# Coordinate rings and birational charts

aa r X i v : . [ m a t h . R T ] F e b COORDINATE RINGS AND BIRATIONAL CHARTS

Sergey Fomin and George Lusztig

Abstract.

Let G be a semisimple simply connected complex algebraic group. Let U be the unipotent radical of a Borel subgroup in G . We describe the coordinaterings of U (resp., G/U , G ) in terms of two (resp., four, eight) birational chartsintroduced in [L94, L19] in connection with the study of total positivity. Introduction

Let G be a simply connected, almost simple algebraic group over C . Fix amaximal torus T of G and a pair B + , B − of opposite Borel subgroups containing T ,with unipotent radicals U + , U − . Let ν = dim( U + ) and r = dim( T ). For anirreducible quasi-aﬃne variety X over C , we denote by O ( X ) the algebra of regularfunctions X −→ C , and let [ O ( X )] be the quotient ﬁeld of O ( X ).In this paper, we show (see Theorems 0.3, 4.2 and 5.2) that the algebra O ( U + )(resp., O ( G/U − ) and O ( G )) can be completely described in terms of two (resp.,four and eight) birational charts C ν −→ U + (resp., C ν × ( C ∗ ) r −→ G/U − and C ν × ( C ∗ ) r × C ν −→ G ) which were introduced in [L94],[L19] in connection withthe study of total positivity.Theorem 0.3 provides a proof of a conjecture in [L19, 6.1(a)]. Theorem 4.2(resp., Theorem 5.2) establishes a weak form of a conjecture in [L19, 6.3(a)] (resp.,[L19, 6.2(a)]) in which only two birational charts, instead of four (resp., eight), wereused. The proof of Theorem 0.3 given in Section 3 relies on the results in [BZ97]and [FZ99] that describe the inverse of the charts for U + in terms of “generalizedminors.” Theorems 4.2 and 5.2 are proved in Sections 4 and 5, respectively, usingreduction to the case of U + . In particular, our proof of Theorem 5.2 does not usethe more complete results on generalized minors in [FZ99]. (The latter techniquewould have allowed to decrease the number of charts from eight to two, but thenthe two charts used would not be canonical, unlike the eight that we consider here.) Mathematics Subject Classiﬁcation . Primary 22E46. Secondary 20G20, 14M15.

Key words and phrases.

Lie group, coordinate ring, birational chart, generalized minor.This work was supported by NSF grants DMS-1664722 (S. F.) and DMS-1855773 (G. L.) andby a Simons Fellowship (S. F.). Typeset by

AMS -TEX SERGEY FOMIN AND GEORGE LUSZTIG

In order to state our main result (Theorem 0.3), we will need to introduce somenotation.Let U + i ( i ∈ I ) be the simple root subgroups of U + , and let U − i ( i ∈ I ) be thecorresponding root subgroups of U − ; here I is a ﬁnite indexing set. We assumethat for any i ∈ I we are given isomorphisms of algebraic groups x i : C ∼ −→ U + i and y i : C ∼ −→ U − i such that ( T, B + , B − , x i , y i ; i ∈ I ) is a pinning for G . Deﬁnition 0.1.

Let I ∗ be the set of all pairs ( i, j ) ∈ I × I such that any elementin U i commutes with any element in U j . There is a unique (up to a labelingconvention) partition I = I ⊔ I into two disjoint subsets such that I × I ⊂ I ∗ and I × I ⊂ I ∗ . Let r = ♯ ( I ) and r = ♯ ( I ) be the cardinalities of I and I .It is known that h = 2 ν/r is an integer (the Coxeter number).For ε ∈ Z , we deﬁne [ ε ] ∈ { , } by ε ≡ [ ε ] mod 2. With this notation, we have ν = r [ ε ] + r [ ε +1] + · · · + r [ ε + h − | {z } h terms . (If h is even, this follows from r + r = r ; if h is odd, we use that r = r = r/ ε ∈ { , } , let us ﬁx the ordering of the elements of I ε : I ε = { i ε , i ε , . . . , i εr ε } . We then deﬁne the sequence j ε ∈ I ν (a distinguished reduced expression) by(0.1.1) j ε = ( j ε , j ε , . . . , j εν )= ( i [ ε ]1 , i [ ε ]2 , . . . , i [ ε ] r [ ε ] , i [ ε +1]1 , i [ ε +1]2 , . . . , i [ ε +1] r [ ε +1] ,i [ ε +2]1 , i [ ε +2]2 , . . . , i [ ε +2] r [ ε +2] , . . . , i [ ε + h − , i [ ε + h − , . . . , i [ ε + h − r [ ε + h − ) . (The upper indices are not exponents.) Thus, the ﬁrst r [ ε ] terms of j ε are theelements of I [ ε ] in their order, the next r [ ε +1] terms are the elements of I [ ε +1] intheir order, and these patterns keep alternating until we accumulate ν entries.For a sequence of indices i = ( i , i , . . . , i n ) ∈ I n of length n ≥

0, we deﬁne themap f i : C n −→ U + by(0.1.2) f i ( a , a , . . . , a n ) = x i ( a ) x i ( a ) . . . x i n ( a n ) . In particular, one can choose i = j ε for ε ∈ { , } , as in (0.1.1) above. Thefollowing fact is well known: Proposition 0.2.

The maps f j , f j are birational isomorphisms from C ν to U + . Proposition 0.2 can be deduced from the proof of [L94, 2.7] using (1.3.1) below;it can also be deduced from [BZ97]. See also 3.12(d).By Proposition 0.2, each map f j ε ( ε ∈ { , } ) induces an isomorphism of ﬁelds f ∗ j ε : [ O ( U + )] ∼ −→ [ O ( C ν )]. OORDINATE RINGS AND BIRATIONAL CHARTS 3

Theorem 0.3.

An element φ ∈ [ O ( U + )] belongs to O ( U + ) if and only if therational function f ∗ j ε ( φ ) ∈ [ O ( C ν )] belongs to O ( C ν ) for ε = 0 and for ε = 1 . The proof of Theorem 0.3 is given in Section 3.The instances of Theorem 0.3 for G of types A and A have been veriﬁed in[L19, Section 6.1]. In the rest of this section, we work out the latter case in detail. Example 0.4.

Let G = SL ( C ), with T , B + , and B − its subgroups of diagonal,upper-triangular, and low-triangular matrices, respectively. Then U + = u = u u u u u u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u , u , u , u , u , u ∈ C ,O ( U + ) = C [ u , u , u , u , u , u ] ,r = 3 ,I = { , , } ,x ( a ) = a , x ( a ) = a

00 0 1 00 0 0 1 , x ( a ) = a ,ν = 6 ,h = 4 . We set I = { } and I = { , } . Then r = 1, r = 2, and j = (2 , , , , , , j = (1 , , , , , ,f j ( a , a , a , a , a , a ) = a

00 0 1 00 0 0 1 a · · · a = a + a a a a a a a + a a a + a a + a a a + a , (0.4.1) f j ( b , b , b , b , b , b ) = b b · · · b

00 0 1 00 0 0 1 = b + b b b + b b + b b b b b b + b b b b + b . (0.4.2) SERGEY FOMIN AND GEORGE LUSZTIG

Proposition 0.2 asserts that each of the 6 parameters a , a , a , a , a , a (resp., b , b , b , b , b , b ) can be expressed as a rational function in the 6 matrix entries u ij (1 ≤ i < j ≤

4) of the unipotent upper-triangular matrix u = ( u ij ) = f j ( a , a , a , a , a , a )(resp., f j ( b , b , b , b , b , b )). For example,(0.4.3) a = u u − u u u u − u , a = u u − u u u − u , a = u u − u u ,a = u ( u u − u ) u u − u , a = u − u u − u u u − u , a = u u . (For explicit formulas for matrices of arbitrary size, see [BFZ96, Theorem 1.4].)Any rational function φ ( u ) = φ ( u , u , u , u , u , u ) ∈ [ O ( U + )]can be rewritten in terms of the parameters a i (resp., b i ), by substituting theappropriate expressions for the u ij from (0.4.1)–(0.4.2):(0.4.4) φ ( u ) = φ ( a + a , a a , a a a , a + a , a a + a a + a a , a + a )= φ ( b + b , b b + b b + b b , b b b , b + b , b b , b + b ) . Theorem 0.3 asserts that φ is a polynomial in the variables u ij if and only if bothfunctions in the parameters a i (resp., b i ) appearing in (0.4.4) are polynomial.Theorem 0.3 can also be restated entirely in terms of the parameters a i and b i .As observed in [L94] (in a more general setting of an arbitrary pair of reducedexpressions), the birational map relating the ν -tuples ( a i ) and ( b j ) to each othercan be obtained as a composition of simple birational transformations associatedto individual braid moves. In our example, calculations based on those rules yieldthe following formulas expressing a , a , a , a , a , a in terms of b , b , b , b , b , b :(0.4.5) a = b b b b R , a = RQ , a = RP , a = P QR , a = b b b Q , a = b b b P , where(0.4.6) P = b b + b b + b b ,Q = b b + b b + b b ,R = b b b + b b b + b b b + b b b + b b b Theorem 0.3 (in this example) says that a polynomial Φ( a , a , a , a , a , a ) liesin the subring C [ a + a , a a , a a a , a + a , a a + a a + a a , a + a ] ⊂ C [ a , a , . . . , a ](cf. (0.4.1)) if and only if substituting (0.4.5)–(0.4.6) into Φ( a , a , a , a , a , a )produces a polynomial (rather that merely a rational function) in b , b , . . . , b .(An alternative criterion would be to substitute (0.4.3) into Φ( a , a , a , a , a , a )and verify that the result lies in C [ u , u , . . . , u ].) OORDINATE RINGS AND BIRATIONAL CHARTS 5

1. Preliminaries on the Weyl group and weights

Let ι : G −→ G be the unique automorphism of G such that ι ( x i ( a )) = y i ( a ), ι ( y i ( a )) = x i ( a ) for i ∈ I, a ∈ C and ι ( t ) = t − for t ∈ T . We have ι = 1.Let Y = Hom( C ∗ , T ) and X = Hom( T, C ∗ ). We write the operation in each ofthese groups as addition. Let h , i : Y × X −→ Z be the obvious perfect pairing.For i ∈ I , let α i ∈ X be the simple root corresponding to U i and let ˇ α i be thecorresponding simple coroot. Let X + = { λ ∈ X | h ˇ α i , λ i ≥ ∀ i ∈ I } . For i ∈ I ,the fundamental weight ω i ∈ X is deﬁned by the condition h ˇ α j , ω i i = δ ij for j ∈ I .We have ω i ∈ X + .For i ∈ I , we denote by P i the (parabolic) subgroup of G generated by B + together with S j ∈ I −{ i } U − j . For i ∈ I deﬁne s i : Y −→ Y by χ χ − h χ, α i i ˇ α i . Let W be the subgroupof Aut( Y ) generated by { s i ; i ∈ I } . This is a Coxeter group with the simplereﬂections { s i | i ∈ I } and with the length function that we denote by w

7→ | w | .Let w ∈ W be the unique element such that | w | = ν , the maximal possiblelength. Now W acts on X by the rule h χ, w ( λ ) i = h w − ( χ ) , λ i for χ ∈ Y , λ ∈ X .For i ∈ I , let W ω i be the W -orbit of the weight ω i in X and let W I −{ i } be thesubgroup of W generated by { s j ; j ∈ I − { i }} . This is exactly the stabilizer of ω i with respect to the W -action on X .Let N T be the normalizer of T in G . Now N T /T acts in an obvious way on Y .This gives an embedding of N T /T ֒ → Aut( Y ) that identiﬁes N T /T with W .For i ∈ I , set ˙ s i = x i (1) y i ( − x i (1) ∈ N T and ¨ s i = y i (1) x i ( − y i (1) ∈ N T .We extend this to deﬁne representatives ˙ w ∈ N T and ¨ w ∈ N T for all w ∈ W byrequiring that for any w, w ′ , w ′′ ∈ W satisfying w ′′ = ww ′ and | w ′′ | = | w | + | w ′ | ,we have ˙ w ′′ = ˙ w ˙ w ′ and ¨ w ′′ = ¨ w ¨ w ′ .For ε ∈ { , } , we set z ε = Y i ∈ I ε s i ∈ W. (Here the factors commute, so the order does not matter.) Lemma 1.3 (see [Bo, Chapter V, §

6, Ex. 2]) . We have (1.3.1) | z [ ε ] z [ ε +1] . . . z [ ε + h − | = | z [ ε ] | + | z [ ε +1] | + · · · + | z [ ε + h − | = ν. It follows that, if I ′ ⊂ I [ ε ] , I ′′ ⊂ I [ ε + l +1] , w ′ = Y i ∈ I ′ s i , w ′′ = Y i ∈ I ′′ s i , then (1.3.2) | wz [ ε +1] . . . z [ ε + l ] w ′ | = | w | + | z [ ε +1] | + | z [ ε +2] | + · · · + | z [ ε + l ] | + | w ′ | , provided either (a) l ∈ { , , . . . , h − } or (b) w = 1 and l ∈ { , , . . . , h − } , or (c) w ′ = 1 and l ∈ { , , . . . , h − } . SERGEY FOMIN AND GEORGE LUSZTIG

We denote Y ′ = { ω i | i ∈ I } , (1.4.1) Y ′′ = { w ω i | i ∈ I } . (1.4.2)If γ ∈ Y ′ , then h ˇ α j , γ i ≥ j ∈ I . If γ ∈ Y ′′ , then h ˇ α j , γ i ≤ j ∈ I . Fix ε ∈ { , } . Recall that j ε = ( j ε , j ε , . . . , j εν ) was deﬁned in (0.1.1). For k ∈ { , , . . . , ν } , we set γ εk = s j εν . . . s j εk +1 s j εk ω j εk , ˜ γ εk = s j εν . . . s j εk +2 s j εk +1 ω j εk . In order to represent γ εk and ˜ γ εk more explicitly, we will need to introduce someadditional notation. For l ∈ { , , . . . , h − } and i ∈ I [ ε + h − l ] , let v εl,i = z [ ε + h − z [ ε + h − · · · z [ ε + h − l +1] s i ω i . Let X ε ⊔ X ε . . . ⊔ X εh be the partition of { , , . . . , ν } given by X ε = { , , . . . , r [ ε ] } , X ε = { r [ ε ] + 1 , r [ ε ] + 2 , . . . , r [ ε ] + r [ ε +1] } , X ε = { r [ ε ] + r [ ε +1] + 1 , r [ ε ] + r [ ε +1] + 2 , . . . , r [ ε ] + r [ ε +1] + r [ ε +2] } , · · · · · · · · · · · · Since s j s j ′ = s j ′ s j for j, j ′ in the same I ε and s j ω j ′ = ω j ′ if j = j ′ , we see that γ εk = v εl,j εk if l ∈ { , , . . . , h − } , k ∈ X εl +2 ⊂ { r + 1 , r + 2 , . . . , ν } , ˜ γ εk = v εl,j εk if l ∈ { , , . . . , h − } , k ∈ X εl ⊂ { , , , . . . , ν − h } ˜ γ εk ∈ Y ′ if k ∈ X εh − ⊔ X εh = { ν − h + 1 , ν − h + 2 , . . . , ν } ,γ εk ∈ Y ′′ if k ∈ X ε ⊔ X ε = { , , . . . , r } . For k, k ′ in { , , . . . , ν } such that ( j εk , j εk ′ ) / ∈ I ∗ (see Deﬁnition 0.1), we set γ εk,k ′ = s j εν . . . s j εk +1 s j εk ω j εk ′ . From the deﬁnitions we see that under these assumptions,(a) γ εk,k ′ is either equal to one of the elements γ εk ′′ or lies in Y ′ . OORDINATE RINGS AND BIRATIONAL CHARTS 7

Lemma 1.6.

Let γ = v εl,i where ε ∈ { , } , i ∈ I [ ε + h − l ] , l ∈ { , , . . . , h − } . (a) If j ∈ I [ ε + h ] , then h ˇ α j , γ i ≥ . (b) There exists j ∈ I [ ε + h +1] such that h ˇ α j , γ i < .Proof. Let us prove (a). We have h ˇ α j , γ i = h s i z [ ε + h − l +1] . . . z [ ε + h − ˇ α j , ω i i . To show that this is nonnegative, it suﬃces to prove that(c) s i z [ ε + h − l +1] . . . z [ ε + h − ˇ α j is a positive coroot.We have | s i z [ ε + h − l +1] · · · z [ ε + h − | = | s i | + | z [ ε + h − l +1] | + · · · + | z [ ε + h − | . (Use i ∈ I [ ε + j − l ] and (1.3.2) which holds since l − ≤ h − | s i z [ ε + h − l +1] · · · z [ ε + h − s j | = | s i | + | z [ ε + h − l +1] | + · · · + | z [ ε + h − | + | s j | . The latter follows from (1.3.2) since l − ≤ h −

2. This proves (a).Now suppose that (b) does not hold. Then by (a), we have h ˇ α j , γ i ≥ j ∈ I . Therefore γ ∈ X + . Since γ ∈ W ω i , we have γ = ω i . Hence z [ ε + h − · · · z [ ε + h − l +1] s i is in the stabilizer of ω i , i.e., in W I −{ i } . This contradicts | z [ ε + h − · · · z [ ε + h − l +1] s i | = | z [ ε + h − | + · · · + | z [ ε + h − l +1] | + | s i | which holds by (1.3.2). (cid:3) Lemma 1.7.

Let ε ∈ { , } , i ∈ I [ ε + h − l ] , l ∈ { , , . . . , h − } . Let w ∈ W be theunique element of minimal length in { w ∈ W | w ω i = v εl,i } . (a) If j ∈ I [ ε + h ] , then | s j w | > | w | . (b) There exists j ∈ I [ ε + h +1] such that | s j w | < | w | .Proof. Assume that j ∈ I satisﬁes | s j w | < | w | . Then | w − s j | < | w − | , and using[BZ97, Proposition 2.6] we see that h ˇ α j , v εl,i i <

0. Now using Lemma 1.6(a), wededuce that j / ∈ I [ ε + h ] , proving (a). Now suppose (b) does not hold. Then by (a), | s j w | > | w | for all j ∈ I . Hence w = 1 and v εl,i = ω i so that h ˇ α j , v ελ,i i ≥ j ∈ I . This contradicts Lemma 1.6(b). (cid:3) Let ε ∈ { , } . Denote(1.8.1) Y ε = { v εl,i | i ∈ I [ ε + h − l ] , l ∈ { , , . . . , h − }} . We are going to show that(a) all the weights in Y ε are distinct.To prove this, suppose that v εl,i = v εl ′ ,i ′ where i ∈ I [ ε + h − l ] , i ′ ∈ I [ ε + h − l ′ ] , and l, l ′ ∈ { , , . . . , h − } . Then W ω i = W ω i ′ and therefore ω i = ω i ′ and so i = i ′ . SERGEY FOMIN AND GEORGE LUSZTIG

Suppose that l = l ′ . Without loss of generality, we may assume that l > l ′ .Setting e = l − l ′ ≥ z [ ε + h − l + e ] z [ ε + h − l + e − · · · z [ ε + h − l +1] s i ω i = s i ω i . Hence(b) s i z [ ε + h − l + e ] z [ ε + h − l + e − . . . z [ ε + h − l +1] s i ∈ W I −{ i } .From (1.3.2) we see that | s i z [ ε + h − l + e ] z [ ε + h − l + e − . . . z [ ε + h − l +1] s i | = | s i z [ ε + h − l + e ] | + | z [ ε + h − l + e − | + · · · + | z [ ε + h − l +1] | + | s i | which contradicts (b). Hence l = l ′ and (a) is proved.We note that(1.8.2) ♯ ( Y ε ) = h − X l =1 r [ ε + h − l ] = h X l =1 r [ ε + h − l ] − r [ ε +1] − r [ ε ] = ν − r. Lemma 1.9.

With the notation introduced in (1.4.1) , (1.4.2) , (1.8.1) , we have Y ε ∩ Y ′ = ∅ and Y ε ∩ Y ′′ = ∅ (assuming G is not of type A ).Proof. If γ ∈ Y ε , then h ˇ α j , γ i < j by Lemma 1.6(b); thus γ / ∈ Y ′ by 1.4.Assume that v εl,i = w ω j for some l ∈ { , , . . . , h − } , i ∈ I [ ε + h − l ] , j ∈ I . Then ω i and ω j are in the same W -orbit. Hence i = j and we have z [ ε + h − z [ ε + h − · · · z [ ε + h − l +1] s i ω i = z [ ε + h − z [ ε + h − · · · z [ ε + h − l +1] z [ ε + h − l ] · · · z [ ε ] ω i . This implies that s i z [ ε + h − l ] · · · z [ ε ] ω i = ω i , i.e., s i z [ ε + h − l ] · · · z [ ε ] lies is in the stabi-lizer of ω i , that is, in W I −{ i } . So any reduced expression of it does not contain s i .If l ≥

2, this contradicts | s i z [ ε + h − l ] · · · z [ ε ] | = | s i | + | z [ ε + h − l ] | + · · · + | z [ ε ] | . since h − l + 1 + 1 ≤ h . Therefore l = 1 and moreover any reduced expression of s i w does not contain s i . But this cannot happen if G is of type other than A .Indeed, for some ε ∈ { , } , z [ ε + h − z [ ε + h − · · · z [ ε + h − l +1] z [ ε + h − l ] · · · z [ ε ] gives a reduced expression of w such that s i appears in the ﬁrst group z [ ε + h − . If s i does not appear in any other group, then there are only two factors and h = 2.But h > A . (cid:3) OORDINATE RINGS AND BIRATIONAL CHARTS 9

2. An irreducibility property

In this section, we prove the following result.

Proposition 2.1.

Let w, w ′ ∈ W . The set U + ∩ ( B − ˙ w ′ B + ˙ w − ) is empty if w ′ w ;it is smooth and irreducible, of dimension ν − | w ′ | , if w ′ ≤ w . For y ∈ W , let U + y = { u ∈ U + , ˙ y − u ˙ y ∈ U − } ,U + y = { u ∈ U + , ˙ y − u ˙ y ∈ U + } . The multiplication map U + y × U + y ∼ −→ U + is an isomorphism of varieties. For x ∈ G and a subgroup C of G , we shall write x C instead of xCx − . For w ∈ W , we shall write w C instead of ˙ w C .We denote by B the variety of Borel subgroups in G . For B ′ , B ′′ ∈ B , there isa unique w ∈ W such that for some x ′ , x ′′ in G we have B ′ = x ′ B + , B ′′ = x ′′ B + , x ′− x ′′ ∈ B + ˙ wB + ; we then write w = pos( B ′ , B ′′ ).For z, z ′ ∈ W , we denote R z,z ′ = { B ∈ B | pos( B − , B ) = z ′ , pos( B, B + ) = z − w } . It is known [KL79] that R z,z ′ is nonempty if and only if z ≤ z ′ . We show: Proposition 2.4. If z ≤ z ′ then R z,z ′ is smooth, irreducible of dimension | z ′ |−| z | .Proof. We shall adapt an argument in [L98, 1.4] by replacing R by C . The set˜ R z,z ′ = { B ∈ B | pos( B − , B ) = z ′ , pos( B, w z B − ) = w } . is an open nonempty subset in { B | pos( B − , B ) = z ′ } ∼ = C | z ′ | . Hence it is smoothirreducible of dimension | z ′ | . Clearly, the map ( B, u ) u B is an isomorphism R z,z ′ × ( U − ∩ w z U − ) ∼ −→ ˜ R z,z ′ . Now the claim follows since U − ∩ w z U − ∼ = C | z | . (cid:3) A result related to Proposition 2.4 holds for the analogue of R z,z ′ over aﬁnite ﬁeld F q . By [KL79], the number of F q -rational points in this analogue isgiven by the polynomial R z,z ′ in loc.cit. evaluated at q . By the inductive formulain loc.cit. , the latter is monic of degree | z ′ | − | z | . Proof of Proposition 2.1.

Setting B = x B + , we can reformulate Proposition 2.4as the statement that { xB + ∈ G/B + | pos( B − , x B + ) = z ′ , pos( x B + , B + ) = z − w } = (( U + w z ˙ B + ) ∩ ( B − ( w z ′− )˙ B + )) /B + = ( U + w z ( w z )˙) ∩ ( B − ( w z ′− )˙ B + ) is smooth, irreducible of dimension | z ′ | − | z | if z ≤ z ′ , and is empty if z z ′ .Replacing here w z, w z ′ by w, w ′ we deduce that ( U + w ˙ w ) ∩ ( B − ˙ w ′ B + ) is smooth,irreducible of dimension | w | − | w ′ | if w ′ ≤ w , and is empty if w ′ w .Using 2.2, we see that the map( U + w ˙ w ) ∩ ( B − ˙ w ′ B + ) × U + w −→ ( U + ˙ w ) ∩ ( B − ˙ w ′ B + )given by ( u ′ ˙ w, u ′′ ) u ′ u ′′ ˙ w with u ′ ∈ U + w such that u ′ ˙ w ∈ B − ˙ w ′ B + and u ′′ ∈ U + w is an isomorphism of varieties. Since U + w ∼ = C ν −| w | , we conclude that( U + ˙ w ) ∩ ( B − ˙ w ′ B + ) is smooth, irreducible of dimension ν − | w ′ | if w ′ ≤ w , and isempty if w ′ w . This completes the proof of Proposition 2.1. (cid:3)

3. Proof of Theorem 0.3

When G is of type A , we have j = j and the theorem is trivial. For the restof this section, we assume that G is not of type A . Fix i ∈ I . Let V i = { f ∈ O ( G ) | f ( utg ) = ω i ( t ) f ( g ) ∀ u ∈ U − , t ∈ T, g ∈ G } . The group G acts on V i by g : f g f where ( g f )( g ) = f ( gg ). There isa unique f ∈ V i such that f ( gu ) = f ( g ) for all g ∈ G, u ∈ U + and such that f (1) = 1. We denote it by ∆. (Note that ∆ depends on the choice of i .)We show that ∆( ˙ s i ) = 0. Setting g c = y i ( − c ) ˇ α i ( c − ) x i ( c ) for c ∈ C ∗ , we seethat lim c →∞ g c = ˙ s i in G . We have ∆( g c ) = ω i ( ˇ α i ( c − )) = c − , so ∆( ˙ s i ) = lim c →∞ c − = 0.It follows that ∆ vanishes on U − ˙ s i B + , hence also on the closure(a) Z = U − ˙ s i B + = ∪ w ; s i ≤ w U − ˙ wB + = ∪ w ∈ W − W I −{ i } U − dwB + = G − ( U − P i ).The function ∆ is preserved (up to a nonzero scalar) by the action of P i on V i .Hence ∆ takes only nonzero values on the open subset U − P i of G , implying that(b) Z = { g ∈ G ; ∆( g ) = 0 } . Deﬁnition 3.2.

Let i ∈ I and γ ∈ W ω i . Following [BZ97], we set ∆ γ = ¨ w ∆,where w ∈ W is such that wω i = γ . This does not depend on the choice of w . Inparticular, ∆ ω i = ∆.Let ∆ + γ be the restriction of ∆ γ to U + . For u ∈ U + , we have ∆ + γ ( u ) = ∆( u ¨ w ),with w as above. (Note that ∆ + γ is not identically zero on U + . Otherwise we wouldhave ∆( U − B + ¨ w ) = 0; but U − B + ¨ w is dense in G ; hence ∆ = 0, a contradiction.)We will also use the notation(3.2.1) Z γ = { u ∈ U + | ∆ + γ ( u ) = 0 } . OORDINATE RINGS AND BIRATIONAL CHARTS 11

Lemma 3.3.

Let i ∈ I , w ∈ W , and γ = wω i . Then (a) Z γ = S y ∈ W − W I −{ i } ( U + ∩ ( U − ˙ yB + ˙ w − )) ; (b) if s i w then Z γ is empty; (c) if s i ≤ w , then Z γ is the closure of U + ∩ ( U − ˙ s i B + ˙ w − ) (a smooth irreduciblevariety of dimension ν − ).Proof. Using 3.1(a),(b), we get Z γ = { u ∈ U + ; ∆( u ¨ w ) = 0 } = { u ∈ U + ; u ¨ w ∈ Z } = { u ∈ U + | u ¨ w ∈ ∪ y ∈ W − W I −{ i } U − dyB + } , and (a) follows. (We used that B + ¨ w − = B + ˙ w − .)By Proposition 2.1, U + ∩ ( U − ˙ s i B + ˙ w − ) is smooth irreducible of dimension ν − s i ≤ w and is empty if s i w . Moreover if y satisﬁes s i < y , thenthe same Proposition shows that U + ∩ ( U − ˙ yB + ˙ w − ) is either empty or irreducibleof dimension ν − | y | ≤ ν −

2. Since, by Krull’s theorem, Z γ is either empty or ofpure dimension ν −

1, the statements (b) and (c) follow. (cid:3)

Lemma 3.4.

Let ε ∈ { , } , γ ∈ Y ε . Then: (a) Z γ (see (3.2.1) ) is an irreducible variety of dimension ν − ; (b) for any j ∈ I [ ε + h ] and any c ∈ C we have Z γ x j ( c ) ⊂ Z γ ; (c) there exists j ∈ I [ ε + h +1] such that for some c ∈ C we have Z γ x j ( c )

6⊂ Z γ .Proof of (a) . We write γ = wω i with i ∈ I, w ∈ W . By Lemma 1.9, we have γ / ∈ Y ′ hence w / ∈ W I −{ i } so that s i ≤ w . Now (a) follows from Lemma 3.3(c). (cid:3) Proof of (b) . We write γ = wω i where i ∈ I and w ∈ W is the unique element ofminimal length in { w ∈ W | w ω i = γ } . Using Lemma 3.3(c), we see that it isenough to show that for j, c as in (b) we have( U + ∩ ( U − ˙ yB + ˙ w − )) x j ( c ) ⊂ U + ∩ ( U − ˙ yB + ˙ w − )for any y ∈ W − W I −{ i } . This follows from ˙ w − x j ( c ) ∈ U + ˙ w − which in turnfollows from | s j w | > | w | (see 1.7(a)) or equivalently | w − s j | > | w − | . (cid:3) Proof of (c) . Suppose that (c) does not hold. Using (b), we see that for any j ∈ I and any c ∈ C we have Z γ x j ( c ) ⊂ Z γ . Since the elements x j ( c ) for various j, c generate the group U + , it follows that Z γ U + ⊂ Z γ . Since Z γ = ∅ , we concludethat Z γ = U + . This contradicts Lemma 3.3(b),(c). (cid:3) Lemma 3.5.

Let γ ∈ Y and γ ′ ∈ Y . Then every irreducible component of Z γ ∩ Z γ ′ has dimension ≤ ν − .Proof. By Lemma 3.4(c) with ε = 0, there exist j ∈ I [ h +1] and c ∈ C such that Z γ x j ( c )

6⊂ Z γ . By Lemma 3.4(b) with ε = 1, we have Z γ ′ x j ( c ) ⊂ Z γ ′ . Therefore Z γ = Z γ ′ . Since Z γ , Z γ ′ are irreducible of dimension ν −

1, the lemma follows. (cid:3)

Consider the partition(3.6.1) U + = G z ∈ W U + ( z )where U + ( z ) = U + ∩ B − ˙ zB − is smooth and irreducible of dimension | z | (cf. Proposition 2.1 with ( w, w ′ ) replacedby ( w , zw )). Furthermore, the closure of U + ( z ) in U + is equal to G z ′ ; z ′ ≤ z U + ( z ′ ).It follows that U + ( w ) is open dense in U + . For z ∈ W , we set U − ( z ) = U − ∩ B + ˙ zB + = ι ( U + ( z ))(see 1.1 for the deﬁnition of ι ). Then U − = G z ∈ W U − ( z )and U − ( w ) is open dense in U − .Let A : U + ( w ) ∼ −→ U + ( w )be the composition U + ( w ) ∼ −→ { B ∈ B | pos( B + , B ) = pos( B, B − ) = w } ∼ −→ U − ( w ) ∼ −→ U + ( w )where the ﬁrst isomorphism is u u B − , the second isomorphism is the inverse of u ′ u ′ B + , and the third isomorphism is the restriction of ι .We will show that A is an involution. For u ∈ U + , we have u B − = ι ( A ( u )) B + .Replacing u by A ( u ) we obtain A ( u ) B − = ι ( A ( u )) B + . Applying ι , we obtain ι ( A ( u )) B + = A ( u ) B − , i.e., u B − = A ( u ) B − . Hence u = A ( u ) and A = 1. Let ε ∈ { , } . We denote V ε = { u ∈ U + | ∆ + γ εk ( u ) = 0 , ∆ +˜ γ εk ( u ) = 0 , k = { , , . . . , ν }} . This set is open in U + . It is also nonempty, since each of ∆ + γ εk , ∆ +˜ γ εk is not identicallyzero on U + . We denote V ε ∗ = { u ∈ U + ( w ) | A ( u ) ∈ V ε } = U + ( w ) ∩ A − ( V ε ) } . This set is open in U + . It is also nonempty, as it is the intersection of two opennonempty subsets of U + . We shall need the following result from [BZ97], [FZ99]: OORDINATE RINGS AND BIRATIONAL CHARTS 13

Lemma 3.8.

The map f j ε : C ν −→ U + restricts to an isomorphism ( C ∗ ) ν ∼ −→ V ε ∗ . Using the results in 1.5, we see that(3.9.1) V ε ∗ = { u ∈ U + ( w ) | ∆ + γ ( Au ) = 0 for all γ ∈ Y ε ∪ Y ′ ∪ Y ′′ } . If γ = ω i with i ∈ I , then ∆ + γ is the function u ∆ ω i ( u ) = ∆ ω i (1) = 1(a constant function). If γ = w ω i with i ∈ I , then ∆ + γ ( u ) = 0 for any u ∈ U + ( w ).(Indeed, writing u = u ′ ˙ w b ′ with u ′ ∈ U − , b ′ ∈ B − , so that u ˙ w = u ′ tu with t ∈ T, u ∈ U + , we have ∆ + γ ( u ) = ∆ ω i ( u ˙ w ) = ∆ ω i ( u ′ tu ) = ω i ( t ) = 0.) It followsthat Y ′ and Y ′′ can be eliminated from (3.9.1), and we conclude that(3.9.2) V ε ∗ = { u ∈ U + ( w ) | ∆ + γ ( Au ) = 0 for all γ ∈ Y ε } . Lemma 3.10. dim( U + ( w ) − ( V ∗ ∪ V ∗ )) ≤ ν − .Proof. From (3.9.2), we obtain U + ( w ) − ( V ∗ ∪ V ∗ ) = [ ( γ,γ ′ ) ∈ Y × Y A ( U + ( w ) ∩ Z γ ∩ Z γ ′ ) . It remains to use that dim( Z γ ∩ Z γ ′ ) ≤ ν − γ, γ ′ ) ∈ Y × Y (see 3.5). (cid:3) Lemma 3.11.

Let i = ( i , i , . . . , i n ) ∈ I n be a reduced expression, that is, theelement w = s i . . . s i n ∈ W has length n . Let ′ f i : ( C ∗ ) n −→ U + be the restriction of the map f i in (0.1.2) . Then ′ f i is an isomorphism of ( C ∗ ) n onto an open subset ′ U + i of U + ( w ) .Proof. Induction on n . For n = 0, the result is obvious. Assume that n ≥

1. Let i ′ = ( i , i , . . . , i n − ) ∈ I n − and let w ′ = s i . . . s i n − . The map U + ( w ′ ) × C ∗ −→ U + ( w )( u ′ , c ) u ′ x i n ( c )is an isomorphism of U + ( w ′ ) × C ∗ onto an open subset of U + ( w ). It restricts toan isomorphism of ′ U + i ′ × C ∗ onto an open subset ′ U + i of U + ( w ). (cid:3) Let ε ∈ { , } and let i ∈ I [ ε + h +1] . Deﬁne k ∈ X ǫh (in the notation of 1.5) by j εk = i . Let C νi (resp. ′ C νi ) be the subset of C ν consisting of all ( a , a , . . . , a ν ) suchthat a l ∈ C ∗ for l = k whereas a k ∈ C (resp. a k = 0). By restricting f j ε : C ν −→ U + to C νi (resp. ′ C νi ), we obtain maps f j ε ; i : C νi −→ U + and ′ f j ε ; i : C νi −→ U + .It follows from Lemma 3.11 that(a) ′ f j ε ; i is an isomorphism of ′ C νi onto an open subset ′ U + j ε ; i of U + ( w s i ). We next prove that(b) f j ε ; i is an isomorphism of C νi onto an open subset U + j ε ; i of U + ( w ) ∪ U + ( w s i )containing ′ U + j ε ; i . Proof.

The map U + ( w s i ) × C −→ U + , ( u ′ , c ) u ′ x i n ( c ), is an isomorphism of U + ( w s i ) × C onto an open subset of U + ( w ) ∪ U + ( w s i ). It restricts to anisomorphism of ′ U + j ε ; i × C onto an open subset U + j ε ; i of U + ( w ) ∪ U + ( w s i ). (cid:3) The following is a special case of Lemma 3.11:(c) ′ f j ε is an isomorphism of ( C ∗ ) ν onto an open subset ′ U + j ε of U + ( w ).From (c), we deduce that(d) f j ε is a birational isomorphism from C ν to U + . Lemma 3.13.

Let U be the open subset of U + deﬁned by (3.13.1) U = V ∗ ∪ V ∗ ∪ [ ε ∈{ , } ,i ∈ I [ ε + h +1] U + j ε ; i . Then dim( U + − U ) ≤ ν − .Proof. Using the partition (3.6.1), it is enough to show that(3.13.2) dim( U + ( z ) ∩ ( U + − U )) ≤ ν − z ∈ W .Case 1: z = w . We have U + ( w ) ∩ ( U + − U ) ⊂ U + ( w ) − ( V ∗ ∪ V ∗ ) . Thereforedim( U + ( w ) ∩ ( U + − U )) ≤ dim( U + ( w ) − ( V ∗ ∪ V ∗ )) ≤ ν − z = w s i with i ∈ I . Deﬁne ε ∈ { , } by i ∈ I [ ε + h +1] . Then U + ( w s i ) ∩ ( U + − U ) ⊂ U + ( w s i ) ∩ ( U + − U + j ε ; i ) ⊂ U + ( w s i ) − ′ U + j ε ; i . The last diﬀerence has dimension ≤ ν − U + ( w s i ) is irreducibleof dimension ν − ′ U + j ε ; i is a nonempty open subset of U + ( w s i ).Case 3: z is not of the form w or w s i . Then | z | ≤ ν −

2. Thereforedim( U + ( z )) ≤ ν − (cid:3) Proof of Theorem 0.3.

The “only if” part of Theorem 0.3 is obvious. Letus prove the “if” statement. Consider φ ∈ [ O ( U + )] such that f ∗ j ε ( φ ) ∈ [ O ( C ν )]belongs to O ( C ν ) for ε = 0 and for ε = 1. From our assumption we see that φ | V ε ∗ isregular for ε ∈ { , } (see Lemma 3.8) and that φ | U + j ε ; i is regular for ε ∈ { , } and i ∈ I [ ε + h +1] (see 3.12(b)). Hence φ is regular on U . Using this and Lemma 3.13,we conclude that φ is regular on U + . Theorem 0.3 is proved. (cid:3) OORDINATE RINGS AND BIRATIONAL CHARTS 15

4. The study of O ( G/U − ) For i = ( i , i , . . . , i ν ) ∈ I ν , we deﬁne the maps f i ;+ : C ν × T −→ G/U − f i ; − : C ν × T −→ G/U − by f i ;+ ( a , a , . . . , a ν , t ) = x i ( a ) x i ( a ) . . . x i ν ( a ν ) t U − ,f i ; − ( a , a , . . . , a ν , t ) = y i ( a ) y i ( a ) . . . y i ν ( a ν ) t ˙ w U − . Of particular interest to us are the cases where i = j ε , for ε ∈ { , } , as in (0.1.1).Proposition 0.2 implies that both f j ε ;+ and f j ε ; − are birational isomorphisms from C ν × T to G/U − . Consequently the maps f ∗ j ε ;+ and f ∗ j ε ; − are well-deﬁned isomor-phisms [ O ( G/U − )] ∼ −→ [ O ( C ν × T )]. Theorem 4.2.

An element φ ∈ [ O ( G/U − )] belongs to O ( G/U − ) if and only ifeach of the four rational functions f ∗ j ;+ ( φ ) , f ∗ j ;+ ( φ ) , f ∗ j ; − ( φ ) , f ∗ j ; − ( φ ) ∈ [ O ( C ν × T )] belongs to O ( C ν × T ) . The proof of Theorem 4.2 will rely on the following statement.

Lemma 4.3.

We have (4.3.1) dim(

G/U − − (( U + T U − ∪ U − T ˙ w U − ) /U − )) ≤ dim( G/U − ) − . Proof.

The inequality (4.3.1) is equivalent todim(

G/B − − (( U + B − ∪ ( U − ˙ w B − ) /B − )) ≤ dim( G/B − ) − , which is equivalent to the inequalitydim( B − ( { B ∈ B | pos( B, B + ) = w } ∪ { B ∈ B ; pos( B, B − ) = w } ) ≤ dim B − z ∈ W − { } and z ′ ∈ W − { w } , we havedim( { B ∈ B | pos( B − , B ) = z ′ , pos( B, B + ) = z − w } ) ≤ ν − . The last claim follows from Proposition 2.4 since | z ′ | − | z | ≤ ν − (cid:3) Proof of Theorem 4.2.

The “only if” statement in the theorem is obvious.Let us prove the “if” statement. Thus, let φ ∈ [ O ( G/U − )] be such that the fourconditions in the theorem are satisﬁed. We need to show that φ ∈ O ( G/U − ).Suppose G is of type A . Then φ is regular on ( U + T U − ∪ U − T ˙ w U − ) /U − .Hence by (4.3.1), it is regular on G/U − , and we are done. In the rest of the proof, we assume that G is of type other than A .We ﬁrst show that φ regular on the open subset U + T U − /U − of G/U − . Withthe notation as in 3.7 and 3.12, we see as in the proof in 3.14 that φ is regular oneach of the following open subsets of U + T U − /U − : • V ε T U − /U − , for ε ∈ { , } ; • U + j ε ; i T U − /U − , for ε ∈ { , } and i ∈ I [ ε + h +1] .Hence φ is regular on the union of these subsets, i.e., on U T U − /U − (here U ⊂ U + is as in (3.13.1)). By Lemma 3.13, we havedim(( U + T U − − U T U − ) /U − ) ≤ ν + r − G/U − ) − . Since φ is regular on U T U − /U − , it follows that φ is regular on U + T U − /U − .We next show that φ is regular on the open subset U − T ˙ w U − /U − of G/U − .We denote V ε −∗ = ι ( V ε ∗ ) ⊂ U − (cf. 3.7) and U − j ε ; i = ι ( U + j ε ; i ) ⊂ U − (cf. 3.12).As in the proof in 3.14 (with U + replaced by U − ), we see that φ is regular oneach of the following open subsets of U − T ˙ w U − /U − : • V ε − T ˙ w U − /U − , for ε ∈ { , } ; • U − j ε ; i T ˙ w U − /U − , for ε ∈ { , } and i ∈ I [ ε + h +1] .Hence φ is regular on the union of these subsets, i.e., on U − T ˙ w U − /U − where U − = ι ( U ) ⊂ U − . By Lemma 3.13 (with U − instead of U + ), we havedim(( U − T ˙ w U − − U − T ˙ w U − ) /U − ) ≤ ν + r − G/U − ) − . Since φ is regular on U − T ˙ w U − /U − , it follows that φ is regular on U − T ˙ w U − /U − .Thus φ is regular on the open subset (( U + T U − ) ∪ ( U − T ˙ w U + )) /U − of G/U − .Using this and Lemma 4.3, we conclude that φ is regular on G/U − , as desired. (cid:3)

5. The study of O ( G ) For i = ( i , i , . . . , i ν ) ∈ I ν and i ′ = ( i ′ , i ′ , . . . , i ′ ν ) ∈ I ν , we deﬁne the maps f i , i ′ ; ± : C ν × T × C ν −→ G,f i , i ′ ; ∓ : C ν × T × C ν −→ G by f i , i ′ ; ± ( a , a , . . . , a ν , t, b , b , . . . , b ν )= x i ( a ) x i ( a ) . . . x i ν ( a ν ) t y i ′ ( b ) y i ′ ( b ) . . . y i ′ ν ( b ν ) ,f i , i ′ ; ∓ ( a , a , . . . , a ν , t, b , b , . . . , b ν )= y i ( a ) y i ( a ) . . . y i ν ( a ν ) t − x i ′ ( b ) x i ′ ( b ) . . . x i ′ ν ( b ν ) . Thus, f i , i ′ ; ∓ = ιf i , i ′ ; ± .Let j ε , for ε ∈ { , } , be as in (0.1.1). From Proposition 0.2 one can deduce thatfor each of the four possible pairs ( ε, ε ′ ) ∈ { , } × { , } , both maps f j ε , j ε ′ ; ± and f j ε , j ε ′ ; ∓ are birational isomorphisms from C ν × T × C ν to G . It follows that both f ∗ j ε , j ε ′ ; ± and f ∗ j ε , j ε ′ ; ∓ are well deﬁned isomorphisms [ O ( G )] ∼ −→ [ O ( C ν × T × C ν )]. OORDINATE RINGS AND BIRATIONAL CHARTS 17

Theorem 5.2.

An element φ ∈ [ O ( G )] belongs to O ( G ) if and only if for each ofthe four possible pairs ( ε, ε ′ ) ∈ { , } × { , } , both rational functions f ∗ j ε , j ε ′ ; ± ( φ ) , f ∗ j ε , j ε ′ ; ∓ ( φ ) ∈ [ O ( C ν × T × C ν )] belong to O ( C ν × T × C ν ) . The proof of Theorem 5.1 will rely on the following statement.

Lemma 5.3. dim( G − (( U + T U − ) ∪ ( U − T U + ))) ≤ dim( G ) − .Proof. Using the Bruhat decomposition, we obtain: G − (( U + T U − ) ∪ ( U − T U + )) = ( G − ( B + U − )) ∩ ( G − ( B − U + ))= (cid:16) [ w ∈ W −{ } B + ˙ wU − (cid:17) ∩ (cid:16) [ w ′ ∈ W −{ } B − ˙ w ′ U + (cid:17) = [ w,w ′ in W −{ } ( B + ˙ wU − ) ∩ ( B − ˙ w ′ U + ) . It is therefore enough to show that for any w = 1 and w ′ = 1, we have(5.3.1) dim(( B + ˙ wU − ) ∩ ( B − ˙ w ′ U + )) ≤ dim( G ) − . This is clear if either B + ˙ wU − or B − ˙ w ′ U + has dimension ≤ dim( G ) −

2. Thuswe can assume that dim( B + ˙ wU − ) = dim( B − ˙ w ′ U + ) = dim( G ) − | w | = | w ′ | = 1. Then both B + ˙ wU − and B − ˙ w ′ U + (closures in G ) are irreducibleof dimension dim( G ) −

1. If B + ˙ wU − = B − ˙ w ′ U + , thendim( B + ˙ wU − ∩ B − ˙ w ′ U + ) ≤ dim( G ) − , implying (5.3.1). Thus we may assume that B + ˙ wU − = B − ˙ w ′ U + . By our assump-tion, w = s i for some i ∈ I . For any c ∈ C we have y i ( c ) B − ˙ w ′ U + ⊂ B − ˙ w ′ U + hence y i ( c ) B − ˙ w ′ U + ⊂ B − ˙ w ′ U + . Using our assumption, we also deduce that y i ( c ) B + ˙ s i U − ⊂ B + ˙ s i U − for any c ∈ C . We have B + ˙ s i U − = B + ( s i w )˙ U + ˙ w − .For c ∈ C ∗ , we have y i ( c ) B + ˙ s i U − ⊂ B + ˙ s i B + B + ( s i w )˙ U + ˙ w − ⊂ B + ˙ w B + ˙ w − = B + U − and this is disjoint from B + ˙ s i U − . (We have used that | s i ( s i w ) | = | s i | + | s i w | .)This contradicts the inclusion y i ( c ) B + ˙ s i U − ⊂ B + ˙ s i U − . (cid:3) Proof of Theorem 5.2.

The “only if” statement in Theorem 5.2 is obvious. Letus prove the “if” statement. Consider φ ∈ [ O ( G )] such that the eight conditionsin Theorem 5.2 are satisﬁed. We need to show that φ ∈ O ( G ). Suppose that G is of type A . Then φ is regular on U + T U − ∪ U − T U + . Henceby Lemma 5.3, it is regular on G , and we are done.In the rest of the proof, we assume that G is of type other than A .We will ﬁrst show that φ is a regular function on the open set U + T U − .From our assumptions we see—as in the proof in 3.14—that (using the samenotation as 4.4) φ is regular on each of the following open subsets of U + T U − : • V ε ∗ T V ε ′ −∗ , for ε, ε ′ ∈ { , } ; • V ε ∗ T U − j ε ; i , for ε ∈ { , } and i ∈ I [ ε + h +1] ; • U + j ε ; i T V ε ′ −∗ , for ε, ε ′ ∈ { , } and i ∈ I [ ε + h +1] ; • U + j ε ; i T U − j ε ′ ; i ′ , for ε, ε ′ ∈ { , } , i ∈ I [ ε + h +1] , and i ′ ∈ I [ ε ′ + h +1] .Hence φ is regular on the union of these subsets, i.e., on U T U − (where U ⊂ U + isgiven by (3.13.1) and U − = ι ( U ) ⊂ U − ). We have U + T U − − U T U − = (( U + − U ) T U − ) ∪ ( U T ( U − − U − )) . By Lemma 3.13, we havedim(( U + − U ) T U − ) ≤ ν − r + ν = dim( G ) − U T ( U − − U − )) ≤ dim( G ) − . It follows that dim( U + T U − − U T U − ) ≤ dim( G ) − . Since φ is regular on U T U − , we conclude that φ is regular on U + T U − . An entirelysimilar argument shows that φ is regular on U − T U + . It follows that φ is regularon the open subset ( U + T U − ) ∪ ( U − T U + ) of G . Together with Lemma 5.3, thisimplies that φ is regular on G . (cid:3) References [BFZ96] A. Berenstein, S. Fomin and A. Zelevinsky,

Parametrizations of canonical bases andtotally positive matrices , Adv. Math. (1996), 49–149.[BZ97] A. Berenstein and A. Zelevinsky,

Total positivity in Schubert varieties , Comm. Math.Helv. (1997), 128–166.[Bo] N. Bourbaki, Groupes et alg`ebres de Lie, Chapitres 4–6 , Hermann, Paris, 1968.[FZ99] S. Fomin and A. Zelevinsky,

Double Bruhat cells and total positivity , Jour. Amer.Math. Soc. (1999), 335–380.[KL79] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras ,Inv. Math. (1979), 165–184.[L94] G. Lusztig, Total positivity in reductive groups , Lie theory and geometry, Progr. inMath., vol. 123, Birkh¨auser, 1994, pp. 531–568.[L98] G. Lusztig,

Introduction to total positivity , Positivity in Lie theory: open problems,ed. J. Hilgert et al., de Gruyter, 1998, pp. 133–145.[L19] G. Lusztig,

Total positivity in reductive groups,II , Bull. Inst. Math. Acad. Sinica (2019), 403–460.(2019), 403–460.