⊂ Yhsl2 is not the standard embedding  :
sl → gl given by e → E ,f → E ,h → E −E . In fact, ı| = θ ◦, where
2 2 12 21 11 22 sl2
θ ∈ Aut(gl ) is the Chevalley involution given by
2
θ(E ) = E (3.2)
ij ij
with 1 = 2 and 2 = 1.
2we follow thesign conventions of [22].
8 S.GAUTAMANDV.TOLEDANOLAREDO
3.3. The trigonometric Casimir connection of gl [28]. Let T ⊂ GL be the
2 2
maximal torus consisting of diagonal matrices and t its Lie algebra. The trigono-
metric Casimir connection of gl2 is the Yhgl2–valued connection on T given by
d(ε −ε )
∇gl2 = d−h 1 2 κ−dε A −dε A (3.3)
C eε1−ε2 −1 1 1 2 2
where b
(1) {ε ,ε } is the basis of t∗ given by ε (E ) = δ and {dε } are the corre-
1 2 i jj ij i
sponding translation–invariant 1–forms on T.
(2) The elements A1,A2 ∈ Yhgl2 are given by
A = 2t(2) −h(t(1))2−ht(1)
1 11 11 11
A = 2t(2) −h(t(1))2−ht(1)−hκ
2 22 22 22
(r) (r)
Let the symmetric group S2 act on Yhgl2 by σ(tij )= tσ(i),σ(j), and regard Yhsl2 as
embedded in Yhgl2 via 3.2.
Theorem. [28, §5]
gl
(1) The trigonometric Casimir connection ∇ 2 is a flat, S –equivariant con-
C 2
nection on the trivial vector bundle T ×Yhgl2.
(2) The restriction of ∇gl2 to the maximal tobrus H of SL is the trigonometric
C 2
Casimir connection ∇sl2 of sl .
C 2
b
3.4. Evaluation homomorbphism. The Yangian Yhgl2 admits a one–parameter
family of algebra homomorphisms eva : Yhgl2 → Ugl2[h] labelled by a ∈ C[h], and
given by
ev (t(r)) = ar−1E
a ij ij
Note that this expression continues to make sense, and to define a homomorphism
Yhgl2 → Ugl2[h] if a is a central element in Ugl2[h].
The evaluation homomorphism of Y~(gl2) does not restrict to the one for Yhsl2
defined in 2.3. However, the following holds
Lemma. If the evaluation points are related by
h
r = s+ (I +1) (3.4)
2
the following diagram is commutative
Yhsl2 ı // Yhgl2
IIeIvIsIIIII$$ zzuuuuuθu◦ueuvur
Ugl [h]
2
where θ is the Chevalley involution 3.2.
MONODROMY OF THE TRIGONOMETRIC CASIMIR CONNECTION FOR sl2 9
Proof. This clearly holds for the generators e0,f0,ξ0 of Yhsl2, and follows for the
element t by comparing
1
h h
ev (ı(t )) = (E −E )(−r+ (I +1))− κ
r 1 11 22
2 2
where we used (3.1), with ev (t )= sh−hκ/2. (cid:3)
s 1
3.5. Now let (s ,...,s ) ∈ C[h]k and set r = s + h(I +1) for any 1 ≤ a ≤ k, as
1 k a a 2
in (3.4). Consider the algebra homomorphism
evr = evr1⊗···⊗evrk◦∆(k) : Yhgl2 → Ugl⊗2k[h]
Proposition. [28, Prop. 5.6]
(1) The image of ∇gl2 under the evaluation homomorphism ev is the Ugl⊗k[h]–
C r 2
valued connection given by
b
d(ε −ε )
∇gl2 = d−h 1 2 ∆(k)(κ)−dε A −ε A (3.5)
C,s eε1−ε2 −1 1 1 2 2
where
b
A = (2s E +hE E )(a) +2h E(a)E(b)
1 a 11 11 22 12 21
a a<b
X X
A = (2s E +hE E )(a) −2h E(a)E(b) −h κ(a)
2 a 22 11 22 12 21
a a<b a
X X X
(2) The restriction of ∇gl2 to H ⊂ T is the image of the Usl⊗k[h]–valued con-
C,s 2
nection ∇sl2 of Proposition 2.4 under the Chevalley involution θ⊗k.
C,s
b
Proof. (1) By 3b.1, ∆(t(2)) =t(2)⊗1+1⊗t(2)+h t(1)⊗t(1), which implies that
ii ii ii i′ ii′ i′i
∆(k)(t(2))= (t(2))(a) +h (t(1))(a)(t(1))P(b) +h (t(1))(a)(t(1))(b)
ii ii iı ıi ii ii
a a<b a<b
X X X
where 1 = 2,2 = 1. Since ∆(k)(t(1))2 = 2 (t(1))(a)(t(1))(b) + ((t(1))(a))2, this
ii a<b ii ii a ii
yields
P P
ev 2t(2) −h(t(1))2−ht(1) = (E (2r −h(E +1)))(a) +2h (t(1))(a)(t(1))(b)
r ii ii ii ii a ii iı ıi
(cid:16) (cid:17) Xa Xa<b
h
Substituting r = s + (I +1) yields the claimed formula for A = ev (A ). The
a a 2 1 r 1
formula for A follows from the above, and the fact that
2
∆(k)(κ) = κ(a) +2 (t(1))(a)(t(1))(b) +(t(1))(a)(t(1))(b)
12 21 21 12
Xa Xa<b(cid:16) (cid:17)
(2) is a direct consequence of Proposition 3.3 and Lemma 3.4. (cid:3)
Remarks.
10 S.GAUTAMANDV.TOLEDANOLAREDO
0 i
(1) SincetheChevalleyinvolutionisgivenbyconjugatingbythematrix ∈
i 0
(cid:18) (cid:19)
SL , the application of θ⊗k to the connection ∇sl2 yields a connection with
2 C,s
the same monodromy.
(2) Asshownin[28,§5.15],theconnection∇gl2 coinbcides,moduloabelianterms,
C,k
with the trigonometric dynamical differential equations for gl considered in
2
[25]. b
4. Affine braid groups
4.1. Set
B = π (H /W) and B = π (T /W)
SL2 1 reg GL2 1 reg
The following is well known [7, 21, 12]
Proposition.
(1) B is the affine braid group of type A , and hence admits the presentation
SL2 1
B = hS ,S | no relations i
SL2 0 1
(2) B can be realised as the subgroup of the Artinbraid group on three strands
GL2
B , consisting of braids where the first strand is fixed. It has the presentation
3
B = hX ,b|bX bX = X bX bi
GL2 1 1 1 1 1
4.2. We describe the generators S ,S ,b,X below, together with the inclusion
0 1 1
B ⊂ B stemming from the W–equivariant embedding H ⊂ T .
SL2 GL2 reg reg
Identify to this end the tori H and T with C× and (C×)2 respectively, by
z 0 z 0
z → and (z ,z ) → 1
0 z−1 1 2 0 z
2
(cid:18) (cid:19) (cid:18) (cid:19)
Intermsoftheseidentifications, theinclusionH ⊂T isgivenbyz 7→ (z,z−1). More-
over, H ⊂ H and T ⊂ T are identified with C×\{±1} and Y (C×) respectively,
reg reg 2
where the latter is the configuration space of two ordered points in C×.
4.3. ThegeneratorsS ,S ofB maybedescribedasfollows[23,29,30]. Identify
0 1 SL2
theLiealgebra hof H withC by mappinghto1. Theexponential mapexp(2πι−) :
h → H maps ha-reg to H , where
reg
1
ha-reg = h\ {α = n}∼= C\ Z
2
n∈Z
[
The affine Weyl group W of type A is generated by the affine (real) reflections
aff 1
s ,s through the points u = 1/2 and u = 0 respectively. W is isomorphic to
0 1 aff
Z ⋉Z, with the generator s of Z acting on h as the reflection u → −u and the
2 1 2
generatorτ = s s ofZasthetranslationu → u+1. Thuswehavetheidentification
0 1
exp(2πι−) :ha-reg/W ∼= H /W (4.1)
aff reg
Fix now a base point (say u= 1/4) in ha-reg lying in the interval (0,1/2). Then, the
generators S are represented by the loops in ha-reg/W given in Figure 4.1. These
i aff
correspond, via the identification (4.1) to the loops in H shown in Figure 4.2.
reg