Modular Lie algebras and the Gelfand-Kirillov conjecture
aa r X i v : . [ m a t h . R T ] J u l MODULAR LIE ALGEBRAS AND THE GELFAND–KIRILLOVCONJECTURE
ALEXANDER PREMET
Abstract.
Let g be a finite dimensional simple Lie algebra over and algebraicallyclosed field K of characteristic 0. Let g Z be a Chevalley Z -form of g and g k = g Z ⊗ Z k ,where k is the algebraic closure of F p . Let G k be a simple, simply connected algebraic k -group with Lie( G k ) = g k . In this paper, we apply recent results of Rudolf Tangeon the fraction field of the centre of the universal enveloping algebra U ( g k ) toshow that if the Gelfand–Kirillov conjecture (from 1966) holds for g , then for all p ≫ k ( g k ) on the dual space g k is purely transcendental overits subfield k ( g k ) G k . Very recently, it was proved by Colliot-Th´el`ene–Kunyavski˘ı–Popov–Reichstein that the function field K ( g ) is not purely transcendental over itssubfield K ( g ) g provided that g is of type B n , n ≥
3, D n , n ≥
4, E , E , E or F .We prove a modular version of this result (valid for p ≫
0) and use it to show that,in characteristic 0, the Gelfand–Kirilov conjecture fails for the simple Lie algebrasof the above types. In other words, if g of type B n , n ≥
3, D n , n ≥
4, E , E , E or F , then the Lie field of g is more complicated than expected. Introduction and preliminaries1.1.
Let K be an algebraically closed field. Given a Lie algebra L over K we denoteby U ( L ) the universal enveloping algebra of L . Since U ( L ) is a Noetherian domain,it admits a field of fraction which we shall denote by D ( L ). Let A r ( K ) denote the r -th Weyl algebra over K (it is generated over K by 2 r generators u , . . . , u r , v , . . . , v r subject to the relations [ u i , u j ] = [ v i , v j ] = 0 and [ u i , v j ] = δ ij for all i, j ≤ r ). Givena collection of free variables y , . . . , y s we define A r,s ( K ) := A r ( K ) ⊗ K [ y , . . . , y s ] . Being a Noetherian domain the algebra A r,s ( K ) also admits a field of fractions denoted D r,s ( K ).In [19], Gelfand and Kirillov put forward the following Hypoth`ese fondamentale : The Gelfand–Kirillov conjecture. If char( K ) = 0 and L is the Lie algebra ofan algebraic K -group, then D ( L ) ∼ = D r,s ( K ) for some r, s depending on L .If the Gelfand–Kirillov conjecture holds for L , then necessarily s = index L = tr . deg( Z ( D ( L )) , r = 12 (dim L − index L ) , where Z ( D ( L )) is the centre of D ( L ); see [32] for more detail.In [19], the conjecture was settled for nilpotent Lie algebras, sl n and gl n . In 1973,the conjecture was confirmed in the solvable case independently by Borho [7], Joseph[22] and McConnell [28]. In 1979, Nghiem considered the semi-direct products of sl n , Mathematics Subject Classification (2000 revision ). Primary 17B35. Secondary 17B20, 17B50. p n and so n with their standard modules and proved the conjecture for those; see[30]. A breakthrough in the general case came in 1996 when Jacques Alev, AlfonsOoms and Michel Van den Bergh constructed a series of counterexamples to theconjecture, focusing on semi-direct products of the form L = Lie( H ) ⋉ V where H is a simple algebraic group and V is a rational H -module admitting a trivial genericstabilizer ( V is regarded as an abelian ideal of L ). The smallest known counterexampleis the 9-dimensional semi-direct product of sl with a direct sum of two copies of theadjoint module. In [2], Alev, Ooms and Van den Bergh proved that the conjectureholds in dimension ≤ simple Lie algebra L = sl n remained a complete mystery until now.It suffices to say that the answer is unknown already for L = sp . A weaker positiveresult in the case of L simple was obtained by Gelfand and Kirillov in 1968. Theyproved in [20] that there exists a finite field extension F of the centre Z ( D ( L )) suchthat the field of fractions of D ( L ) ⊗ Z ( D ( L )) F is isomorphic to D N,l ( K ), where l isthe rank of L and N = (dim L − l ). It is conjectured in [1] that such a weakenedversion of the conjecture should hold for any algebraic Lie algebra L . At the oppositeextreme, it was proved in [15] for L simple that the obvious analogue of the Gelfand–Kirillov conjecture holds for the fraction fields of the largest primitive quotients of U ( L ). As the author first learned from Jacques Alev, the Gelfand–Kirillov conjecturemakes perfect sense in the case where the base field K has characteristic p > L = g p is the Lie algebra of a simple, simply connected algebraic K -group G p .The Lie algebra g p = Lie( G p ) carries a canonical p -th power map x x [ p ] equi-variant under the adjoint action of G p . The elements x p − x [ p ] with x ∈ g generate alarge subalgebra of the centre Z ( g p ) of the universal enveloping algebra U ( g ), calledthe p -centre of U ( g p ) and denoted Z p ( g ). It follows from the PBW theorem that U ( g p ) is free module of finite rank over Z p ( g p ). Let Q ( g p ) denote the field of fractionsof Z ( g p ). It is well known that under very mild assumptions on G p one has that D ( g p ) ∼ = U ( g p ) ⊗ Z ( g p ) Q ( g p ) is a central division algebra of dimension p n − l over thefield Q ( g p ), where n = dim g p and l = rk G p ; see [42, 26] for more detail.It is known (and easily seen) that if the Gelfand–Kirillov conjecture hold for g p ,then the field Q ( g p ) is purely transcendental over K and the order of the similarityclass of D ( g p ) in the Brauer group Br( Q ( g p )) equals p ; see [33, 3] for more detail. Atthe Durham Symposium on Quantum Groups in July 1999, Alev asked the authorwhether the field Q ( g p ) is purely transcendental over K . The question was, no doubt,motivated by the Gelfand–Kirillov conjecture.In [33], Rudolf Tange and the author answered Alev’s question in affirmative for g p = gl n and for g p = sl n with p ∤ n . Using our result Jean-Marie Bois was ableto confirm the modular Gelfand–Kirillov conjecture in these cases; see [3]. Recently,Tange [37] solved Alev’s problem for any simple, simply connected group G p subject o some (very mild) assumptions on p . In [37], he also proved that the centre Z ( g p )is a unique factorisation domain, thus confirming an earlier conjecture of Braun–Hajarnavis; see [8, Conjecture E]. Let g be a characteristic 0 counterpart of g p , a simple Lie algebra which hasthe same root system as G p . Although proving the Gelfand–Kirillov conjecture for g p would probably have little impact on its validity for g (apart from some heuristicevidence), it turns out that disproving the conjecture for g p for almost all p is sufficientfor refuting the original conjecture for g .In what follows we assume that K is an algebraically closed field of characteristic 0and denote by k the algebraic closure of the prime field F p . We let g Z be a Chevalley Z -form associated with a minimal admissible lattice in g and set g k := g Z ⊗ Z k . Then g ∼ = g Z ⊗ Z K and g k = Lie( G k ) for some simple, simply connected algebraic k -group G k of the same type as g .In Section 2 we prove a reduction theorem which states that if the Gelfand–Kirillovconjecture holds for g , then it holds for g k for almost all p . In Section 3, we applyTange’s results [37] to show that if the modular Gelfand–Kirillov conjecture holds for g k , then the field k ( g k ) of rational functions on g k is purely transcendental over thefield of invariants k ( g k ) G k .Incidently, it was recently proved by Jean-Lois Colliot-Th´el`ene, Boris Kunyavski˘ı,Vladimir Popov and Zinovy Reichstein that if the function field K ( g ) is purely tran-scendental over the field of invariants K ( g ) g , then g is of type A n , C n or G ; see [13,Thm. 0.2(b)]. In Section 4, we prove a modular version of this result valid for p ≫ Theorem 1.1.
Let g be a finite dimensional simple Lie algebra over an algebraicallyclosed field K of characteristic . If D ( g ) ∼ = D r,s ( K ) for some r, s , then g is of type A n , C n or G . This shows that the original Gelfand–Kirillov conjecture does not hold for simpleLie algebras of types B n , n ≥
3, D n , n ≥
4, E , E , E and F . It seems plausibleto the author that the conjecture does hold for simple Lie algebras of type C. Thesupporting evidence for that comes from [13, Thm. 0.2(a)] which says that in typeC the field K ( g ) is purely transcendental over its subfield K ( g ) g . Some of the resultsobtained in [30] might be useful for proving the conjecture in type C. Acknowledgement.
Part of this work was done during my stay at the Isaac NewtonInstitute (Cambridge) in May–June 2009. I would like to thank the Institute forwarm hospitality and excellent working conditions. The results of this paper wereannounced in my talk at the final conference of the EPSRC Programme “AlgebraicLie Theory” held at the INI in June 2009. I would like to thank Boris Kunyavski˘ı,Vladimir Popov, Andrei Rapinchuk and Rudolf Tange for some very helpful emailcorrespondence.
2. The Gelfand–Kirillov conjecture and its modular analogues2.1.
In this paper we treat the Gelfand–Kirillov conjecture as a noncommutativeversion of a purity problem for field extensions. In order to reduce it to a classicalpurity problem, as studied in birational invariant theory, we seek a passage to finite haracteristics. As a first step, we make a transit from g to g k ensuring in advancethat the validity of the Gelfand-Kirillov conjecture for g implies that for g k . Since U ( g k ) is a finite module over its center Z ( g k ), the field of fractions D ( g k ) is a finitedimensional central division algebra over the fraction field Q ( g k ) of Z ( g k ). Thisenables us to apply recent results of Tange [37] on the rationality of Q ( g k ) to reducethe original problem about the structure of D ( g ) to the purity problem for the fieldextension k ( g k ) / k ( g k ) G k . In this subsection we prove our reduction theorem:
Theorem 2.1.
If the Gelfand–Kirillov conjecture holds for g , then it holds for g k forall p ≫ , where k is the algebraic closure of F p .Proof. (A) Choose a Chevalley basis B = { x , . . . , x n } of g Z and denote by U d ( g ) the d -th component of the canonical filtration of U ( g ). If the field of fractions D ( g ) isisomorphic to Frac (cid:0) A N ⊗ Z ( g ) (cid:1) , where N is the number of positive roots of g , thenthere exist w , . . . , w N ∈ D ( g ) such that[ w i , w j ] = [ w N + i , w N + j ] = 0 (1 ≤ i, j ≤ N );(1) [ w i , w N + j ] = δ i,j (1 ≤ i, j ≤ N );(2) Q k · x k = P k , (1 ≤ k ≤ n )(3)for some nonzero polynomials P i , Q i in w , . . . , w N with coefficients in Z ( g ) (here (3)follows from the fact that the monomials w a w a · · · w a N N with a i ∈ Z + form a basisof the k -subalgebra of D ( g ) generated by w , . . . , w N ). Since w i = v − i u i for some nonzero elements u i , v i ∈ U d ( i ) ( g ), we can rewrite (1) and (2) as follows v − i u i · v − j u j = v − j u j · v − i u i ;(4) v − N + i u N + i · v − N + j u N + j = v − N + j u N + j · v − N + i u N + i ;(5) v − i u i · v − N + j u N + j − v − N + j u N + j · v − i u i = δ i,j (1 ≤ i, j ≤ N ) . (6)As the nonzero elements of U ( g ) form an Ore set, there are nonzero elements v i,j , u i,j ∈ U d ( i,j ) ( g ) such that(7) v i,j u i = u i,j v j (1 ≤ i, j ≤ N ) . Thus we can rewrite (4), (5) and (6) in the form v − i v − i,j · u i,j u j = v − j v − j,i · u j,i u i (1 ≤ i, j ≤ N or N ≤ i, j ≤ N )(8) v − i v − i,N + j · u i,N + j u N + j = δ ij + v − N + j v − N + j,i · u N + j,i u N + i (1 ≤ i, j ≤ N ) . (9)By the same reasoning, there exist nonzero elements a i,j , b i,j ∈ U d ( i,j ) ( g ) such that(10) a i,j v i,j v i = b i,j v j,i v j (1 ≤ i, j ≤ N ) . Since v i,j v i ( v j,i v j ) − = a − i,j b i,j , it is straightforward to see that (8) and (9) can be rewritten as a i,j u i,j u j = b i,j u j,i u i (1 ≤ i, j ≤ N or N ≤ i, j ≤ N )(11) a i,N + j u i,N + j u N + j = δ ij a i,N + j v i,N + j v i + b i,N + j u N + j,i u i (1 ≤ i, j ≤ N ) . (12) or an m -tuple i = ( i (1) , i (2) , . . . , i ( m )) with 1 ≤ i (1) ≤ i (2) ≤ · · · ≤ i ( m ) ≤ N and m ≥ nonzero elements u i (1) ,...,i ( k ) , v i (1) ,...,i ( k ) ∈ U d ( i ) ( g ),where 3 ≤ k ≤ m , such that(13) v i (1) ,...,i ( k ) u i (1) ,...,i ( k − u i ( k − = u i (1) ,...,i ( k ) v i ( k ) . Write w i := w i (1) · w i (2) · . . . · w i ( m ) = Q mk =1 v − i ( k ) u i ( k ) . Then w i = v − i (1) u i (1) · v − i (2) u i (2) · Y mk =3 v − i ( k ) u i ( k ) = v − i (1) v − i (1) ,i (2) u i (1) ,i (2) u i (2) · v − i (3) u i (3) · Q mk =4 v − i ( k ) u i ( k ) = v − i (1) v − i (1) ,i (2) v − i (1) ,i (2) ,i (3) u i (1) ,i (2) ,i (3) u i (3) · Q mk =4 v − i ( k ) u i ( k ) = · · · = (cid:16)Y mk =1 v i (1) ,...,i ( m − k +1) (cid:17) − · u i (1) ,...,i ( m ) u i ( m ) . We now put v i := Q mk =1 v i (1) ,...,i ( m − k +1) and u i := u i (1) ,...,i ( m ) u i ( m ) .Let { i (1) , . . . , i ( r ) } be the set of all tuples as above with P mℓ =1 i ( ℓ ) ≤ M , where M = max { deg P i , deg Q i | ≤ i ≤ n } . Clearly, P k = P rj =1 λ j,k w i ( j ) and Q k = P rj =1 µ j,k w i ( j ) for some λ j,k , µ j,k ∈ Z ( g ), where 1 ≤ k ≤ n . The above discussionthen shows that P k = P rj =1 λ j,k v − i ( j ) u i ( j ) and Q k = P ri =1 µ j,k v − i ( j ) u i ( j ) .It is well known that Z ( g ), the centre of U ( g ), is freely generated over K by l = rk g elements ψ , . . . , ψ l ∈ U ( g Z ). Moreover, for p ≫ U ( g k ) G k ⊂ Z ( g k ) with respect to the adjoint action of G k is freely generated over k by ψ , . . . , ψ l ,the images of ψ , . . . , ψ l in U ( g k ) = U ( g Z ) ⊗ Z k ; see [21, 9.6] for instance.We can write λ j,k = X a λ j,k ( a , . . . , a l ) ψ a · · · ψ a l l and µ j,k = X a µ j,k ( a , . . . , a l ) ψ a · · · ψ a l l for some scalars λ j,k ( a , . . . , a l ) , µ j,k ( a , . . . , a l ) ∈ K , where the summation runs overfinitely many l -tuples a = ( a , . . . , a l ) ∈ Z l + .There exist nonzero c i ( j ); k , d i ( j ); k ∈ U d ( i ( j ) ,k ) ( g ) such that(14) u i ( j ) x k d i ( j ); k = v i ( j ) c i ( j ); k (1 ≤ j ≤ r, ≤ k ≤ n ) . Since P k = Q k x k , we have that(15) r X j =1 λ j,k v − i ( j ) u i ( j ) = r X i =1 µ j,k c i ( j ); k d − i ( j ); k (1 ≤ k ≤ n ) . Set v i ( j ) (0) := v i ( j ) , u i ( j ) (0) = u i ( j ) , c i ( j ); k (0) := c i ( j ); k , d i ( j ); k (0) := d i ( j ); k . For eachpair ( j, s ) of positive integers satisfying r ≥ j > s > nonzero elements v i ( j ) ( s ) , u i ( j ) ( s ) , c i ( j ); k ( s ) , d i ( j ); k ( s ) ∈ U d ( i ( j ) ,k,s ) ( g ) such that v i ( j ) ( s ) v i ( s ) ( s −
1) = u i ( j ) ( s ) v i ( j ) ( s − d i ( j ); k ( s − c i ( j ); k ( s ) = d i ( s ); k ( s − d i ( j ); k ( s ) . (17) ultiplying both sides of (15) by v i (1) on the left and by d i (1) ,k on the right we obtain(after applying (16) and (17) with s = 1) that0 = λ ,k u i (1) d i (1); k − µ ,k v i (1) c i (1); k + r X j =2 (cid:0) λ j,k v i (1) v − i ( j ) u i ( j ) d i (1); k − µ j,k v i (1) c i ( j ); k d − i ( j ); k d i (1); k (cid:1) = λ ,k u i (1) d i (1); k − µ ,k v i (1) c i (1); k + r X j =2 (cid:0) λ j,k v i ( j ) (1) − u i ( j ) (1) u i ( j ) d i (1); k − µ j,k v i (1) c i ( j ); k c i ( j ); k (1) d i (1); k (1) − (cid:1) . Multiplying both sides of this equality by v i (2) (1) on the left and by d i (2); k (1) on theright and applying (16) and (17) with s = 2 we get0 = λ ,k v i (2) (1) u i (1) d i (1); k d i (2); k (1) − µ ,k v i (2) (1) v i (1) c i (1); k d i (2); k (1)+ λ ,k u i (2) (1) u i (1) d i (1); k d i (2); k (1) − µ ,k v i (2) (1) v i (1) c i (2); k c i (2); k (1)+ r X j =3 λ j,k v i ( j ) (2) − u i ( j ) (2) u i ( j ) (1) u i ( j ) d i (1); k d i (2); k (1) − r X j =3 µ j,k v i (2) (1) v i (1) c i ( j ); k c i ( j ); k (1) c i ( j ); k (2) d i (2); k (2) − . Repeating this process r times we get rid of all denominators and arrive at theequality (cid:16) r X j =1 λ j,k r − j Y ℓ =1 v i ( r − ℓ +1) ( r − ℓ ) j Y ℓ =1 u j ( j − ℓ +1) ( j − ℓ ) (cid:17) r Y ℓ =1 d i ( ℓ ); k ( ℓ −
1) =(18) = (cid:16) r Y ℓ =1 v i ( r − ℓ +1) ( r − ℓ ) (cid:17)(cid:16) r X j =1 µ j,k j Y ℓ =1 c i ( ℓ ); k ( ℓ − r Y ℓ = j +1 d i ( ℓ ); k ( ℓ − (cid:17) (at the ℓ -th step of the process we multiply the the preceding equality by v i ( ℓ ) ( ℓ − d i ( ℓ ); k ( ℓ −
1) on the right and then apply (16) and (17) with s = ℓ ).(B) In part (A) we have introduced certain nonzero elements(19) u i , v i , u i,j , v i,j , a i,j , b i,j , u i (1) ,...,i ( s ) , v i (1) ,...,i ( s ) , u i ( j ) ( ℓ ) , v i ( j ) ( ℓ ) , c i ( j ); k ( ℓ ) , d i ( j ); k ( ℓ )in U ( g ) with i, j, k, s, i ( j ) , ℓ ranging over finite sets of indices. These elements satisfyalgebraic equations (7), (10), (11), (12), (13), (14), (16) and (17). We have alsointroduced, for 1 ≤ k ≤ r , two nonzero finite collections of scalars { λ j,k ( a , . . . , a l ) } and { µ j,k ( a , . . . , a l ) } in K linked with the elements (19) by equation (18).The procedure described in part (A) shows that the above data can be parametrisedby the points of a locally closed subset of an affine space A D K , where D is sufficientlylarge. More precisely, there exist finite sets F and G of polynomials in D variableswith coefficients in Z such that a point x ∈ A D K lies in our locally closed set if andonly if and f ( x ) = 0 for all f ∈ F and g ( x ) = 0 for some g ∈ G . Let e X denote thezero locus of the set F in A D K . uppose the Gelfand–Kirillov conjecture holds for g . Then there exists x ∈ e X ( K )such that g ( x ) = 0 for some g ∈ G . We set X := { x ∈ e X | g ( x ) = 0 } , a nonemptyprincipal open subset of e X . As X is an affine variety defined over the algebraic closure Q of the field of rationals, we have that X ( Q ) = ∅ . Hence there is a finitely generated Z -subalgebra A of Q for which X ( A ) = ∅ . There are an algebraic number field K and a nonzero d ∈ Z such that A ⊂ O K [ d − ], where O K denotes the ring of algebraicintegers of K . Since the map Spec( O K ) → Spec( Z ) induced by inclusion Z ֒ → O K issurjective, it must be that X ( k ) = ∅ for every prime p ∈ N with p ∤ d (recall that k stands for the algebraic closure of F p ).(C) When X ( k ) = ∅ , we can find nonzero elements u i , v i , u i,j , v i,j , a i,j , b i,j , u i (1) ,...,i ( s ) , v i (1) ,...,i ( s ) , u i ( j ) ( ℓ ) , v i ( j ) ( ℓ ) , c i ( j ); k ( ℓ ) , d i ( j ); k ( ℓ ) ∈ U ( g k )satisfying (7), (10), (11), (12), (13), (14), (16), (17) and nonzero collections of scalars { λ j,k ( a , . . . , a l ) } and { µ j,k ( a , . . . , a l ) } in k for which the modular version of (18)holds. As all steps of the procedure described in part (A) are reversible and thenonzero elements of U ( g k ) still form an Ore set, this enables us to find w , . . . , w N ∈ Frac U ( g k ) and nonzero polynomials P , . . . , P n and Q , . . . , Q n in w , . . . , w N withcoefficients in the invariant algebra U ( g k ) G k for which the modular versions of (1), (2)and (3) hold. Since the images of x , . . . , x n in g k generate Frac U ( g k ) as a skew-field,applying [3, Lem. 1.2.3] shows that the Gelfand–Kirillov conjecture holds for g k forall p ≫ (cid:3)
3. The Gelfand–Kirillov conjecture and purity of field extensions3.1.
In this section we investigate the modular situation under the assumption that p ≫
0. We are going to apply recent results of Rudolf Tange [37] on the Zassenhausvariety of g k to show that if the Gelfand–Kirillov conjecture folds for g k , then the field k ( g ∗ k ) = Frac S ( g k ) is purely transcendental over its subfield k ( g ∗ k ) G k = Frac S ( g k ) G k .To explain Tange’s results in detail we need a geometric description of the Zassenhausvariety of g k . We follow the exposition in [37] very closely.Recall that the Zassenhaus variety Z of g k is defined as the maximal spectrumof the centre Z ( g k ) of U ( g k ). The Lie algebra g k = Lie( G k ) carries a natural p -thpower map x x [ p ] equivariant under the adjoint action of G k . We denote by Z p ( g k )the p -centre of U ( g k ); it is generated as a k -algebra by all η ( x ) := x p − x [ p ] with x ∈ g k . It follows easily from the PBW theorem that Z p ( g k ) is a polynomial algebrain η ( x ) , . . . , η ( x n ) and U ( g k ) is a free Z p ( g k )-module of rank p n . This implies that Z ( g k ) is is a Noetherian domain of Krull dimension n = dim g k , thus showing that Z is an irreducible n -dimensional affine variety. By an old result of Zassenhaus [42],the variety Z is normal. To ease notation we often identify the elements of g Z with their images in g k = g Z ⊗ Z k . Recall that B = { x , . . . , x n } is a Chevalley basis of g Z . Then there is amaximal torus of T ⊂ G defined and split over Q , such that B = { h α | α ∈ Π } ∪ { e α | α ∈ Φ } , where Φ is the root system of G with respect to T and Π is a basis of simple roots inΦ (we adopt the standard convention that h α = (d α ∨ )(1) where d α ∨ is the differential t 1 of the coroot α ∨ : k × → T , and e α is a generator of the Z -module g Z ∩ g α , where g α is the α -root space of g with respect to T ).Set t := Lie( T ) and denote by T k the maximal torus of G k obtained from T bybase change. Set t k := Lie( T k ), and identify the dual space t ∗ k with the subspace of g ∗ k consisting of all linear functions χ on g k with χ ( e α ) = 0 for all α ∈ Φ. We write X ( T k ) for the group of rational characters of T k and denote by W the Weyl group N G k ( T k ) /Z G k ( T k ). This group is generated by reflections s α with α ∈ Φ and it actsnaturally on both t k and t ∗ k .Let Φ + be the positive system of Φ containing Π and let ρ = P α ∈ Φ + α . Then d ρ is and F p -linear combination of the d α ’s with α ∈ Π. To ease notation we write ρ instead of d ρ . The dot action of W on t ∗ k is defined as follows: w • χ = ( χ + ρ ) − ρ ( ∀ w ∈ W, χ ∈ t ∗ k ) . The induced dot action of W on S ( t k ) has the property that s α • t = s α ( t ) − α ( t ) forall t ∈ t k and α ∈ Π. There exists a unique algebra isomorphism γ : S ( t k ) ∼ −→ S ( t k )such that γ ( t ) = t − ρ ( t ) for all t ∈ t k . The dot action of W is related to naturalaction of W on S ( t k ) by the rule w • = γ − ◦ w ◦ γ for all w ∈ W , which gives rise toan isomorphism of invariant algebras γ : S ( t k ) W • ∼ −→ S ( t k ) W .Put Φ − = − Φ + and write n ± k for the k -span of the e α ’s with α ∈ Φ ± . Then S ( g k ) = S ( n − k ) ⊗ k S ( t k ) ⊗ k S ( n + k ) and U ( g k ) = U ( n − k ) ⊗ k U ( t k ) ⊗ k U ( n + k ) as vectorspaces. Write S + ( g k ) and U + ( g k ) for the augmentation ideals of S ( g k ) and U ( g k ),respectively, and denote by Ψ (resp., ˜Ψ) the linear map from S ( g k ) onto S ( t k ) (resp.,from U ( g k ) onto U ( t k ) = S ( t k )) taking u ⊗ h ⊗ v with u ∈ S ( n − k ), h ∈ S ( t k ), v ∈ S ( n + k )(resp., u ∈ U ( n − k ), h ∈ U ( t k ), v ∈ U ( n + k )) to u hv , where x is the scalar part of x ∈ S ( g k ) (resp., x ∈ U ( g k )) with respect to the decomposition S ( g k ) = k ⊕ S + ( g k )(resp., U ( g k ) = k ⊕ U + ( g k )). Note that the map Ψ is an algebra epimorphism andso is the restriction of ˜Ψ to U ( g k ) T k .For g ∈ G k , x ∈ g k , χ ∈ g ∗ k we write g · x for (Ad g )( x ) and g · χ for (Ad ∗ g )( χ ).Since p ≫
0, the Chevalley restriction theorem holds for g k , that is, the restriction ofΨ to S ( g k ) G k induces an isomorphism of invariant algebras(20) Ψ : S ( g k ) G k ∼ −→ S ( t k ) W . As p is large, we can argue as in the proof of Proposition 2.1 in [39] to deduce thatthe restriction of ˜Ψ to U ( g k ) G k ⊂ U ( g k ) T k induces an algebra isomorphism(21) ˜Ψ : U ( g k ) G k ∼ −→ S ( t k ) W • (in fact, this holds under very mild assumptions on p ; see [26, Lem. 5.4]). As the Killing form κ of g k is nondegenerate for almost primes p , we may identifythe G k -modules g k and g ∗ k by means of Killing isomorphism κ : g k ∋ x κ ( x, · ) ∈ g ∗ k .If χ = κ ( x, · ) ∈ g ∗ k and x = x s + x n is the Jordan–Chevalley decomposition of x inthe restricted Lie algebra g k , then we define χ s := κ ( x s , · ) and χ n := κ ( x n , · ). Wecall χ s and χ n the semisimple and nilpotent part of χ . Denote by ( t k ) reg the set of allregular elements of t and put ( t ∗ k ) reg := κ (cid:0) ( t k ) reg (cid:1) . The elements of ( t ∗ k ) reg are called regular linear functions on t . Note that χ ∈ ( t ∗ k ) reg if and only if χ = κ ( t, · ) forsome t ∈ t k whose centraliser in g k equals t k . It follows that χ ∈ ( t ∗ k ) reg if and only if χ ( h α ) = 0 for all α ∈ Φ. In view of [36, Cor. 2.6], this implies that χ ∈ ( t ∗ k ) reg if and nly if the stabiliser of χ in W is trivial. As a consequence, Z G k ( χ ) = T k for every χ ∈ ( t ∗ k ) reg .Denote by ( g k ) rs the set of all regular semisimple elements of g k . Since everysemisimple element of g k lies in the Lie algebra of a maximal torus of G k and allmaximal tori of G k are conjugate, we have the equality ( g k ) rs = G k · ( t k ) reg ; see [24, §
13] or [5, 4.5]. We set ( g ∗ k ) rs := κ (cid:0) G k · ( t k ) reg (cid:1) and call the elements of ( g ∗ k ) rs regularsemisimple linear functions on g k .Now define ¯ H := Q α ∈ Φ h α , an element of S ( t k ) W , and pick H ∈ S ( g k ) G k such thatΨ( H ) = ¯ H . It is well known (and easy to see when p ≫
0) that for all χ ∈ g ∗ k and f ∈ S ( g k ) G k one has f ( χ ) = f ( χ s ). As ( g ∗ k ) rs = G k · ( t ∗ k ) reg and χ ∈ ( t ∗ k ) reg if and onlyif ¯ H ( χ ) = 0, the G k -conjugacy of maximal toral subalgebras of g k implies that(22) ( g ∗ k ) rs = { χ ∈ g ∗ k | H ( χ ) = 0 } is a principal Zariski open subset of g ∗ k . The Weyl group W acts on the affine variety( G k /T k ) × ( t ∗ k ) reg by the rule w ( gT k , λ ) = ( gw − T k , w ( λ )) and this action commuteswith the left regular action of G k on the first factor. It follows from [4, Prop. II. 6.6and Thm. AG. 17.3] that the coadjoint action-morphism gives rise to a G k -equivariantisomorphism of affine algebraic varieties (cid:0) ( G k /T k ) × ( t ∗ k ) reg (cid:1) /W ∼ −→ ( g ∗ k ) rs ;see [37, 1.3] for more detail. For a vector space V over k the the Frobenius twist V (1) is defined as the vectorspace over k with the same underlying abelian group as V and with scalar multipli-cation given by λ · v := λ /p v for all v ∈ V and λ ∈ k . The polynomial functions on V (1) are the p -th powers of those on V . The identity map V → V (1) is a bijectiveclosed morphism of affine varieties, called the Frobenius morphism . The image of asubset Y ⊆ V under this morphism is denoted by Y (1) . The Frobenius twist of a k -algebra V is defined similarly: the scalar multiplication is modified as above, butthe product in V is unchanged. If V has an F p -structure and G k acts on V as algebraautomorphisms via a rational representation ρ : G k → GL( V ) defined over F p , then G k also acts on V (1) (as algebra automorphisms) via the rational representation ρ ◦ Fr,where Fr is the Frobenius endomorphism of G k . This action coincides with the onegiven by composing ρ with the Frobenius endomorphism of GL( V ) associated by the F p -structure of V .The preceding remark applies in the case where V = S ( g k ) and ρ : G k → GL( V ) isthe rational G k -action by algebra automorphisms extending the adjoint action of G k .The F p -structure of S ( g k ) is given by the canonical isomorphism S ( g k ) ∼ = S ( g F p ) ⊗ F p k where g F p = g Z ⊗ Z F p . Thus, there is a k -algebra isomorphism φ : S ( g k ) (1) ∼ → S ( g k )such that φ ( g · f ) = ( g Fr )( φ ( f )) for all g ∈ G k and f ∈ S ( g k ) (1) .The rule g ⋆ f := φ − ( g ( φ ( f ))) defines a rational action of G k on S ( g k ) (1) = k [( g (1) k ) ∗ ] ∼ = k [( g ∗ k ) (1) )]. In [37], the induced action of G k on ( g ∗ k ) (1) is called thethe star action . By construction, it has the property that(23) g Fr ⋆ χ = g · χ (cid:0) ∀ g ∈ G k , χ ∈ ( g ∗ k ) (1) (cid:1) . It was first observed in [27] that the algebra map η : S ( g k ) (1) = S ( g (1) k ) → Z ( g k )sending x ∈ g k to η ( x ) ∈ Z p ( g k ) is a G k -equivariant algebra isomorphism. One checks asily that η ◦ Ψ = ˜Ψ ◦ η . Also, γ ( η ( t )) = η ( t ) for all t ∈ t k , which stems from thefact that ρ ( t [ p ] ) = ρ ( t ) p . In [37], Tange introduced a principal open subset Z rs of Z and showed that it isisomorphic to a principal open subset of g ∗ k contained in ( g ∗ k ) rs . In order to explainhis construction in detail we need a more explicit description of the variety Z .Recall from Sect. 2 that Z ( g k ) G k = U ( g k ) G k = k [ ψ , . . . , ψ l ] is a polynomial algebrain l variables, where ψ , . . . , ψ l are the images in U ( g k ) of algebraically independentgenerators ψ , . . . , ψ l of Z ( g ) contained in U ( g Z ). In view of (21) and properties of γ , this implies that both S ( t k ) W • and S ( t k ) W are polynomial algebras in l variables.It is worth mentioning that the map in (20) gives rise to a natural isomorphism S ( g (1) k ) G k ∼ −→ S ( t (1) k ) W .By Veldkamp’s theorem, Z ( g k ) ∼ = Z p ( g k ) ⊗ Z p ( g k ) G k U ( g k ) G k and, moreover, Z p ( g k ) is a free Z p ( g k )-module with basis { ψ a · · · ψ a l l | ≤ a i ≤ p − } ;see [39]. A geometric interpretation of Veldkamp’s theorem is given in [29]. Following[37, 1.6] we let ξ : t ∗ k → ( t (1) k ) ∗ be the morphism induced by η : S ( t (1) k ) → U ( t k ) = S ( t k )and let ζ : ( g (1) k ) ∗ → ( t (1) k ) ∗ /W be the morphism associated with the composite k [( t (1) k ) ∗ ] W ∼ −→ k [( g (1) k ) ∗ ] G k ֒ → k [( g (1) k ) ∗ ] , where the first isomorphism is induced by Ψ − . Let π : ( t (1) k ) ∗ → ( t (1) k ) ∗ /W and π • : t ∗ k → t ∗ k /W • be the quotient morphisms. Note that ξ ( λ )( t ) = λ ( t ) p − λ ( t [ p ] ).If λ lies in the F p -span of Π, then λ ( t ) p = λ ( t [ p ] ) for all t ∈ t k because h [ p ] α = h α for all α ∈ Φ. Thus, ξ ( λ ) = 0 in that case. Applying this with λ = ρ , we seethat ξ ( w • λ ) = ξ ( w ( λ )) = w ( ξ ( λ )) for all t ∈ t k and w ∈ W . Also, ζ ( χ ) = π ( χ ′ s ),where χ ′ s is a G k -conjugate of χ s that lies in ( t (1) k ) ∗ (it is important here that π ( χ ′ s )is independent of the choice of χ ′ s , which follows from the fact that the intersectionof ( t (1) k ) ∗ with G k · χ is a single W -orbit in ( t (1) k ) ∗ ). Finally, define ν : ( g ∗ k ) (1) ∼ → ( g (1) k ) ∗ by setting ν ( χ ) = χ p for all χ ∈ ( g ∗ k ) (1) . By [29, Cor. 3], there is a canonical G k -equivariant isomorphism(24) Z ∼ −→ ( g ∗ k ) (1) × ( t ∗ k ) (1) /W t ∗ k /W • where the G k -action on the fibre product is given by from the coadjoint action on thefirst factor, the morphism t ∗ k /W • −→ ( t (1) k ) ∗ /W is induced by ξ and the morphism( g ∗ ) (1) → ( t (1) k ) ∗ /W is the composite of ν and ζ . In what follows we identify Z with a closed subset of the affine space ( g ∗ k ) (1) × t ∗ k /W • by means of isomorphism (24). Note that ( χ, π • ( λ )) ∈ ( g ∗ k ) (1) × t ∗ k /W • belongsto Z if and only if there exists w ∈ W such that λ ( t ) p − λ ( t [ p ] ) = w ( χ ′ s ) p ( ∀ t ∈ t k )where χ ′ s ∈ t ∗ k ∩ ( G k · χ s ).Recall that G k operates on ( g ∗ k ) (1) via the star action (23). From the above discus-sion it follows that this action gives rise to the star action on the Zassenhaus variety via:(25) g ⋆ ( χ, π • ( λ )) := ( g ⋆ χ, π • ( λ )) (cid:0) ∀ ( χ, π • ( λ )) ∈ Z (cid:1) . Following [37, Sect. 2], we now define Z rs := pr − (cid:0) ( g ∗ rs ) (1) (cid:1) , where pr : Z → ( g ∗ k ) (1) is the first projection. In view of (22) it is straightforward to see that Z rs = (cid:8) ( χ, π • ( λ )) ∈ Z | H p ( χ ) = 0 (cid:9) is a nonempty principal open subset of Z .Set ¯ F := Q α ∈ Φ ( h pα − h α ), an element of S ( t k ) W , and pick F ∈ S ( g k ) G k withΨ( F ) = ¯ F . Note that H | F because ¯ H divides ¯ F . Define( t ∗ k ) ′ rs := { χ ∈ t ∗ k | ¯ F ( χ ) = 0 } and ( g ∗ k ) ′ rs := { χ ∈ g ∗ k | F ( χ ) = 0 } . Clearly, ( t ∗ k ) ′ rs consists of all χ ∈ t ∗ k with χ ( h α ) F p for all α ∈ Φ. The precedingremark shows that ( g ∗ k ) ′ rs is a principal open subset of g ∗ k contained in the principalopen set ( g ∗ k ) rs = G k · ( t ∗ k ) rs . Therefore, ( g ∗ k ) ′ rs = G k · ( t ∗ k ) ′ rs . By [37, Thm. 1], there isan isomorphism of algebraic varieties β : Z rs ∼ −→ ( g ∗ k ) ′ rs which intertwines the staraction of G k on Z with the coadjoint action in the following sense:(26) β ( g ⋆ ( χ, π • ( λ ))) = g · β (( χ, π • ( λ ))) (cid:0) ∀ ( χ, π • ( λ )) ∈ Z rs (cid:1) . For 1 ≤ i ≤ l set ϕ i := gr ψ i , a homogeneous element of S ( g Z ) = gr U ( g Z ).Since p ≫
0, we may also assume that the elements ϕ , . . . , ϕ l generate the invariantalgebra S ( g ) g and their images ϕ , . . . , ϕ l in S ( g k ) = S ( g Z ) ⊗ Z k generate S ( g k ) G k .We are now ready to prove the main result of this section: Theorem 3.1.
If the Gelfand–Kirillov conjecture holds for g , then for all p ≫ thefield of rational functions k ( g ∗ k ) = Frac S ( g k ) is purely transcendental over its subfield k ( g ∗ k ) G k = k ( ϕ , . . . , ϕ l ) .Proof. Suppose the Gelfand–Kirillov conjecture holds for g . Theorem 2.1 then saysthat it holds for g k for all p ≫
0. More precisely, it follows from the proof of Theo-rem 2.1 that D ( g k ) is generated as a skew-field by ψ , . . . , ψ l ∈ Z ( g k ) and elements w , . . . , w N which satisfy relations (1) and (2) (here N = | Φ + | ). For 1 ≤ i ≤ N set z i := w pi . Since D ( g k ) ∼ = D N, l ( k ) as k -algebras, the elements z , . . . , z N are centralin Frac U ( g k ). Moreover, the centre of D ( g k ) is k ( z , . . . , z N , ψ , . . . , ψ l ) and theelements z , . . . , z N , ψ , . . . , ψ l are algebraically independent; see [3, 1.1.3] for moredetail.On the other hand, it is well known that in the modular case D ( g k ) is the centrallocalisation of U ( g k ) by the set Z p ( g k ) × of nonzero elements of Z p ( g k ). Likewise,the centre of D ( g k ) is the localisation of Z ( g k ) by the set Z p ( g k ) × . It follows thatthe centre of D ( g k ) equals Q ( g k ) = Q p [ ψ , . . . , ψ l ], where Q p is the field of fractionsof Z p ( g k ). Since the Q p -vector space Q ( g k ) has a basis consisting of monomials in ψ , . . . , ψ l , it is straightforward to see that the field of invariants Q ( g k ) G k coincideswith Q G k p [ ψ , . . . , ψ l ]. As Z p ( g k ) is a polynomial algebra and the connected group G k coincides with its derived subgroup, we have that Q G k p = Frac Z p ( g k ) G k . This showsthat Q ( g k ) G k = Frac Z ( g k ) G k = k ( ψ , . . . , ψ l ). We thus deduce that(27) k ( Z ) = Q ( g k ) = k ( z , . . . , z N , ψ , . . . , ψ l ) = k ( Z ) G k (cid:0) z , . . . , z N (cid:1) s purely transcendental over the field of invariants k ( Z ) G k = k ( ψ , . . . , ψ l ).Recall that in our geometric realisation (24) the ordinary action of G k on Z is givenby g · ( χ, π • ( λ )) = ( g · χ, π • ( λ )) for all g ∈ G k and all ( χ, π • ( λ )) ∈ Z . Since in (24) weregard χ as an element of ( g ∗ k ) (1) , comparing this with (23) and (25) yields that everyorbit with respect to the ordinary action of G k on Z is an orbit of G k with respect tothe star action and vice versa. From this it follows that both actions have the samerational invariants.The comorphism of β − : ( g ∗ k ) ′ rs ∼ −→ Z rs induces a field isomorphism between k ( Z ) and k ( g ∗ k ); we call it b . Combining (26) with the preceding remark one ob-serves that b sends the subfield k ( Z ) G k = k ( ψ , . . . , ψ l ) onto k ( g ∗ k ) G k . But then(27) shows that k ( g ∗ k ) = k ( g ∗ k ) G k (cid:0) b ( z ) , . . . , b ( z N ) (cid:1) is purely transcendental over k ( g ∗ k ) G k = k ( ϕ , . . . , ϕ l ). This completes the proof. (cid:3) Remark . Combining Theorem 3.1 with the Killing isomorphism κ : g k ∼ → g ∗ k wesee that for all p ≫ k ( g k ) is purely transcendentalover its subfield k ( g k ) G k .
4. Purity, generic tori and base change4.1.
We keep the notation introduced in Sections 2 and 3 and assume that char( k ) = p ≫
0. Recall that { x , . . . , x n } = { h α | α ∈ Π } ∪ { e α | α ∈ Φ } is a Chevalleybasis of g Z and we identify the x i ’s with their images in g k . Write Π = { α , . . . , α l } and let { X , . . . , X l } and { X α | α ∈ Φ } be two sets of independent variables. Set K := Q ( X , . . . , X l ) and e K := K ( X α | α ∈ Φ) and denote by K p an algebraic closureof k ( X , . . . , X l ). Write e K p := K p ( X α | α ∈ Φ) and denote by K p an algebraic closureof e K p . To ease notation we shall assume that K is an algebraic closure of e K (this willcause no confusion).Given a field F we write g F for the Lie algebra g Z ⊗ Z F over F and denote by G F the simple, simply connected algebraic F -group with Lie algebra g F . Let e t := P li =1 X i h α i and e x := P α ∈ Φ X α e α . Since the X i ’s are algebraically independent, e t is a regular semisimple element of g K contained in g Z [ X , . . . , X l ]. Its image e t p := e t ⊗ ∈ g Z [ X , . . . , X l ] ⊗ Z k is a regular semisimple element of g K p . The image of e x in g Z [ X α | α ∈ Φ] ⊗ Z k is denoted by e x p . Set e y := e t + e x and e y p := e t p + e x p . These areregular semisimple elements of g e K and g e K p , respectively.Write G p for the group G K p and g p for its Lie algebra g K p . Given a closed subgroup H of G p defined over e K p we write H p for the group H ( K p ). Set T gen := Z G ( e y ) and T gen p := Z G p ( e y p ). It follows from [5, 4.3] that T gen and T gen p are maximal tori of G and G p defined over e K and e K p , respectively. Let t gen := Lie( T gen ) and t gen p := Lie( T gen p ). Let T p be the variety of maximal toral subalgebras of g p . As all maximal toralsubalgebras of g p are conjugate under G p and the normaliser N p of t p := t k ⊗ k K p in G p is a reductive group, T p ∼ = G p /N p is an affine algebraic variety. It follows froma well known result of Grothendieck [16, Exp. XIV, Thm. 6.2] that the variety T p is K p -rational. More precisely, let m p be orthogonal complement to t p with respectto the Killing form of g K p . A natural K p -defined birational isomorphism between T p and m p can be obtained as follows; see [5, 7.9]: he set T ◦ p of all h ∈ T p with h ∩ m p = 0 is open, nonempty in T p and the set m ◦ p := { m ∈ m p | e t p + m ∈ ( g p ) rs and ad( e t p + m ) | m p is injective } is open, nonempty in m p . For every m ∈ m ◦ p the centralizer of e t p + m in g p is anelement of T ◦ p . Since for every h ∈ T ◦ p there exists a unique m = m ( h ) ∈ m p with e t p + m ∈ h , the map µ : T ◦ p −→ m ◦ p , h m ( h ) , gives rise to a K p -defined birationalisomorphism between T p and m p . The K p -defined birational map µ enables us toidentify the field K p ( T p ) with K p ( m p ) ∼ = K p ( X α | α ∈ Φ) = e K p . It is straightforwardto see that e x p ∈ m ◦ p and µ ( t gen p ) = e t p + e x p = e y p . Since the field K p ( e x p ) = K p ( e y p ) isnothing but e K p , we now deduce that t gen p is a generic point of the K p -variety T p . Recall that ϕ , . . . , ϕ l are free generators of S ( g ) g contained in S ( g Z ) and suchthat S ( g k ) G k = k [ ϕ , . . . , ϕ l ], where ϕ i = ϕ i ⊗ ∈ S ( g Z ) ⊗ Z k = S ( g k ). We identifythe G k -modules g k and g ∗ k by means the Killing isomorphism κ ; see Remark 3.1. Thus,we may regard ϕ , . . . , ϕ l as free generators of the invariant algebra k [ g k ] G k .Let Y p be the fibre ϕ − ( e y p ) of the adjoint quotient map ϕ : g p −→ g p //G p , thatis, Y p := { y ∈ g p | ϕ i ( y ) = ϕ i ( e y p ) for all 1 ≤ i ≤ l } . As p ≫
0, all fibres of ϕ are irreducible complete intersections of dimension n − l in the affine space g p ; see [39] for more detail. Since e y p is regular semisimple, theorbit G p · e y p is Zariski closed in g p and dense in Y p . This shows that Y p = G p · e y p is a smooth variety and the defining ideal of Y p is generated by the regular functions ϕ − ϕ ( e y p ) , . . . , ϕ l − ϕ l ( e y p ). Since e y p is regular semisimple, the orbit map G p → Y p isseparable. Applying [4, Prop. II. 6.6], we now deduce that the e K p -varieties G p /T gen p and Y p are e K p -isomorphic (recall from (4.1) that T gen p is the centraliser of e y p in G p ). Our next result is inspired by [13, Thm. 4.9]. The argument in [13] exploits thenotion of versality of (
G, S )-fibrations introduced in [13, Sect. 3] and seems to rely onthe characteristic zero hypothesis (see the footnote on p. 20 of [13]). Our argument isdifferent and it works under very mild assumptions on the characteristic of the basefield.
Proposition 4.1.
If the field k ( g ∗ k ) is purely transcendental over k ( g ∗ k ) G k , then thehomogeneous space G p /T gen p is e K p -birational to an affine space.Proof. If k ( g ∗ k ) is purely transcendental over k ( g ∗ k ) G k , then there exist F , . . . , F N ∈ k ( g k ) such that k ( g ) = k ( F , . . . , F n , ϕ , . . . , ϕ l ). Then the rational map F : g k A N k × A l k taking y ∈ g k to F ( y ) := (cid:0) ( F ( y ) , . . . , F N ( y ) , ( ϕ ( y ) , . . . , ϕ l ( y ) (cid:1) ∈ A N k × A l k induces a k -isomorphism F : U ∼ → V between a k -defined nonempty open subset U of g k and a k -defined nonempty open subset V of A N k × A l k .Since f ( e y p ) = 0 for every nonzero f ∈ k [ g k ] and U = g k \ Z for some Zariskiclosed Z ( g k defined over k , we see that e y p ∈ U p := g p \ Z ( K p ). Likewise, V = (cid:0) A N k × A l k (cid:1) \ Z ′ for some Zariski closed subset Z ′ ( A N k × A l k defined over k . We set V p := (cid:0) A N K p × A l K p (cid:1) \ Z ′ ( K p ) and observe that F gives rise to a k -defined isomorphismbetween U p and V p . ut Y ◦ p := Y p ∩ U p . As e y p ∈ Y ◦ p , we see that Y ◦ p is a nonempty closed subset of U p defined over e K p . Furthermore, dim Y ◦ p = n − l = 2 N . Therefore, F ( Y ◦ p ) is a2 N -dimensional nonempty closed subset of V p . On the other hand, it is immediatefrom the definition of F and our discussion in (4.3) that F ( Y ◦ p ) ⊆ A N K p × pt. Thisimplies that F ( Y ◦ p ) is e K p -isomorphic to a Zariski open subset of A N K p defined over e K p . Since Y p is e K p -isomorphic to G p /T gen p by our discussion in (4.3) and F ( Y ◦ p ) is e K p -isomorphic to Y ◦ p , we conclude that the homogeneous space G p /T gen p is rationalover e K p . (cid:3) In order to adapt the proof of the crucial Theorem 6.3 from [13] to our modularsetting we need a smooth projective model of G p /T gen p defined over e K p , that is, asmooth projective e K p -variety Y cp together with an open embedding G p /T gen p ֒ → Y cp defined over e K p . Proposition 4.2.
For all p ≫ the variety G p /T gen p has a smooth projective modeldefined over e K p .Proof. Let ϕ : g → g //G be the adjoint quotient map and set Y := ϕ − ( e y ). Arguingas in (4.3) we observe that Y = G · e y is a smooth variety and the defining ideal of Y is generated by the regular functions ϕ i − ϕ i ( e y ), where 1 ≤ i ≤ l . Our discussion in(4.1), (4.2) and (4.3) now shows that there exists a finitely generated Z -subalgebra R of e K = K ( T ) and an affine flat scheme Y of finite type over S := Spec( R ) such that Y = Y × S Spec( K ) and Y p = Y × S Spec( K p ) ( ∀ p ≫ . By Hironaka’s theorem on resolution of singularities there exists a smooth pro-jective K ( T )-variety Y c ⊆ P d K and an open immersion ω : Y → Y c defined over K ( T ). Let Γ ω denote the graph of ω . Since all projective schemes are separated,Γ ω = { ( y, ω ( y )) | y ∈ Y } is a closed subset of Y × P d K ; see [35, p. 47]. As K ( T ) is aperfect field, Γ ω is defined over K ( T ); see [4, AG, §
14] for detail.Let e R be a finitely generated Z -subalgebra of K ( T ) containing R and all elementswhich we need to define ω , Y c , Γ ω , and the field isomorphism K ( Y ) ∼ → K ( Y c ) inducedby the rational inverse of the comorphism ω ∗ . Then we obtain a projective scheme Y c of finite type over e S := Spec( e R ) and an e S -morphism e ω : Y → Y c whose basechange to Spec( e K ) is ω : Y → Y c . We also obtain an e S -subscheme e S -subscheme Γ e ω of Y × e S P d e R such that Γ ω = Γ e ω × e S Spec( K ( T )) . By localising further as necessary wemay assume that the scheme Y c is smooth over e S and the schemes Y , Y c and Γ e ω areflat over e S . We let π : Γ e ω → Y denote the first projection.Given a closed point s ∈ e S and an e S -scheme V we write κ ( s ) for the residue field ofthe local ring of s and V s for the scheme-theoretical fibre V × e S Spec( κ ( s )). It followsfrom the above discussion that for every closed point s ∈ e S the schemes Y s and Y cs are smooth and the base change e ω s : Y s → Y cs is birational.If A is the affine coordinate ring of Y , then Γ ω ⊆ Y × P d e R corresponds to a graded A -algebra B = B ⊕ B ⊕ B ⊕ . . . with B = A generated over A by d + 1 elements.By [18, Thm. 14.8], there is an ideal J of A such that for every prime ideal P of A the algebra Frac ( A/P ) ⊗ A B has positive Krull dimension if and only if P ⊇ J . We enote by Y ′ the closed subscheme of Y corresponding to the ideal J . Then for everyclosed point s ∈ e S we have that x ∈ Y ′ s if and only if the fibre π − s ( x ) of the basechange π s : (Γ e ω ) s → Y s has positive dimension.Set Y ′ := Y ′ × e S Spec( K ). Since Γ ω is closed and ω is injective, the set Y ′ red ( K ) isempty. In conjunction with Hilbert’s Nullstellensatz this implies that the ideal J ⊗ e R e K of A ⊗ e R e K = e K [ Y ] coincides with e K [ Y ]. Then P ki =1 c i q i = 1 for some q , . . . , q k ∈ J and c , . . . , c k ∈ e K = Frac e R . Localising e R further, we may assume that all c i ’s arein e R . Then the above discussion shows that for every closed point s ∈ e S the reducedfibres of e ω s : Y s → Y cs are finite.Since e R is a Noetherian domain whose field of fractions is e K = K ( X α | α ∈ Φ),for every p ≫ s ∈ Spec( e R ) with κ ( s ) = F p ( X , . . . , X l ) (cid:0) X α | α ∈ Φ (cid:1) .The discussion in (4.3) shows that for each such s the scheme Y s × Spec( κ ( s )) Spec( K p )is nothing but Y p . Since Y p is reduced, the base change e ω s : Y s → Y cs gives riseto a natural morphism ω p : Y p → (cid:0) Y cs × Spec( κ ( s )) Spec( K p ) (cid:1) red . We denote by Y cp theirreducible component of the reduced scheme (cid:0) Y cs × Spec( κ ( s )) Spec( K p ) (cid:1) red that contains ω p ( Y p ).Since Y cs is smooth, projective, so too is Y cp . Furthermore, our earlier remarks inthe proof imply that ω p is a e K p -defined birational morphism of algebraic varieties andall fibres of ω p are finite. The variety Y cp is smooth, hence normal. Applying Zariski’sMain Theorem to the quasi-finite birational morphism ω p : Y p → Y cp , we now deducethat ω p is an open embedding; see [31, Cor. 1(i)] for the statement and a short proofof the result we need (it is worth mentioning that [31] is available on the web).Since the variety Y p is defined over e K p by our discussion in (4.3), it follows from [4,AG, 14.5] that so is Y cp = ω p ( Y p ). But then the composite G p /T gen p ∼ → Y p ω p −→ Y cp isa smooth projective model of G p /T gen p defined over e K p = K p ( T p ). (cid:3) Remark . Since the variety Y cp is projective and its open set ω p ( Y p ) ∼ = Y p is affine,all irreducible components of the complement D := Y cp \ ω p ( Y p ) have codimension 1in Y c ; see [23, Ch. 2]. Remark . Let T be a maximal F -torus in a split connected reductive algebraic F -group G . If T is F -split, then there exist Borel subgroups B + and B − in G defined over F and such that B + ∩ B − = T . Thus, in the F -split case the variety( G / B + ) × ( G / B − ) provides a natural smooth projective model of the homogeneousspace G / T (this was pointed out to me by Panyushev and Serganova). Unfortunately,it is not clear how to adapt this construction to the case of a non-split maximal torus.It would be very interesting to find an explicit F ( T )-defined smooth projective modelof the homogeneous space G / T for an arbitrary maximal F -torus T of G . It is known that in characteristic 0 the generic torus T gen ⊂ G splits over afinite Galois extension of e K whose group acts on the weight lattice X ( T gen ) as theWeyl group W . This result is sometimes attributed to ´E. Cartan; see [9, 10]. Modernproofs can be found in [40, 41] and in the “dismissed appendix” to [13] written byJ.-L. Colliot-Th´el`ene; see [11].The rest of the paper relies on a modular version of this result. In order to applythe arguments from [40, 41] in the characteristic p case one needs to know that the orphism α : G/T × T → H is separable and the second projection π : H → Y is birational (notation of loc. cit. ). This was checked earlier by Vladimir Popov andAndrei Rapinchuk. The proof below was outlined to the author by Andrei Rapinchuk. Proposition 4.3.
There exists a finite Galois extension L/ e K p with group W whichsplits the e K p -torus T gen p and acts on the weight lattice of T gen p in the standard way.Proof. (1) Following [40] we set H p := ( gN p , gtg − ) | g ∈ G p , t ∈ T p } ⊂ ( G p /N p ) × G p . By [25, p. 10], the set H p is Zariski closed in ( G p /N p ) × G p . By the definition of H p , the K p -morphism α : ( G p /T p ) × T p −→ H p taking ( gT p , t ) to ( gN p , gtg − ) issurjective. We can write α as the composition α ◦ α , where α : G p × T p −→ ( G p /T p ) × G p , ( g, t ) ( gT p , gtg − ); α : ( G p /T p ) × G p −→ ( G p /N p ) × G p , ( gT p , x ) ( gN p , x ) . The morphism α is an ´etale Galois cover. We need to show that the second projection π : H p → G p is a separable morphism. Define α : ( G p /T p ) × T p −→ G p to be thecomposite of α and π . Then α ( gT p , t ) = gtg − for all ( gT p , t ) ∈ ( G p /T p ) × T p .We compute the differential of α at ( eT p , t ) ∈ ( G p /T p ) × T p , where e is the identityelement of G p and t is any regular element of T p . Write D = K p [ ε ] for the algebra ofdouble numbers over K p , so that ε = 0. Let X be in the K p -span of the e γ ’s whichwe identify with g p / t p , the tangent space of G p /T p at eT p . If Y ∈ Lie( T p ) = t p , then t Y lies in the tangent space to T p at t and (d α ) ( eT p , t ) ( X, t Y ) is the coefficientof ε of the element ( e + εX )( t + εt Y )( e − εX ) = t + ε ( t Y + Xt − t X ) ∈ D. Multiplying this by t − to move everything back to the identity, we get(d α ) ( eT p , t ) ( X, t Y ) = Y + t − Xt − X = Y + (cid:0) Ad( t − ) − Id (cid:1) ( X ) . Since t ∈ T p is regular, we see that the image of (d α ) ( X, t Y ) has dimension n =dim G p . So α = α ◦ π is a separable morphism and hence so is π .(2) Write G rs p for the set of all regular semisimple elements in G p , and set T rs p := G p ∩ T p . As the group G p is simply connected it follows from Steinberg’s restrictiontheorem that there exists a regular invariant function f ∈ k [ G k ] G k such that G rs p = { g ∈ G p | f ( g ) = 0 } . Hence G rs p is a principal Zariski open subset in G p . In particular,the varieties G rs p and T rs p are smooth and affine.Let H rs p := { ( gN p , gtg − ) | g ∈ G p , t ∈ T rs p } . The Weyl group W acts on ( G p /T p ) × T p by the rule ( gT p , t ) w = ( g · w T p , · w − t · w ) . It is straightforward to see that the set H rs p is W -stable, the restriction of π to H rs p is bijective and the fibres of α are W -orbits. Also, H rs p = (cid:0) ( G p /N p ) × G rs p (cid:1) ∩ H p is aprincipal Zariski open subset of H p .By part (1), the restriction of π to H rs p is separable. So π : H rs p → G rs p is a bijectiveseparable morphism of affine varieties. Therefore, it is birational. As the variety G rs p is smooth, hence normal, Zariski’s Main Theorem now yields that π : H rs p → G rs p isa K p -isomorphism. But the H rs p is an affine normal variety, and we can apply [4,Prop. II. 6.6] to conclude that α : ( G p /T p ) × T rs p −→ H rs p is the geometric quotientfor the action of W . e have the following commutative diagram, where π is the first projection and β is the canonical map.( G p /T p ) × T rs p α −−−→ H rs pp y y π G p /T p β −−−→ G p /N p ∼ ←−−− T p , (28)where p and π are the first projections and β is the quotient morphism. All thesemaps are defined over k ⊂ K p .(3) Recall from (4.2) that e y p identifies with a generic point of T p ∼ = G p /N p in thesense that the fields K p ( e y p ) = e K p and K p ( G p /N p ) are K p -isomorphic. The algebra of K p -defined regular functions of the fibre β − ( e y p ) is(29) K p [ G p /T p ] ⊗ K p [ G p /N p ] K p ( G p /N p ) ∼ = K p ( G p /T p ) , showing that K p ( β − ( e y p )) = K p ( G p /T p ) is a Galois extension of e K p = K p ( G p /N p )with Galois group W .Since T gen p = Z G p ( e y p ) is defined over e K p , it contains a e K p -rational regular element;we call it s . (This follows from the fact that T gen p ( e K p ) is dense in T gen p ; see [16]). Let( gN p , s ) = π − ( s ). Since π is a K p -isomorphism, we have that ( gN p , s ) ∈ H rs p ( e K p ).If P ⊂ K p is a finite Galois extension of e K p which splits T gen p , then the split P -tori T p and T gen p are conjugate by a P -rational element of G p ; see [6, 4.21, 8.2]. Putdifferently, ( gN p , s ) = α ( hT p , s ′ ) for some P -point ( hT p , s ′ ) of ( G p /T p ) × T rs p . But then(28) shows that e y p ∈ β (( G p /T p )( P )). Conversely, if L is a finite Galois extension of e K p such that e y p ∈ β (( G p /T p )( L )), then (28) yields that( gN p , s ) = π − ( s ) ∈ α (( G p /T p ) × T p )( L )) . Hence L splits T gen p . Applying this with L = K p ( G p /T p ) and taking into account(29) one can observe that L = K p ( G p /T p ) is a minimal splitting field for T gen p andGal( L/ e K p ) = W . Indeed, since L W = e K p and W acts faithfully on L , the L -algebra L ⊗ L W L is isomorphic to a direct sum of | W | copies of L . Moreover, it followsfrom the normal basis theorem by comparing W -invariants that if F/ e K p is a Galoisextension contained in L , then L ⊗ L W F is isomorphic as an F -algebra to a directsum | W | copies of F if and only if F = L .By the minimality of the splitting field L , the Galois group of L/ e K p acts faith-fully on the weight lattice X ( T gen p ) giving a natural injective group homomorphism τ : Gal( L/ e K p ) → Aut(Φ). Since the group G p is e K p -split, the image of Gal( L/ e K p )under τ is contained in W ⊆ Aut(Φ); see [38, 2.3]. As τ is injective, this shows that W = Gal( L/ e K p ) acts on X ( T gen p ) in the standard way. (cid:3) It what follows we shall assume without loss of generality that our algebraicclosure K p of e K p = K p ( G p /N p ) contains L = K p ( G p /T p ). The result below (which iscrucial for us) has been proved in [13] under the assumption that the base field hascharacteristic 0; compare [13, Thm 6.3(b)]. Although it follows from a more generalresult obtained in [12], the proof given in [13] is self-contained modulo [34, Thm. 4],[14, Prop. 2.1.1] and [14, Prop. 2.A.1]. ecall that a free Z -module of finite rank acted upon by a group Γ is called a permutation lattice if it has a Z -basis whose elements are permuted by Γ. Proposition 4.4.
If the homogeneous space Y p = G p /T gen p is e K p -rational, then thereexists a short exact sequence of Γ -lattices −→ P −→ P −→ X ( T gen p ) −→ with P and P permutation lattices over Γ = Gal( L/ e K p ) .Proof. The proof repeats almost verbatim the argument in [13, p. 23]; we sketch itfor the convenience of the reader.By Proposition 4.2, the variety Y p has a smooth projective model Y cp defined over e K p . The open immersion Y p ⊂ Y cp gives rise to an exact sequence of Galois lattices(30) 0 −→ K p [ Y p ] × / K × p −→ Div ∞ Y cp −→ Pic Y cp −→ Pic Y p −→ , where Div ∞ Y cp is the free abelian group on the irreducible components of the excep-tional divisor D = Y cp \ Y p ; see Remark 4.1. Since [34, Thm. 4] and [14, Prop. 2.1.1]hold in any characteristic, we can repeat the argument in [13, p. 23] to obtain that K p [ Y p ] × = K × p and Pic Y p ∼ = X ( T gen p ) as Galois modules.Since [14, Prop. 2.A.1] hold in any characteristic, we can apply it to the e K p -rationalhomogeneous space Y p to deduce that the Galois lattice Pic Y cp has the property that Q ⊕ Pic Y cp ∼ = Q for some permutation Galois lattices Q and Q . As P := Div ∞ Y cp is a permutation Galois lattice as well, the short exact sequence0 −→ P −→ Pic Y cp −→ X ( T gen p ) −→ → P → P → X ( T gen p ) → P = P ⊕ Q and P = Q ⊕ Pic Y cp being permutation Galois lattices. (cid:3) We denote by P (Φ) the weight lattice of the root system Φ. Corollary 4.1.
If the Gelfand–Kirillov conjecture holds for g , then there exists ashort exact sequence of W -modules (31) 0 −→ P −→ P −→ P (Φ) −→ with P and P permutation W -lattices.Proof. Combining Theorem 3.1 and Proposition 4.1 we see that if the Gelfand–Kirillovconjecture holds for g , then for all p ≫ Y p = G p /T gen p isrational over e K p . As G p is simply connected, X ( T gen p ) ∼ = P (Φ) as W -modules. Nowthe result follows by applying Propositions 4.3 and 4.4 (cid:3) We are finally ready for the main result of this paper.
Theorem 4.1.
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E-mail address : [email protected]@maths.man.ac.uk