A generalization of Steinberg's cross-section

Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element w of minimal length r a subvariety V of G isomorphic to an affine space of dimension r which meets the regular unipotent class Y in exactly one point. In this paper this is generalized to the case where w is replaced by any elliptic element in the Weyl group of minimal length d in its conjugacy class, V is replaced by a subvariety V' of G isomorphic to an affine space of dimension d and Y is replaced by a unipotent class Y' of codimension d in such a way that the intersection of V' and Y' is finite.

Introduction 0.1.Let G be a connected semisimple algebraic group over an algebraically closed field.Let B, B − be two opposed Borel subgroups of G with unipotent radicals U, U − and let T = B ∩ B − , a maximal torus of G. Let N T be the normalizer of T in G and let W = N T /T be the Weyl group of T , a finite Coxeter group with length function l.For w ∈ W let ẇ be a representative of w in N T .The following result is due to Steinberg [St,8.9](but the proof in loc.cit. is omitted): if w is a Coxeter element of minimal length in W then (i) the conjugation action of U on U ẇU has trivial isotropy groups and (ii) the subset (U ∩ ẇU − ẇ−1 ) ẇ meets any U -orbit on U ẇU in exactly one point; in particular, (iii) the set of U -orbits on U ẇU is naturally an affine space of dimension l(w).
More generally, assuming that w is any elliptic element of W of minimal length in its conjugacy class, it is shown in [L3] that (i) holds and, assuming in addition that G is of classical type, it is shown in [L5] that (iii) holds.In this paper we show for any w as above and any G that (ii) (and hence (iii)) hold, see Theorem 3.6(ii) (actually we take ẇ of a special form but then the result holds in general since any representative of w in N T is of the form t ẇt −1 for some t ∈ T ).We also prove analogous statements in some twisted cases, involving an automorphism of the root system or a Frobenius map (see 3.6) and a version over Z of these statements using the results in [L2] on groups over Z.Note that the proof of (ii) given in this paper uses (as does the proof of (i) in [L3]) a result in [GP,3.2.7] and a weak form of the existence of "good elements" [GM] in an elliptic conjugacy class in W which (for exceptional types) rely on a computer.0.2.Let w be an elliptic element of W which has minimal length in its conjugacy class C and let γ be the unipotent class of G attached to C in [L3].Recall that γ has codimension l(w) in G.As an application of our results we construct (see 4.2(a)) a closed subvariety Σ of G isomorphic to an affine space of dimension l(w) such that Σ ∩ γ is a finite set with a transitive action of a certain finite group whose order is divisible only by the bad primes of G.In the case where C is X.H. supported in part by HKRGC grant 601409; G.L. supported in part by the National Science Foundation Typeset by A M S-T E X 1 the Coxeter class, Σ reduces to Steinberg's cross section [St] which intersects the regular unipotent class in G in exactly one element.0.3.Recently, A. Sevostyanov [Se] proved statements similar to (i),(ii),(iii) in 0.1 for a certain type of Weyl group elements assuming that the ground field is C. It is not clear to us what is the relation of the Weyl group elements considered in [Se] with those considered in this paper.0.4.The following (unpublished) example of N. Spaltenstein, dating from the late 1970's, shows that the statement (i) (for Coxeter elements) in 0.1 can be false if the assumption of minimal length is dropped: the elements ẇ =        0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 x 0 x 0 0 1 x 0 x 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 −1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 −1 0 0 0 0 1 0 0 0 0 0 0 1 of GL 6 (C) (with x ∈ C) satisfy u x y ẇu −1 x = y ẇ; hence if U is the group of upper triangular matrices in GL 6 (C) then in the conjugation action of U on U ẇU , the isotropy group of y ẇ contains the one parameter group {u x ; x ∈ C}.

Polynomial maps of an affine space to itself
1.1.Let C be the class of commutative rings with 1.Let N be an integer ≥ 1.A family (f A ) A∈C of maps f A : A N − → A N is said to be polynomial if there exist (necessarily unique) polynomials with integer coefficients For such a family we define for any A ∈ C an A-algebra homomorphism f * . ., X N ) for all i ∈ [1, N ] (here we view f i as an element of A[X 1 , . . ., X N ] using the obvious ring homomorphism Z − → A).We have the following result.

Proposition 1.2. Assume that f
the ring of p-adic integers (p is a prime number); (c) A is a perfect field; (d) A is the ring of rational numbers which have no p in the denominator (p is a prime number); (e) A = Z.
(ii) If A is an algebraically closed field then f A is an isomorphism of algebraic varieties.
We prove (i).In case (a), A N is a finite set and the result follows.
Assume that A is as in (b).For any s ≥ 1 let A s = Z/p s Z (a finite ring) and let Assume that A is as in (c).Let A ′ be an algebraic closure of A. By [BR] (see also [Ax], [G1, 10.4.11 (The obvious action of γ on A ′N is denoted again by γ.) Using the injectivity of f A ′ we deduce that γ(ξ ′ ) = ξ ′ .Since this holds for any γ and A is perfect, it follows that ξ ′ ∈ A N .We have f A (ξ ′ ) = ξ.Thus f A is surjective, as desired.
Assume that A is as in (d).Let A 0 = Q p , the field of p-adic numbers.We can view A as the intersection of two subrings of A 0 , namely Assume that A = Z.We can view A as ∩ p A p where p runs over the set of prime numbers and A p is the ring denoted by A in (d); the intersection is taken in the 1 is bijective for any algebraically closed field A 1 (see (i)), it is enough to show that the morphism f A is étale (see [G2, 17.9.1]).Let A ′ = A ⊕ A regarded as an A-algebra with multiplication (a, b)(a ′ , b) = (ab, ab ′ + a ′ b).The unit element is 1 = (1, 0).We set T = (0, 1).Then (a, b) = a + bT and T 2 = 0. Then f A ′ is defined.There exist is injective.It follows that for any a * ∈ A N , the N × N -matrix with (j, k)entries ∂X k (a * ) is nonsingular.This shows that f A is étale.This proves (ii).The proposition is proved.
A∈C is a polynomial family and ξ Z = 1.Thus ξ * Z (X i ) = X i for any i (there is at most one element of Z[X 1 , . . ., X N ] with prescribed values at any (x 1 , . . ., x N ) ∈ Z N ).We see that the polynomials with integer coefficients which define ξ are X 1 , . . ., X N .It follows that ξ A = 1 for any A ∈ C; hence f A f ′ A = 1 and f A is a bijection.Also, since ξ * A = 1 we have f ′ A * f * A = 1 for any A hence f * A is an isomorphism.The proposition is proved.

Reductive groups over a ring
2.1.We fix a root datum R as in [L1, 2.2].This consists of two free abelian groups of finite type Y, X with a given perfect pairing , : Y × X → Z and a finite set I with given imbeddings I − → Y (i → i) and I − → X (i → i ′ ) such that i, i ′ = 2 for all i ∈ I and i, j ′ ∈ −N for all i = j in I; in addition, we are given a symmetric bilinear form for all i ∈ I and i, j ′ = 2i • j/i • i for all i = j in I.We assume that the matrix M = (i • j) i,j∈I is positive definite.
Let W be the (finite) subgroup of Aut(X) generated by the involutions s i : x − i, x i ′ (i ∈ I).For i = j in I let n i,j = n j,i be the order of s i s j in W .Note that W is a (finite) Coxeter group with generators S := {s i ; i ∈ I}; let l : W − → N be the standard length function.Let w I be the unique element of maximal length of W .For J ⊂ I let W J be the subgroup of W generated by {s i ; i ∈ J}.Let X be the set of all sequences i 1 , i 2 , . . ., i n in I such that s i 1 s i 2 . . .s i n = w I and l(w I ) = n.
2.2.Now (until the end of 2.10) we fix A ∈ C. Let UA be the A-algebra attached to R and to A (with v = 1) in [L1, 31.1.1],where it is denoted by A U. As in loc.cit.we denote the canonical basis of the A-module UA by Ḃ.For a, b, c ∈ Ḃ we define L2, 1.5]; in particular we have ab = c∈ Ḃ m c a,b c.Let ÛA be the A-module consisting of all formal linear combinations a∈ Ḃ n a a with n a ∈ A. There is a well defined A-algebra structure on ÛA such that L1, 31.1.1].Let ǫ : ÛA − → A be the algebra homomorphism given by a∈ Ḃ n a a → n 1 0 .We identify UA with the subalgebra of ÛA consisting of finite A-linear combinations of elements in Ḃ.
Let f A be the A-algebra with 1 associated to M and A (with v = 1) in [L1, 31.1.1],where it is denoted by A f .Let B be the canonical basis of the A-module f A (see [L1, 31.1.1]).For i ∈ I, c ∈ N, the element θ 2.4.For any i ∈ I, h ∈ A we set ).This follows from the definitions using [L2, 2.3(c)].
For any w ∈ W we set ẇ = ṡi 1 ṡi 2 . . .ṡi k ∈ G A where i 1 , . . ., i k in I are such that s i 1 s i 2 . . .s i k = w, k = l(w).This is well defined, by (a).

2.5.
We have the following result (see [L2, 4.7(a)]): (a) Let w ∈ W and let i ∈ I be such that l(ws The proof of the following result is similar to that of (a): (b) Let z ∈ W and let i ∈ I be such that l( (We use 2.5(a).)From [L2, 4.8(a)] we see that (a) the map where k = l(w).Let (h 1 , h 2 , . . ., h n ) ∈ A n and let u ∈ U A be its image under the map 2.6(a).We have u = u ′ u ′′ where u ′ = r 1 r 2 . . .r k , u ′′ = r k+1 r k+2 . . .r n and then, as we have seen, we have u ∈ w U A .Thus we have the following results: (a) the restriction of the map 2.6(a) to A k (identified with in the obvious way) defines a bijection in the obvious way) defines a bijection A n−k ∼ − → w U A .Using (a),(b) and 2.6 we see also that (c) multiplication defines a bijection Thus the map in (d) is injective.This proves (d).
Combining (d) with (a) and 2.6(a), we obtain a bijection (e) We can reformulate (a) as follows.
The following result is an immediate consequence of (f).
We have . Then (u ẏ1 , u ′ g 2 , g 3 , . . ., g t ) represents ξ and is as in (a) with s = 1.Now assume that s ≥ 2 and that (g 1 , g 2 , . . ., g t ) is a representative of ξ as in (a) with s replaced by s − 1.We have is a representative of ξ as in (a).This completes the inductive proof of (a).
We show: (b) there exist unique elements The existence of these elements folows from (a) with s = t − 1 and 2.7(d).We prove uniqueness.Assume that u Then there exist v 1 , v 2 , . . ., v t−1 in U A such that The first of these equations implies (using 2.7(d)) that v 1 = 1 and u ′ 1 = u 1 .Then the second equation becomes u ′ 2 ẏ2 = u 2 ẏ2 v 2 ; using again 2.7(d) we deduce that v 2 = 1 and u ′ 2 = u 2 .Continuing in this way we get  ∼ − → Ũ (y * ) such that κ y * H = Hκ x * and such that H is compatible with the U A × U A -actions (as in 2.7(i)).Applying 2.7(g) and 2.7(i) with w replaced by x a (resp.y b ) and w 1 , w 2 , . . ., w r replaced by a sequence of simple reflections whose product is a reduced expression of x a (resp.y b ) we see that the general case is reduced to the case where l * as in 2.7(i)).Note that h, h′ are bijections (they are well defined by 2.7(i)).We set H = ( h′ ) −1 h.It is clear that H, H satisfy the requirements of (a).This proves (a).
2.10.Let δ be an automorphism of R that is, a triple consisting of automorphisms δ : Y − → Y , δ : X − → X and a bijection δ : I − → I such that δ(y), δ(x) = y, x for y ∈ Y, x ∈ X, δ is compatible with the imbeddings I − → Y , I − → X and δ(i)•δ(j) = i•j for i, j ∈ I.There is a unique group automorphism of W , w → δ(w) such that δ(s i ) = s δ(i) for all i ∈ I.There is a unique algebra automorphism (preserving 1) of f A , x → δ(x), such that δ(θ There is a unique algebra automorphism (preserving 1) of ÛA , u → δ(u) such that δ( a∈ Ḃ n a a) = a∈ Ḃ n a δ(a) for all functions Ḃ − → A, a → n a .This automorphism restricts to a group automorphism G A − → G A denoted again by δ and to an automorphism of U A .For any i ∈ I, h ∈ H we have δ( 2.11.Let A ∈ C, A ′ ∈ C and let χ : A − → A ′ be a homomorphism of rings preserving 1.There is a unique ring homomorphism (preserving 1) ÛA − → ÛA ′ , u → χ(u) such that χ( a∈ Ḃ n a a) = a∈ Ḃ χ(n a )a for all functions Ḃ − → A, a → n a .This restricts to a group homomorphism G A − → G A ′ denoted again by χ and to a group homomorphism

The main results
3.1.In this section A ∈ C is fixed unless otherwise specified.We also fix an automorphism δ of R as in 2.10 and a ring automorphism χ of A preserving 1.There are induced group automorphisms of G A denoted again by δ, χ (see 2.10, 2.11).These automorphisms commute; we set π = δχ = χδ : G A − → G A ; note that π maps U A onto itself.
Two elements w, w ′ of W are said to be δ-conjugate if w ′ = y −1 wδ(y) for some y ∈ W .The relation of δ-conjugacy is an equivalence relation on W ; the equivalence classes are said to be δ-conjugacy classes.A δ-conjugacy class C in W (or an element of it) is said to be δ-elliptic if C ∩ W J = ∅ for any J I, δ(J) = J.Let C be a δ-elliptic δ-conjugacy class in W .Let C min be the set of elements of minimal length of C.
For any w ∈ W we define a map

3.2.
In this subsection we assume that x, y ∈ W are such that l(xδ(y)) = l(x) + l(y) = l(yx).We show: In the following proof we write U, U x , U y , U δ(y) and Ξ yx A can be identified with (We use 2.7(g), 2.7(i).)The condition that Ξ xδ(y) A is injective can be stated as follows: (a

The condition that Ξ yx
A is injective can be stated as follows: Assume that (a) holds and that the hypothesis of (b) holds.We have Next we assume that (b) holds and that the hypothesis of (a) holds.We have Thus the conclusion of (a) holds.We see that (a) holds if and only if (b) holds.This proves ( * ).

3.3.
In this subsection we assume that x, y ∈ W are such that l(yx) = l(x) + l(y) and we write We show (a) the sets are in natural bijection.

3.4.
In this subsection we assume that x, y ∈ W are such that . We show (a) the sets ).In view of 3.3 applied to x, y and also to δ(y), x we see that to prove (a), it is enough to show: (b) the sets are in natural bijection.here Now (b) follows from the commutative diagram where ι(u, u ′ , g, g ′ ) = (u ′ , π(u), g ′ , π(g)) and ι ′ (g, g ′ ) = (g ′ , π(g)) are bijections.This proves (a).
3.5.We show: (a) if Ξ w A is injective for some w ∈ C min then Ξ w A is injective for any w ∈ C min ; For any w, w ′ in C min there exists a sequence w = w 1 , w 2 , . . ., w r = w ′ in C min such that for any h ∈ [1, r − 1] we have either w h = xδ(y), w h+1 = yx for some x, y as in 3.2 or w h+1 = xδ(y), w h = yx for some x, y as in 3.2 (See [GP], [GKP], [He].)Now (a) follows by applying 3.2( * ) several times.Now assume that for some w ′ ∈ C min , we are given Ξ A ẇ) such that Ξ w A Ξ ′′ = 1.We choose a sequence w = w 1 , w 2 , . . ., w r = w ′ in C min as in the proof of (a).We define a sequence of maps Ξ ′ i : 1 by induction on i as follows.We set Ξ ′ 1 = Ξ ′ .Assuming that Ξ ′ i is defined for some i ∈ [1, r − 1] we define Ξ ′ i+1 so that Ξ ′ i , Ξ ′ i+1 correspond to each other under a bijection as in 3.4.Then the map Ξ ′′ := Ξ ′ r satisfies our requirement.In particular, we see that: A is surjective for some w ∈ C min then Ξ w A is surjective for any w ∈ C min .
Theorem 3.6.Recall that A ∈ C. Let C be a δ-elliptic δ-conjugacy class in W and let w ∈ C min .Then: A is a field, χ has finite order m and the fixed point field A χ is perfect, then Ξ w A is bijective; (iv) if A is an algebraic closure of a finite field F q and χ(x) = x q for all x ∈ A then Ξ w A is bijective; (v) if A is finite and χ is arbitrary then Ξ w A is bijective.
(d) If χ = 1 and A is an algebraically closed field then α w A is an isomorphism of algebraic varieties.We note that (α w A ′ ) A ′ ∈C can be viewed as a polynomial family of injective maps A ′n − → A ′n (A ′ ∈ C, n as in 2.1).Hence the result follows from 1.2(ii).

Applications
4.1.In this section we assume that A is an algebraically closed field.We write G, U, w U, U w , T instead of G A , U A , w U A , U w A , T A .By [L2, 4.11], G is naturally a connected reductive algebraic group over A with root datum R and U is the unipotent radical of a Borel subgroup B * of G with maximal torus T normalized by each ṡi .We assume that G is semisimple or equivalenty that {i ′ ; i ∈ I} span a subgroup of finite index in X.Let δ be an automorphism of R (necessarily of finite order, say c).The corresponding group automorphism δ : G − → G (see 2.10) preserves the algebraic group structure and has finite order c.Let Ĝ be the semidirect product of G with the cyclic group of order c with generator d such that dxd −1 = δ(x) for all x ∈ G. Then Ĝ is an algebraic group with identity component G. Let B be the variety of Borel subgroups of G.For each w ∈ W let O w be the set of all (B, B Define π w : Bw − → B w by (g, g ′w U ) → (g, g ′ B * g ′−1 ).
In the remainder of this section we assume that C is a δ-elliptic δ-conjugay class in W and that w ∈ C min .Then π w is a principal bundle with group T w = {t 1 ∈ T ; ẇ−1 t ẇ = dtd −1 }, a finite abelian group (see loc.cit.); the group T w acts on Bw by t : (g, g ′w U ) → (g, g ′ t −1w U ). Now G acts on B w by x : (g, B) → (xgx −1 , xBx −1 ) and on Bw by x : (g, g ′w U ) → (xgx −1 , xg ′w U ).We show: (a) Let O be a G-orbit in Bw .There is a unique v ∈ U δ(w) such that ( ẇvd, w U ) ∈ O.
Clearly O contains an element of the form ( ẇud, w U ) where u ∈ U .We first show the existence of v.It is enough to show that for some z But this follows from 3.14(c) with χ = 1 and w replaced by w ′ .Now assume that ( ẇvd δ( ẇ) −1 .Using 3.14(b) (appplied to δ(w) instead of w) we see that u ′ = 1 and v = v ′ .This completes the proof of (a).
We can reformulate (a) as follows.
(b) The closed subvariety {( ẇvd, w U ); v ∈ U δ(w) } of Bw meets each G-orbit in Bw in exactly one point.Hence the space of G-orbits in Bw can be identified with the affine space U δ(w) .We show: (c) The closed subvariety {( ẇvd, B * ); v ∈ U δ(w) } of B w is isomorphic to U δ(w) ; its intersection with any G-orbit in B w is a single T w -orbit (for the restriction of the G-action), hence is a finite nonempty set.The first assertion of (c) is obvious.Now let Ō be a G-orbit in B w .Let There exists a G-orbit O in Bw such that π w (O) = Ō.By (b) we can find v ∈ U δ(w) such that ( ẇvd, w U ) ∈ O. Then ( ẇvd, B * ) = π w ( ẇvd, w U ) ∈ Ō so that Z = ∅.If ( ẇvd, B * ) ∈ Z and t ∈ T w then (t ẇvdt −1 , B * ) ∈ Ō and (t ẇvdt −1 , B * ) = ( ẇv ′′ d, w U ) where Thus (t ẇvdt −1 , B * ) ∈ Z so that T w acts on Z.
We can reformulate (c) as follows.
(d) The closed subvariety {( ẇvd, B * ); v ∈ U δ(w) } of B w meets each G-orbit in B w in exactly one T w -orbit.Hence the space of G-orbits in B w can be identified with the orbit space of the affine space U δ(w) under an action of the finite group T w .Statements like the last sentence in (b) and (d) were proved in [L5, 0.4(a)] assuming that G is almost simple of type A, B, C or D. The extension to exceptional types is new.

4.2.
In this subsection we assume that δ = 1 so that d = 1.Let γ be the unipotent class of G attached to C in [L3].Recall from loc.cit.that γ has codimension l(w) in G.The following result exhibits a closed subvariety of G isomorphic to the affine space A l(w) which intersects γ in a finite set.
(a) The closed subvariety Σ := ẇU w of G is isomorphic to U w and Σ ∩ γ is a single T w -orbit (for the conjugation action), hence is a finite nonempty set.

4.3.
In this subsection we assume that A, F q , χ are as in 3.6(iv).Then π = δχ : G − → G is the Frobenius map for an F q -rational structure on G.As in [DL], we set Xw = {g ′w U ∈ G/ w U ; g ′−1 π(g ′ ) ∈ ẇU }.
Now the finite group G π := {g ∈ G; π(g) = g} acts on Xw by by x : g ′w U → xg ′w U .Let w U \\U be the set of orbits of the w U -action on U given by u 1 : u → ẇ−1 u 1 ẇuπ(u 1 ) −1 .According to [DL,1.12],we have a bijection G π \ Xw ∼ − → w U \\U , g ′w U → ẇ−1 g ′−1 π(g ′ ) with inverse induced by u → g ′w U where g ′ ∈ G, g ′−1 π(g ′ ) = ẇu.Under the substitution ẇ−1 u 1 ẇ = u ′ , the w U -action above on U becomes the w −1 U -action on U given by u 2 : u → u ′ uδ( ẇ)p(u ′ ) −1 δ( ẇ) −1 .Using 3.14(c) for δ(w) instead of w we see that the space of orbits of this action can be identified with U δ(w) .Thus we have the following result.
(a) The space of orbits of G π on Xw is quasi-isomorphic to the affine space U δ(w) .A statement like (a) was proved in [L5] assuming that G is almost simple of type A, B, C or D and δ = 1.The extension to general G is new.