CROSS-SECTIONS, QUOTIENTS, AND REPRESENTATION RINGS OF SEMISIMPLE ALGEBRAIC GROUPS

VLADIMIR L. POPOV^{∗}

“Is Steinberg’s theorem […] only true for simply connected groups […]? What happens for GP(1), for instance ? Is there a rational section of G over I(G) (“invariants ”) in this case ? […] Is it true that I(G) is a rational variety […] ?”

A. Grothendieck, Letter to J.-P. Serre, January 15, 1969, [GS, pp. 240–241].

Abstract. Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny τ: Ĝ → G is bijective. In particular, for char k = 0, the converse to Steinberg’s theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]^{G} of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]^{G} and that of the representation ring of G and answer two Grothendieck’s questions on constructing the generating sets of k[G]^{G}. We prove the existence of a rational cross-section in any G (for char k = 0, this has been proved earlier in [CTKPR]). We also prove that the existence of a rational section of the quotient morphism for G is equivalent to the existence of a rational W -equivariant map T −−→ G/T where T is a maximal torus of G and W the Weyl group. We show that both properties hold if the isogeny τ is central.

1. Introduction

Below all algebraic varieties are taken over an algebraically closed field k. We use the standard notation and conventions of [Bor] and [Sp].

Let G be a connected semisimple algebraic group, G ≠ {e}. Let (G//G, π_{G}) be a categoricalRMR quotient for the conjugating action of G on itself, i.e., G//G is an affine variety and

π_{G}: G ⟶ G//G

a surjective morphism such that π_{G}^{∗}(k[G//G]) is the algebra k[G]^{G} of class functions on G. Every fiber of π_{G} is then the closure of a regular conjugacy class (i.e., that of the maximal dimension) and such classes in general position are closed [Ste_{1}, Theorem 6.11, Cor. 6.13, and Sect. 2.14].

Definition 1.1. A closed irreducible subvariety S of G is called a cross-section (of the collection of fibers of π_{G}) in G if S intersects at a single point every fiber of π_{G}.

Date: August 20, 2009. 2000 Mathematics Subject Classification. 14M99, 14L30, 14R20, 14L24, 17B45 . Key words and phrases. Semisimple algebraic group, conjugating action, cross-section, quotient,

representation ring.

∗ Supported by grants ���� �����������, ��������������, and the program Contemporary Problems of Theoretical Mathematics of the Russian Academy of Sciences, Branch of Mathematics.

The elements of S are the “canonical forms” of the elements of a dense constructible subset of G with respect to conjugation. The image of any section of π_{G} (i.e., a morphism σ: G//G → G such that π_{G} ∘ σ = id_{G//G}) is the example of such S and, for char k = 0, every cross-section in G is obtained in this manner (see Remark 1 in Section 6).

In 1965 Steinberg gave an explicit construction of a section of π_{G} for every simply connected semisimple group G (see his celebrated paper [Ste_{1}]). Its image is the cross-section that intersects every regular conjugacy class and does not intersect other conjugacy classes.

In this paper we explore what happens in the general case, i.e., when G is not necessarily simply connected. In this case the following two facts about cross-sections in G for char k = 0 are known.

First, by [CTKPR, Theorem 0.3] in every connected semisimple algebraic group G there is a rational section of π_{G}, i.e., a section over a dense open subset of G//G (local section).

Second, by Kostant’s theorem [K, Theorem 0.10] there is an infinitesimal counterpart of Steinberg’s cross-section: for the adjoint action of G on its Lie algebra Lie G, there is a closed irreducible subvariety in Lie G that intersects every regular G-orbit at a single point.

In order to formulate our result consider the universal covering of G, i.e., an isogeny

τ: Ĝ ⟶ G

such that Ĝ is a simply connected semisimple algebraic group and the composition of τ with every projective rational representation of G lifts to a linear one of Ĝ.

We prove the following

Theorem 1.2. Let G be a connected semisimple algebraic group.

(i) The following properties are equivalent: (a) there is a cross-section in G;

(b) the isogeny τ is bijective. (ii) If σ: G//G → G is a section of π_{G}, then the cross-section σ(G//G) in G intersects every regular conjugacy class and does not intersect other conjugacy classes.

Remark 1.3. The isogeny τ is bijective if and only if it is either an isomorphism or purely inseparable (radical). The latter holds if and only if char k = p > 0 and p divides the order of the fundamental group of G.

Statement (ii) of the next corollary answers a question posed in [CTKPR, p. 4].

Corollary 1.4. Let G be a connected semisimple algebraic group.

(i) If a section of π_{G} exists, then τ is bijective.

(ii) For char k = 0, the following properties are equivalent:

(a) there is a section of π_{G};

(b) there is a cross-section in G;

(c) G is simply connected.

Theorem 1.2 is proved in Section 2.

One can show (see below Lemma 3.1) that if a cross-section in G exists, then, at least for char k = 0, the variety G//G is smooth (the converse is not true). The known criterion of smoothness of G//G (Theorem 3.2) may be interpreted as that of the existence of rk G generators of k[G]^{G}. In Section 3 we consider the general case and describe a minimal generating set of k[G]^{G} and singularities of G//G for any G. This is based on the property that actually G//G is a toric variety of a maximal torus T of G. In particular, it also implies the affirmative answer to Grothendieck’s question cited in the epigraph:

Corollary 3.7. G//G and T/W are the rational varieties.

Here W is the Weyl group of G, i.e., the quotient of T in its normalizer N_{G}(T), acting on T via conjugation.

Parallel to this we describe a minimal generating set of the representation ring R(G) of G. Note that finding generators of R(G) attracted people’s attention during long time, in particular, because of the bearing on the K-theory (cf., e.g., [Hus, Chap. 13] where the generators of R(G) are found for some classical G’s utilizing the ad hoc bulky arguments; see also [A]). Singularities of G//G attracted the attention as well (see [Sl, Sects. 3.15, 4.5]).

The precise formulations of these results are given below in Theorems 3.5, 3.9 and Lemma 3.10.

Constructing the generating sets of k[G]^{G} is the topic of yet two Grothendieck’s questions asked in [GS, p. 241]. In Section 4 we answer the first question in the negative and the second in the positive.

In Section 5 we consider rational sections of π_{G} and rational cross-sections in G, i.e., irreducible closed subsets S of G that intersect at a single point every fiber of π_{G} over a point of a dense open subset of G//G. The closure of the image of a rational section of π_{G} is the example of such S and, for char k = 0, every rational cross-section in G is obtained in this way.

First we show that the existence of a rational section of π_{G} is equivalent to another property. Namely, W also acts on G/T as follows:

w · gT := gw^{.}^{−1}T,  (1)

where w^{.} ∈ N_{G}(T) is a representative of w. We prove

Theorem 1.5. Let G be a connected semisimple algebraic group. The following properties are equivalent:

(i) there is a rational section of π_{G}; (ii) there is a W -equivariant rational map T −−→ G/T.

Then we consider the existence problem and prove the following.

Recall (see [Bor, 22.3]) that the isogeny τ is called central if ker τ lies in the center of G and ker dτ_{e} lies in the center of Lie G.

The next theorem answers the other Grothendieck’s question cited in the epigraph.

Theorem 1.6. Let G be a connected semisimple algebraic group.

(i) There is a rational cross-section in G.

(ii) If the isogeny τ is central, then there is a rational section of π_{G}.

For char k = 0, this theorem has been proved earlier in [CTKPR, Theorem 0.3]. The strategy and the essential part of our proof are the same: we use the relevant characteristic free results from [CTKPR], but bypass Theorem 2.12 from this paper (whose proof is based on the assumption char k = 0) by exploring properties of π_{G} and proving that versality of G holds in any characteristic; this permits us to use Steinberg’s section of πĜ in place of Kostant’s cross-section in Lie G used in [CTKPR].

Theorems 1.5 and 1.6 yield the following

Corollary 1.7. Let G be a connected semisimple algebraic group. If the isogeny τ is central, then there is a W -equivariant rational map T −−→ G/T.

Section 6 contains some remarks, questions, and an example of a cross-section S in G such that π_{G}|_{S} is not separable (hence S is not the image of a section of π_{G}).

Acknowledgements. I am grateful to Vik. Kulikov for the inspiring remark, to A. Parshin for the useful discussion, and to J.-L. Colliot-Thélène and Z. Reichstein for the comments.

2. Cross-sections in G

Fix a choice of Borel subgroup B̂ of Ĝ and maximal torus T̂ ⊂ B̂. Denote by X(T̂) the character lattice of T̂ in additive nonation. For λ ∈ X(T̂), denote by t^{λ} the value of λ: T̂ → G_{m} at t ∈ T̂. Let ϖ_{1}, …, ϖ_{r} ∈ X(T̂) be the system of fundamental weights of T̂ with respect to B̂.

Let ϱ_{i}: Ĝ → GL(V_{i}) be an irreducible representation of Ĝ with ϖ_{i} as the highest weight. Let χ_{ϖ_{i}} ∈ k[Ĝ]Ĝ be the character of ϱ_{i}.

Let Ĉ be the center of Ĝ; it is a finite subgroup of T̂. The conjugating action of Ĝ on itself commutes with the action of Ĉ on Ĝ by left translations. Therefore the latter action descends to Ĝ//Ĝ and

πĜ: Ĝ ⟶ Ĝ//Ĝ

becomes a Ĉ-equivariant morphism.

Endow the r-dimensional affine space A^{r} with the linear action of T̂ by the formula

t · (a_{1}, …, a_{r}) := (t^{ϖ_{1}}a_{1}, …, t^{ϖ_{r}}a_{r}),  t ∈ T̂, (a_{1}, …, a_{r}) ∈ A^{r}.  (2)

Lemma 2.1.

(i) The T̂-stabilizer of the point (1, …, 1) ∈ A^{r} is trivial. In particular, the considered action of T̂ on A^{r} is faithful.

(ii) There is a Ĉ-equivariant isomorphism

λ: Ĝ//Ĝ ⟶^{≃} A^{r}.

Proof. Since ϖ_{1}, …, ϖ_{r} generate X(T̂), we have

⋂_{i=1}^{r}{t ∈ T | t^{ϖ_{i}} = 1} = {e}.  (3)

But (2) entails that the T̂-stabilizer of the point (1, …, 1) coincides with the righthand side of equality (3). This proves (i).

By [Ste_{1}, Theorems 6.1, 6.16] the k-algebra k[Ĝ]Ĝ is freely generated by χ_{ϖ_{1}}, …, χ_{ϖ_{r}} and the morphism

θ: Ĝ ⟶ A^{r}, θ(g) = (χ_{ϖ_{1}}(g), …, χ_{ϖ_{r}}(g)),

is surjective. Hence there is an isomorphism λ: Ĝ//Ĝ ⟶ A^{r} such that the following diagram is commutative:

\xymatrix{ & Ĝ \ar[dl]_{πĜ} \ar[dr]^{θ} \\ Ĝ//Ĝ \ar[rr]^{λ} &  & A^{r}} .  (4)

The morphism θ is Ĉ-equivariant. Indeed, let c ∈ Ĉ. Since ϱ_{i} is irreducible, Schur’s lemma entails that ϱ_{i}(c) = μ_{i,c} id_{V_{i}} for some μ_{i,c} ∈ k. On the other hand, since c ∈ T̂, any highest vector in V_{i} with respect to B̂ is an eigenvector of c with the eigenvalue c^{ϖ_{i}}. Hence μ_{i,c} = c^{ϖ_{i}}. Therefore, for every g ∈ Ĝ, by (2) we have

θ(cg) = (χ_{ϖ_{1}}(cg), …, χ_{ϖ_{r}}(cg))
= (trace (ϱ_{1}(cg)), …, trace (ϱ_{r}(cg)))

= (trace (ϱ_{1}(c)ϱ_{1}(g)), …, trace (ϱ_{1}(c)ϱ_{r}(g)))
= (trace (c^{ϖ_{1}}ϱ_{1}(g)), …, trace (c^{ϖ_{r}}ϱ_{r}(g)))
= (c^{ϖ_{1}}trace (ϱ_{1}(g)), …, c^{ϖ_{r}}trace (ϱ_{r}(g)))
= (c^{ϖ_{1}}χ_{ϖ_{1}}(g), …, c^{ϖ_{r}}χ_{ϖ_{r}}(g))
= c · θ(g),

as claimed.

Since both θ and πĜ are Ĉ-equivariant and πĜ is surjective, commutativity of diagram (4) entails that λ is Ĉ-equivariant as well. This proves (ii). □

Corollary 2.2. Let g be a nonidentity element of Ĉ. Then there is no g-stable cross-section in Ĝ.

Proof. Assume the contrary and let Ŝ be a g-stable cross-section in Ĝ. Since πĜ is Ĉ-equivariant, πĜ|Ŝ: Ŝ → Ĝ//Ĝ is a bijective g-equivariant morphism. As, by Lemma 2.1(ii), there is a point of Ĝ//Ĝ fixed by Ĉ, hence by g, this implies that there is a point of Ŝ fixed by g. But for the action of Ĉ on Ĝ by left translations, the stabilizer of every point is trivial, a contradiction with g ≠ e. □

Given an element h of an algebraic group H, we shall denote its conjugacy class in H by H(h):

H(h) := {shs^{−1} | s ∈ H}.  (5)

Lemma 2.3. Let H and H̃ be connected algebraic groups and let σ: H̃ → H be an isogeny. Then the following properties hold:

(i) σ is a finite morphism;

(ii) σ(H̃(h)) = H(σ(h)) and dim H̃(h) = dim H(σ(h)) for every h ∈ H̃; (iii) if H̃(h) is a regular conjugacy class in H̃ (i.e., that of the maximal dimen-

sion ), then σ(H̃(h)) is a regular conjugacy class in H; (iv) if H and H̃ are semisimple, then for every h ∈ H̃,

σ(π−1 ; H̃(πH̃(h))) = π_{H}^{−1}(π_{H}(σ(h))).

Proof. The varieties H and H̃ are normal (even smooth) and the fiber of σ over every point of H is a finite set whose cardinality does not depend on this point. Hence (cf. [G_{1}, Sect. 2, Cor. 3]) H̃ is the normalization of H in the field of rational functions on H̃ and σ is the normalization map. This proves (i).

The first equality in (ii) holds as σ is an epimorphism of groups. The second follows the first and theorem on dimension of fibers, cf., e.g., [Bor, AG 10.1]. This proves (ii).

As σ is surjective, (iii) follows from (ii).

Since the fibers of πH̃ and π_{H} are the closures of regular conjugacy classes and, by (i), the map σ is closed, (iv) follows from (iii). □

Corollary 2.4. Let G̃ be a connected semisimple algebraic group and let σ: G̃ → G be a bijective isogeny.

(i) If S̃ is a cross-section in G̃, then σ(S̃) is a cross-section in G. (ii) If S is a cross-section in G, then σ^{−1}(S) is a cross-section in G̃.

The same holds if “cross-section” is replaced with “rational cross-section”.

Proof. By Lemma 2.3(i) the bijective map σ is closed. Hence it is a homeomorphism. Both claims follow from this, the definitions of cross-section and rational cross-section, and Lemma 2.3(iv). □

Lemma 2.5. Assume that there is a subgroup Z of Ĉ such that G = Ĝ/Z and τ is the quotient morphism Ĝ → Ĝ/Z. Then there is a morphism

φ: Ĝ//Ĝ ⟶ G//G  (6)

such that (i) (G//G, φ) is a categorical quotient for the action of Z on Ĝ//Ĝ; (ii) the following diagram is commutative:

\xymatrix{Ĝ \ar[r]^{τ} \ar[d]_{πĜ} & G \ar[d]^{π_{G}} \\ Ĝ//Ĝ \ar[r]^{φ} & G//G} ;  (7)

(iii) for every point x ∈ Ĝ//Ĝ, the following equality holds:

τ(π−1 ; Ĝ(x)) = π_{G}^{−1}(φ(x)).  (8)

Proof. As τ^{∗}, π∗ ; Ĝ, and π_{G}^{∗} are injections, there is a unique morphism (6) such that τ^{∗} ∘ π_{G}^{∗} = π∗ ; Ĝ ∘ φ^{∗}, i.e., diagram (7) is commutative.

Consider the action of Ĝ on G via the isogeny τ and the conjugating action of G on itself. The isogeny τ is then Ĝ-equivariant and Ĝ-orbits in G are G-conjugacy classes, so we have k[G]^{G} = k[G]Ĝ. Since the conjugating action of Ĝ on itself commutes with the action of Z by left translations, we have

π∗ ; Ĝ(φ^{∗}(k[G//G])) = τ^{∗}(π_{G}^{∗}(k[G//G])) = τ^{∗}(k[G]^{G}) = τ^{∗}(k[G]Ĝ) = (τ^{∗}(k[G]))Ĝ

= (k[Ĝ]^{Z})Ĝ = (k[Ĝ]Ĝ)^{Z} = (π∗ ; Ĝ(k[Ĝ//Ĝ]))^{Z} = π∗ ; Ĝ(k[Ĝ//Ĝ]^{Z}).

Thus, φ^{∗}(k[G//G]) = k[Ĝ//Ĝ]^{Z}. This proves (i) and (ii). Lemma 2.3(iv) and commutativity of diagram (7) imply (iii). □

Below, given a variety Z, we denote by T_{z,Z} the tangent space of Z at a point z.

Proof of Theorem 1.2. First, we shall prove criterion (i).

1. By Steinberg’s theorem, Ĝ has a cross-section. Hence, by Corollary 2.4, if τ is bijective, then there exists a cross-section in G as well.

So we may assume that τ is not bijective and we then have to prove that there is no cross-section in G. Solving this problem, we may assume that τ is separable. Indeed, if this is not the case, then by [Bor, Prop. 17.9] there exist a connected semisimple algebraic group G̃ and a commutative diagram of isogenies

\xymatrix{Ĝ \ar[rr]^{τ} \ar[dr]_{μ} &  & G \\  & G̃ \ar@{-->}[ur]_{σ}} ,  (9)

where μ is separable and σ is purely inseparable. As σ is bijective, Corollary 2.4 then reduces the problem to proving that there is no cross-section in G̃, i.e., we may replace G by G̃ and τ by μ.

So from now on we may (and shall) assume that τ is a separable isogeny of degree ⩾ 2. This means that there is a nontrivial subgroup Z of Ĉ such that G = Ĝ/Z and τ is the quotient morphism Ĝ → Ĝ/Z.

2. Now, arguing on the contrary, assume that there is a cross-section S in G.

Claim 1. (i) For every point x ∈ Ĝ//Ĝ, the intersection

π−1 ; Ĝ(x) ∩ τ^{−1}(S)  (10)

is a nonempty subset of a single Z-orbit; in particular, it is finite. (ii) There is a nonempty open subset U of Ĝ//Ĝ such that, for every x ∈ U,

intersection (10) is a single point.

Proof of Claim 1. Consider diagram (7). Since S ∩ π_{G}^{−1}(φ(x)) is a single point g, we deduce from (8) that intersection (10) is contained in τ^{−1}(g). This proves (i) as the fibers of τ are Z-orbits.

By Lemma 2.1(i) there is a nonempty open subset U in Ĝ//Ĝ such that the Ĉstabilizer of every point of U is trivial. Take a point x ∈ U. Assume that intersection (10) contains two points g_{1} and g_{2} ≠ g_{1}. By (i) there exists an element z ∈ Z such that g_{2} = zg_{1}. As πĜ is Ĉ-equivariant, x = πĜ(g_{2}) = πĜ(zg_{1}) = z · πĜ(g_{1}) = z · x. Thus, z belongs to the Ĉ-stabilizer of x. The definition of U then implies that z = e. Hence g_{1} = g_{2}, a contradiction. This proves (ii). □

3. Since all the fibers of τ are finite, every irreducible component of τ^{−1}(S) has dimension ⩽ dim S = r and at least one of them has dimension r.

Claim 2. (i) There is a unique r-dimensional irreducible component Ŝ of τ^{−1}(S).

(ii) τ(Ŝ) = S.

Proof of Claim 2. Let Ŝ be an r-dimensional irreducible component of τ^{−1}(S). Then τ(Ŝ) contains an open subset of S. Since τ is closed, this proves (ii).

¿From (ii) we conclude that

π_{G}(τ(Ŝ)) = G//G.  (11)

But by Lemma 2.5 the fibers of φ in commutative diagram (7) are finite. This and (11) imply that πĜ(Ŝ) contains a nonempty open subset of Ĝ//Ĝ.

Let now Ŝ′ be another r-dimensional irreducible components of τ^{−1}(S). Then, as above, πĜ(Ŝ′) contains a nonempty open subset of Ĝ//Ĝ as well. Therefore, πĜ(Ŝ) ∩ πĜ(Ŝ′) contains a nonempty open subset V of Ĝ//Ĝ. We may assume that V ⊆ U for U from Claim 1(ii). The latter then yields that π−1 ; Ĝ(V) ∩ Ŝ = π−1 ; Ĝ(V) ∩ Ŝ′. As

both sides of this equality are the open subsets of respectively Ŝ and Ŝ′, we infer that Ŝ = Ŝ′. This proves (i). □

4. As Ŝ is a unique r-dimensional irreducible component of the Z-stable variety τ^{−1}(S), we conclude that Ŝ is Z-stable. We shall now show that Ŝ is a cross-section in Ĝ. As this property contradicts Corollary 2.2, the proof of (i) will be then completed.

5. Let x be a point of Ĝ//Ĝ. As S is a section of G, the intersection S ∩ π_{G}^{−1}(φ(x)) is a single point g ∈ G. By Claim 2(ii) there is a point ĝ ∈ Ŝ such that τ(ĝ) = g. Commutativity of diagram (7) then entails that x and x̂ := πĜ(ĝ) are in the same fiber of φ. Since the fibers of φ are Z-orbits, there is an element z ∈ Z such that x = z · x̂. As πĜ is Z-equivariant, this yields πĜ(zĝ) = x. But zĝ ∈ Ŝ as Ŝ is Z-stable and ĝ ∈ Ŝ. Hence π−1 ; Ĝ(x) ∩ Ŝ ≠ ∅, i.e.,

πĜ(Ŝ) = Ĝ//Ĝ.  (12)

6. It follows from Claim 1(i),(ii) and (12) that πĜ|Ŝ is the surjective morphism with finite fibers, bijective over an open subset of Ĝ//Ĝ. As Ĝ is normal, Ĝ//Ĝ is normal as well. Let ν: S̃ → Ŝ be the normalization. Then the surjective morphism πĜ|Ŝ ∘ ν: S̃ → Ĝ//Ĝ of normal varieties has finite fibers and is bijective over an open subset of Ĝ//Ĝ. Hence πĜ|Ŝ ∘ ν is bijective (see [G_{1}, Sect. 2, Cor. 2]). Whence πĜ|Ŝ is bijective as well, i.e., Ŝ is a cross-section in Ĝ. This completes the proof of (i).

We now turn to the proof of (ii).

Let S := σ(G//G). Take a point x ∈ S and put y := π_{G}(x). As π_{G}|_{S}: S → G//G is the isomorphism (σ is its inverse), d(π_{G}|_{S})_{x} is the isomorphism as well. Hence (dπ_{G})_{x} is surjective. As dim T_{y,G//G} ⩾ dim G//G = r, this implies that there are functions f_{1}, …, f_{r} ∈ k[G]^{G} such that (df_{1})_{x}, …, (df_{r})_{x} are linearly independent. By [Ste_{1}, Theorem 8.7] this yields that x is regular. As S intersects every fiber of π_{G} at a single point and every such fiber contains a unique regular orbit, this proves (ii). Thus, the proof of Theorem 1.2 comes to a close. □

3. Singularities of G//G and generators of k[G]^{G} and R(G)

The following lemma shows that there is a link between the existence of a crosssection in G and smoothness of G//G.

Lemma 3.1. Let char k = 0. If a surjective morphism α: X → Y of irreducible varieties admits a section σ: Y → X, then smoothness of X implies smoothness of Y.

Proof. Arguing on the contrary, assume that y is a singular point of Y, i.e.,

dim T_{y,Y} > dim Y.  (13)

Put x = σ(y) ∈ X. Since α ∘ σ = id_{Y}, the composition dα_{x} ∘ dσ_{y} is the identity map of T_{y,Y}. Hence dα_{x} is surjective, i.e., rk dα_{x} = dim T_{y,Y}. By (13) this yields

rk dα_{x} > dim Y.  (14)

As char k = 0, there is a dense open subset U of X such that rk dα_{z} = dim Y for every point z ∈ U, see [H, 14.4]. As z ↦ dim ker dα_{z} is the upper semi-continuous function [H, 14.6], we conclude that smoothness of X implies that rk dα_{z} ⩽ dim Y for every point z ∈ X. This contradicts (14). □

This prompts to explore smoothness of G//G. The answer is known:

Theorem 3.2 ([Ste_{3}, §3], [R_{1}, Prop. 4.1], [R_{2}, Prop. 13.3]). Let char k ≠ 2. The following properties are equivalent:

(i) G//G is smooth; (ii) G//G is isomorphic to the affine space A^{r}; (iii) G = G_{1}×⋯×G_{s} where every G_{i} is either a simply connected simple algebraic

group or isomorphic to SO_{n_{i}} for an odd n_{i}.

This criterion of smoothness of G//G may be also interpreted as that of the existence of r generators of the algebra of class functions on G. Below we describe a minimal system of generators of this algebra and singularities of G//G in the general case. This also yields a minimal system of generators of the representation ring of G.

Let B := τ(B̂) and T := τ(T̂). This is respectively a Borel subgroup and a maximal torus of G. We consider the lattice X(T) of characters of T as the sublattice of X(T̂) identifying μ ∈ X(T) with τ^{∗}(μ) ∈ X(T̂). Then X(T̂) is the weight lattice of X(T). The monoid of highest weights of simple Ĝ-modules (with respect to B̂ and T̂) is

D̂ := Nϖ_{1} + ⋯ + Nϖ_{r},  N = {0, 1, 2, …}.  (15)

and that of simple G-modules (with respect to B and T) is

D := D̂ ∩ X(T).  (16)

Let W be the Weyl group of Ĝ, i.e., the quotient of T̂ in its normalizer, acting on T̂ via conjugation. The Weyl group of T is naturally identified with W (see [Bor, Prop. 11.20 and Cor. 2(d) in 13.17]).

If ϖ ∈ D and E(ϖ) is a simple G-module with ϖ as the highest weight, we denote by χ_{ϖ} ∈ k[G]^{G} the character of E(ϖ).

Given a nonzero commutative ring A with identity element and a commutative monoid M, we denote by A[M] the semigroup ring of M over A. We identify A[M] with A ⊗_{Z} Z[M] in the natural way. If S is a submonoid of the multiplicative monoid of A[M] whose elements are linearly independent over A, then the subring of A[M] generated by S is naturally identified with A[S]. In particular, we consider A[X(T)] and A[D] as the subrings of A[X(T̂)]. The former is stable with respect to the natural action of W on Z[X(T̂)]. Using the notation and terminology of Bourbaki [Bou_{2}, VI.3], we denote by e^{μ} the element of Z[X(T̂)] corresponding to μ ∈ X(T̂) and put

S(e^{μ}) := ∑_{ν∈W ·μ} e^{ν} ∈ Z[X(T̂)]^{W} .  (17)

Given an algebraic group H, we denote by R(H) the representation ring of H: its additive group is the Grothendieck group of the category of finite dimensional algebraic H-modules with respect to exact sequences and the multiplication is induced by tensor product of modules. Using τ, we identify R(G) in the natural way with the subring of R(Ĝ).

If E is a finite dimensional algebraic G-module and E_{μ} is its weight space of a weight μ ∈ X(T), then the formal character of E,

ch_{G}[E] := ∑_{μ∈X(T)}(dim E_{μ})e^{μ},  (18)

is an element of Z[X(T)]^{W} depending only on the class [E] of E in R(G). Clearly,

ch_{G}[E ⊗ E′] = ch_{G}[E] ch_{G}[E′].  (19)

According to [Se, 3.6], the homomorphism of Z-modules

ch_{G}: R(G) ⟶ Z[X(T)]^{W} ,  [E] ↦ ch_{G}[E],  (20)

is an isomorphism. By (19) it is an isomorphism of rings.

Definition 3.3. Let ϖ ∈ D̂. We say that an element x ∈ Z[X(T̂)]^{W} is ϖ-sharp if the following property (M) holds:

(M) e^{ϖ} is the unique maximal term of x.

Example 3.4. The elements S(e^{ϖ}) and chĜ[E(ϖ)] are ϖ-sharp (this follows, e.g., from [Bou_{2}, VI.1.6, Prop. 18] and [Hum_{1}, 31.3, Theorem]). □

Property (M) implies that the support of a ϖ-sharp element x lies in ϖ + X(T). This and [Bou_{2}, VI.3.4, formula (6)] yield

x = S(e^{ϖ}) + sum of some S(e^{ϖ′})’s with ϖ′ ∈ D̂, ϖ′ < ϖ.  (21)

By [Bou_{2}, VI.3.2, Lemma 2] if an element x′ is a ϖ′-sharp, then xx′ is (ϖ+ϖ′)-sharp. Now fix a ϖ_{i}-sharp element x_{ϖ_{i}} ∈ Z[X(T̂)]^{W}, i = 1, …, r, and put

x_{ϖ} := x_{ϖ_{1}}^{m_{1}} ⋯ x_{ϖ_{r}}^{m_{r}}  for  ϖ = m_{1}ϖ_{1} + ⋯ + m_{r}ϖ_{r} ∈ D̂.

By [Bou_{2}, VI.3.4, Theorem 1] the set {x_{ϖ} | ϖ ∈ D̂} is then a basis of the Z-module Z[X(T̂)]^{W}. As {e^{μ} | μ ∈ X(T)} is a basis of the Z-module Z[X(T)] and the support of x_{ϖ} lies in ϖ + X(T), we deduce from this and (16) that {x_{ϖ} | ϖ ∈ D} is a basis of the Z-module Z[X(T)]^{W}. Hence the homomorphism of the Z-modules

ϑ: Z[X(T)]^{W} → Z[D],  ϑ(x_{ϖ}) = e^{ϖ} for ϖ ∈ D,  (22)

is an isomorphism. Since x_{ϖ+ϖ′} = x_{ϖ}x_{ϖ′}, it is, in fact, an isomorphism of rings.

As by Dedekind’s theorem {f_{μ}: T → k, t ↦ t^{μ} | μ ∈ X(T̂)} is a basis of the vector space k[T̂] over k, the k-linear map k[T̂] → k[X(T̂)], f_{μ} ↦ e^{μ}, is the isomorphism of kalgebras. We identify them by means of this isomorphism. So we have k[T] = k[X(T)] and

k[T/W] = k[T]^{W} = k[X(T)]^{W} .  (23)

Finally, take into account that by [Ste_{1}, 6.4] the restriction map

res: k[G//G] = k[G]^{G} ⟶ k[T]^{W} ,  res(f) = f|_{T} ,  (24)

is an isomorphism of k-algebras. Summing up, we obtain

Theorem 3.5.

(i) G//G and T/W are the affine toric varieties of T whose algebras of regular functions are isomorphic to k[D].

(ii) In the diagram

k[G//G] ⟶^{res} k[T/W] ⟶^{id⊗ϑ} k[D]

(see (24), (23), (22)) both maps are the isomorphisms of k-algebras. (iii) Let F be the simple subring of k. Then the image of F ⊗_{Z} R(G) in k[G]^{G}

under the composition of the ring isomorphisms

k ⊗_{Z} R(G) ⟶^{id⊗ch_{G}} k[X(T)]^{W} = k[T]^{W} ⟶^{res^{−1}} k[G]^{G}  (25)

is an F -form of k[G//G] isomorphic to F ⊗_{Z}R(G). In particular, if char k = 0, it is a Z-form of k[G//G] isomorphic to R(G).

Remark 3.6. The fact that “multiplicative invariants” of finite reflection groups are semigroup algebras is already in the literature, first implicitly, then explicitly, see the historical account in [L_{1}, Introduction]. Essentially, the main ingredients date back to [Ste_{1}, §6] and [Bou_{2},VI §3].

Since toric varieties are rational, Theorem 3.5(i) yields

Corollary 3.7. G//G and T/W are rational varieties.

In the next statement Theorem 3.5 is applied to finding a minimal system of generators of the algebra k[G]^{G} and that of the ring R(G).

Let H be the Hilbert basis of the monoid D, i.e., the set of all its indecomposable elements:

H = D_{+} \ 2D_{+} where D_{+} := D \ {0}, 2D_{+} := D_{+} + D_{+}.  (26)

The set H is finite, generates D, and every generating set of D contains H (see, e.g., [L_{2}, 3.4]).

Remark 3.8. There is an algorithm for computing H, see [Stu, 13.2] (cf. also Example 3.11 below).

Theorem 3.9.

(i) The cardinality of every generating set of the algebra k[G]^{G} of class functions on G is not less than the cardinality of H. The same holds for every generating set of the representation ring R(G) of G. (ii) {[E(ϖ)] | ϖ ∈ H} is a generating set of the ring R(G).

(iii) {χ_{ϖ} | ϖ ∈ H} is a generating set of the algebra k[G]^{G}.

Proof. (i) Let Y be the affine toric variety of T with k[Y] = k[D]. The linear span I of {e^{ϖ} | ϖ ∈ D_{+}} over k is a maximal T -invariant ideal in k[Y]. Hence I/I^{2} is the cotangent space of Y at the T -fixed point v where I vanishes. As I^{2} is the linear span of {e^{ϖ} | ϖ ∈ 2D_{+}} over k, this and (26) yield

dim T_{v,Y} = dim I/I^{2} = |H|.  (27)

Now take into account that, given an affine algebraic variety X, the algebra k[X] can be generated by d elements if and only if X admits a closed embedding in A^{d}. Hence d ⩾ dim T_{x,X} for every point x ∈ X. This, Theorem 3.5(i),(iii), and (27) prove (i).

(ii) Let μ ∈ D. As H generates D, we deduce from Example 3.4 that there is a μ-sharp monomial M^{μ} in the elements of the set {ch_{G}[E(ϖ)] | ϖ ∈ D}. By (21) we have

M^{μ} = S(e^{μ}) + sum of some S(e^{μ′})’s with μ′ ∈ D, μ′ < μ.  (28)

But {S(e^{μ}) | μ ∈ D} is a basis of the Z-module Z[X(T)]^{W} (see [Bou_{2}, VI.3.4, Lemma 3]). By [Bou_{2}, VI.3.4, Lemma 4] and (28) we then conclude that the set {M^{μ} | μ ∈ D} generates the Z-module Z[X(T)]^{W}. This means that the ring Z[X(T)]^{W} is generated by the set {ch_{G}[E(ϖ)] | ϖ ∈ H}. As (20) is an isomorphism of rings, this proves (ii).

(iii) It follows from (ii) that the set {1 ⊗ [E(ϖ)] | ϖ ∈ H} generates the ring k ⊗_{Z} R(G). But formula (18) shows that χ_{ϖ} is the image of 1 ⊗ E[(ϖ)] under the composition of the ring isomorphisms in diagram (25). This proves (iii). □

Since the Weyl chambers are simplicial cones, Theorem 3.5(i) implies, at least for char k = 0, that G//G and T/W are isomorphic to the quotient of A^{r} by a linear action of a certain finite abelian group (see, e.g., [O, Prop. 1.25]). In particular, G//G and T/W may have only finite quotient singularities. Below this finite group and its action on A^{r} are explicitly described assuming that τ is separable.

The latter assumption means that there is a subgroup Z of Ĉ such that G = Ĝ/Z and τ is the quotient morphism Ĝ → Ĝ/Z. In this situation we have

X(T) = {μ ∈ X(T̂) | c^{μ} = 1 for every c ∈ Z}.  (29)

Consider the T̂-orbit map of the point (1, …, 1) ∈ A^{r}:

ι: T̂ ⟶ A^{r},  ι(t) = t · (1, …, 1).  (30)

The map ι^{∗}: k[A^{r}] → k[T̂] = k[X(T̂)] is an embedding as ι is dominant by Lemma 2.1(i). Let y_{1}, …, y_{r} be the standard coordinate functions on A^{r}. Then (2) and (30) yield

ι^{∗}(y_{i}) = e^{ϖ_{i}}.  (31)

From (2) we deduce that k[A^{r}]^{Z} is the linear span over k of all monomials y^{m_{1}} ⋯ y^{m_{r}} with m_{1}, …, m_{r} ∈ N such that c^{m_{1}ϖ_{i}+···+m_{r}ϖ_{r}} = 1 for every c ∈ Z. By (29) the latter condition is equivalent to the inclusion m_{1}ϖ_{i} + ⋯ + m_{r}ϖ_{r} ∈ X(T). This, (31), (15), and (16) imply that ι^{∗}(k[A^{r}]^{Z}) = k[D]. Thus, taking into account Theorem 3.5, we obtain the isomorphisms of k-algebras

k[A^{r}]^{Z} ⟶^{ι^{∗}} k[D] ⟶^{(id⊗ϑ)^{−1}} k[T/W] ⟶^{(res)^{−1}} k[G//G].

that, in turn, induce the isomorphisms of varieties G//G → T/W → A^{r}/Z.

By means of a special parametrization of T̂ one obtains an explicit description of the elements of Ĉ well adapted for computing k[A^{r}]^{Z}. Since Ĝ = Ĝ_{1} × ⋯ × Ĝ_{s} and Ĉ = Ĉ_{1} × ⋯ × Ĉ_{s} where every Ĝ_{i} is a nontrivial normal simply connected simple subgroup of Ĝ and Ĉ_{i} is the center of Ĝ_{i}, it suffices to describe this parametrization for simple groups Ĝ. The answer is given below in Lemma 3.10.

Namely, let α̂_{1}, …, α̂_{r} ∈ X(T̂) be the system of simple roots of T̂ with respect to B̂ and let α_{i}^{∨}: G_{m} → T̂ be the coroot corresponding to α̂_{i}. Then, for every s ∈ G_{m},

(α_{i}^{∨}(s))^{ϖ_{j}} = \begin{cases}s & if i = j, \\ 1 & if i ≠ j. \end{cases}  (32)

If ⟨ , ⟩ is the natural pairing between the lattices of characters and cocharacters of T̂, we put n_{ij} := ⟨α̂_{i}, α_{j}^{∨}⟩. So (n_{ij})_{i,j=1}^{r} is the Cartan matrix of Ĝ.

By [Ste_{2}, Lemma 28(b),(d) and its Cor. (a)] the map

ν: G_{m}^{r} ⟶ T̂,  ν(s_{1}, …, s_{r}) = α_{1}^{∨}(s_{1}) ⋯ α_{r}^{∨}(s_{r}),  (33)

is an isomorphism of groups and

Ĉ = {α_{1}^{∨}(s_{1}) ⋯ α_{r}^{∨}(s_{r}) | s_{1}^{n_{i1}} ⋯ s_{r}^{n_{ir}} = 1 for every i = 1, …, r}.  (34)

By (32) and (33) we have

(ν(s_{1}, …, s_{r}))^{ϖ_{i}} = (α_{1}^{∨}(s_{1}))^{ϖ_{i}} ⋯ (α_{r}^{∨}(s_{r}))^{ϖ_{i}} = s_{i}.

This and (2) imply that, for every s = (s_{1}, …, s_{r}) ∈ G_{m}^{r} and (a_{1}, …, a_{r}) ∈ A^{r}, we have

ν(s) · (a_{1}, …, a_{r}) = (s_{1}a_{1}, …, s_{r}a_{r}).

Lemma 3.10. For every simple simply connected group Ĝ, the subgroup ν^{−1}(Ĉ) of the torus G_{m}^{r} is described in the following Table 1 (simple roots in (33) are numerated as in [Bou_{2}]):

Table 1.

type of Ĝ  ν^{−1}(Ĉ)
A_{r}  {(s, s^{2}, s^{3}, …, s^{r}) | s^{r+1} = 1}
B_{r}  {(1, …, 1, s^{2}) | s^{2} = 1}
C_{r}  {(s, 1, s, 1, …, s^{r mod 2}) | s^{2} = 1}
D_{r}, r odd  {(s^{2}, 1, s^{2}, 1, …, s^{2}, s, s^{−1}) | s^{4} = 1}
D_{r}, r even  {(s, 1, s, 1, …, s, 1, st, t) | s^{2} = t^{2} = 1}
E_{6}  {(s, 1, s^{−1}, 1, s, s^{−1}) | s^{3} = 1}
E_{7}  {(1, s, 1, 1, s, 1, s) | s^{2} = 1}
E_{8}  {(1, 1, 1, 1, 1, 1, 1, 1)}
F_{4}  {(1, 1, 1, 1)}
G_{2}  {(1, 1)}

Proof. By (34) an element (s_{1}, …, s_{r}) ∈ G_{m}^{r} lies in ν^{−1}(Ĉ) if and only if (s_{1}, …, s_{r}) is a solution of the system of equations

\begin{matrix}x_{1}^{n_{11}} ⋯ x_{r}^{n_{1r}} = 1, \\ . . . . . . . . . . . \\ x_{1}^{n_{r1}} ⋯ x_{r}^{n_{rr}} = 1, \end{matrix}  (35)

where (n_{ij})_{i,j=1}^{r} is the Cartan matrix of Ĝ.

Let, for instance, Ĝ be of type D_{r} for even r. Using the explicit form of the Cartan matrix [Bou_{2}, Planche IV], one immediately verified that every element of C′ := {(s, 1, s, 1, …, s, 1, st, t) | s^{2} = t^{2} = 1} is a solution of (35). Hence, C′ ⊆ ν^{−1}(Ĉ). On the other hand, the fundamental group of the root system of type D_{r} is isomorphic to Z/2Z ⊕ Z/2Z (see [Sp, 8.1.11] and [Bou_{2}, Planche IV]). Hence, the Smith normal form of (n_{ij})_{i,j=1}^{r} is diag (1, …, 1, 2, 2). Therefore, there is a basis β_{1}, …, β_{r} of the coroot lattice of T̂ such that, for (s_{1}, …, s_{r}) ∈ G_{m}^{r}, we have β_{1}(s_{1}) ⋯ β_{r}(s_{r}) ∈ Ĉ if and only if (s_{1}, …, s_{r}) is a solution of the system

x_{1} = 1, …, x_{r−2} = 1, x_{r−1}^{2} = 1, x_{r}^{2} = 1.

This yields |C′| = |Ĉ|; whence C′ = ν^{−1}(Ĉ).

For the groups of the other types the proofs are similar. □

The following examples illustrate how this can be applied to exploring singularities of G//G and finding the minimal generating sets {χ_{ϖ} | ϖ ∈ H} and {[E(ϖ)] | ϖ ∈ H} of, respectively, the algebra of class functions on G and the representation ring of G.

Examples 3.11.

(1) Let Ĝ be of type B_{r} (where B_{1} := A_{1}) and let char k ≠ 2. Table 1 implies that ν^{−1}(Ĉ) is generated by (1, …, 1, −1). Whence k[A^{r}]Ĉ=k[y_{1}, …, y_{r−1}, y_{r}^{2}]. Therefore, for the adjoint G, i.e., for G = SO_{2r+1}, the variety G//G is isomorphic to A^{r} (this agrees with Theorem 3.2) and

H = {ϖ_{1}, …, ϖ_{r−1}, 2ϖ_{r}}.

(2) Let Ĝ be of type D_{r}, let char k ≠ 2, and let Z := {t ∈ Ĉ | t^{ϖ_{1}} = 1}. Then G := Ĝ/Z = SO_{2r}. Table 1 implies that ν^{−1}(Z) is generated by (1, …, 1, −1, −1). Whence k[A^{r}]^{Z} = k[y_{1}, …, y_{r−2}, y_{r−1}^{2}, y_{r}^{2}, y_{r−1}y_{r}]. Therefore, G//G is isomorphic to A^{r−2} × X where X is a nondegenerate quadratic cone in A^{3} and

H = {ϖ_{1}, …, ϖ_{r−2}, 2ϖ_{r−1}, 2ϖ_{r}, ϖ_{r−1} + ϖ_{r}}.

(3) Let Ĝ be of type E_{7} and let char k ≠ 2. Table 1 implies that k[A^{7}]Ĉ is minimally generated by y_{1}, y_{3}, y_{4}, y_{6} and all the monomials of order 2 in y_{2}, y_{5}, y_{7}. Therefore, if G is adjoint, then G//G is isomorphic to A^{4} × Y where Y is the affine cone over the Veronese variety ν_{2}(P^{2}) in P^{5} (in particular, the tangent space of the 7-dimensional variety G//G at the unique fixed point of T, see Theorem 3.5(i) and (27), is 10dimensional) and

H = {ϖ_{1}, ϖ_{3}, ϖ_{4}, ϖ_{6}, 2ϖ_{2}, 2ϖ_{5}, 2ϖ_{7}, ϖ_{2} + ϖ_{5}, ϖ_{2} + ϖ_{7}, ϖ_{5} + ϖ_{7}.}. □

4. Yet two Grothendieck’s questions

Theorem 3.9 describes a minimal generating set of the algebra k[G]^{G} of class functions on G. Constructing the generating sets of k[G]^{G} is the topic of yet two Grothendieck’s questions in [GS, p. 241]:

“[…] When G is an adjoint group, is it possible to generate the affine ring of I(G) with coefficients of the Killing polynomial? In the general case, is it enough to take the coefficients of analogous polynomials for certain linear representations (perhaps arbitrary faithful representations)? […]”

Below we answer these questions.

Let ϱ: G→GL(V) be a finite dimensional linear representation of G. Define the set

C_{ϱ} := {c_{ϱ,i} ∈ k[G] | i = 1, …, dim V }

by the equality

det(xI − ϱ(g)) = ∑_{i=0}^{dim V} x^{dim V −i}c_{ϱ,i}(g) for every g ∈ G,  (36)

where x is a variable. If V =E(ϖ) and ϱ determines the G-module structure of E(ϖ), we put C_{ϖ} := C_{ϱ}.

Clearly, c_{ϱ,i} ∈k[G]^{G} and c_{ϱ,1} is the character of ϱ. Hence by Theorem 3.9(iii)

⋃_{ϖ∈H} C_{ϖ}

is a generating set of the algebra k[G]^{G}. This answers the second Grothendieck’s question in the affirmative.

In order to answer the first one in the negative it is sufficient to find an adjoint G and two elements z_{1}, z_{2} ∈ T such that

(i) z_{1} and z_{2} are not in the same W -orbit; (ii) the spectra of the linear operators Ad_{G} z_{1} and Ad_{G} z_{2} on the vector space

Lie G coincide. Indeed, property (i) implies that there is a function f ∈ k[T]^{W} such that f(z_{1}) ≠ f(z_{2}). Given isomorphism (24), this means that there is a function f̃ ∈ k[G]^{G} such that f̃(z_{1}) ≠ f̃(z_{2}). On the other hand, (36) and property (ii) imply that

c_{Ad_{G},i}(z_{1}) = c_{Ad_{G},i}(z_{2}) for every i.

Therefore, f̃ is not in the subalgebra of k[G]^{G} generated by C_{Ad_{G}}, i.e., the latter is not a generating set of k[G]^{G}.

The following two examples show that one indeed can find G, z_{1}, and z_{2} sharing properties (i) and (ii).

Examples 4.1.

(1) Let G = H × H where H is a connected adjoint semisimple algebraic group. Let T = S × S where S is a maximal torus of H. Let W_{S} be the Weyl group of H naturally acting on S. Take any two elements a, b ∈ S that are not in the same W_{S}-orbit and put z_{1} := (a, b), z_{2} := (b, a) ∈ T. As W = W_{S} × W_{S}, property (i) holds. On the other hand, clearly, for every i = 1, 2, the spectrum of Ad_{G} z_{i} is the union of the spectra of Ad_{H} a and Ad_{H} b; whence property (ii) holds.

(2) In this example G is simple, namely, G = PGL_{3}. Let α_{1}, α_{2} ∈ X(T) be the simple roots of T with respect to B. As the map T → G_{m}^{2}, t ↦ (t^{α_{1}}, t^{α_{2}}), is surjective (in fact, an isomorphism), for every u, v ∈ k, uv ≠ 0, there are z_{1}, z_{2} ∈ T such that z_{1}^{α_{1}} = u, z_{1}^{α_{2}} = v and z_{2}^{α_{1}} = v, z_{2}^{α_{2}} = u. For these z_{1}, z_{2}, property (ii) holds as the set of roots of G with respect to T is {±α_{1}, ±α_{2}, ±(α_{1} + α_{2})}. Now take u and v such that all elements u, u^{−1}, v, v^{−1}, uv, u^{−1}v^{−1} are pairwise different. Then property (i) holds as there are no w ∈ W such that w(α_{1}) = α_{2} and w(α_{2}) = α_{1}.

5. Rational cross-sections

Recall from [Ste_{1}, 2.14, 2.15] that an element x ∈ G is called strongly regular if its centralizer G_{x} is a maximal torus. Such x is regular and semisimple. Strongly regular elements form a dense open subset G_{0} of G stable with respect to the conjugating action of G. Every G-orbit in G_{0} is regular and closed in G. We put

(G//G)_{0} := π_{G}(G_{0})  and  T_{0} := T ∩ G_{0}.

Abusing the notation, we denote π_{G}|_{G_{0}} still by π_{G}:

π_{G}: G_{0} ⟶ (G//G)_{0}.  (37)

Lemma 5.1.

(i) (G//G)_{0} is an open smooth subset of G//G. (ii) π_{G}|_{T_{0}} : T_{0} → (G//G)_{0} is a surjective étale map. (iii) ((G//G)_{0}, π_{G}) is the geometric quotient for the action of G on G_{0}.

Proof. Since G//G is normal and all fibers of π_{G} are of constant dimension and irreducible, π_{G} is an open map (see [Bor, AG.18.4]). Hence (G//G)_{0} is open in G//G.

As every element of G_{0} is semisimple, it is conjugate to an element of T_{0}; whence π_{G}|_{T_{0}} is surjective.

The set T_{0} is open in T and W -stable. For every point t ∈ T_{0}, we have G_{t} = T, hence the W -stabilizer of t is trivial. Thus, the action of W on T_{0} is set theoretically free. Since T is smooth, G//G is normal, and (G//G, π_{G}|_{T}) is the quotient for the action of W on T (see [Ste_{1}, 6.4]), we deduce from [G_{3}, Exp. I, Théorème 9.5(ii)] and [Bou_{1}, V.2.3, Cor. 4] that π_{G}|_{T_{0}} is étale and hence (G//G)_{0} is smooth. This proves (i) and (ii).

By (ii) the map π_{G}: G_{0} → (G//G)_{0} is separable and surjective. As its fibers are G-orbits and (G//G)_{0} is normal, (iii) follows from [Bor, 6.6]. □

The group W acts on G/T ×T_{0} diagonally with the action on the first factor defined by formula (1). The group G acts on G/T × T_{0} via left translations of the first factor. These two actions commute with each other.

Consider the G-equivariant morphism

ψ: G/T × T_{0} ⟶ G_{0},  (gT, t) ↦ gtg^{−1}.  (38)

The proofs of Lemma 5.2 and Corollary 5.4 reproduce that from my letter [P].

Lemma 5.2. ψ is a surjective étale map.

Proof. As every G-orbit in G_{0} intersects T_{0}, surjectivity of ψ follows from (38).

Take a point z ∈ G/T × T_{0}. We shall prove that dψ_{z} is an isomorphism. As G/T × T_{0} and G_{0} are smooth, this is equivalent to proving that ψ is étale at z. Using that ψ is G-equivariant, we may assume that z = (eT, s), s ∈ T_{0}.

Let U_{α} be the one-dimensional unipotent root subgroup of G corresponding to a root α with respect to T and let θ_{α}: G_{a} → U_{α} be the isomorphism of groups such that

tθ_{α}(x)t^{−1} = θ_{α}(t^{α}x) for all t ∈ T, x ∈ G_{a},

see [Bor, IV.13.18]. Put

C_{α} := {(θ_{α}(x)T, s) ∈ G/T × T_{0} | x ∈ G_{a}},
D := {(eT, t) ∈ G/T × T_{0} | t ∈ T_{0}}.

The linear span of all T_{z,C_{α}}’s and T_{z,D} is T_{z,G/T ×T_{0}}. We have

\begin{aligned}ψ(θ_{α}(x)T, s) & = θ_{α}(x)sθ_{α}(x)^{−1} = θ_{α}(x)sθ_{α}(−x) \\ & = θ_{α}(x)θ_{α}(−s^{α}x)s = θ_{α}((1 − s^{α})x)s. \end{aligned}  (39)

Since s is regular, s^{α} ≠ 1. Hence (39) shows that ψ maps the curve C_{α} isomorphically onto the curve

ψ(C_{α}) = {θ_{α}((1 − s^{α})x)s | x ∈ G_{α}}.

Clearly, ψ(D) = T_{0} and ψ|_{D}: D → T_{0} is the isomorphism. But T_{e,G} is the linear span of T_{e,T} and the tangent spaces of the curves {θ_{α}(x) | x ∈ G_{α}} at e. Hence T_{s,G} is the linear span of T_{s,T} and the tangent spaces at s of the right translations of these curves by s. This implies the claim of the lemma. □

Corollary 5.3. ψ is separable.

Corollary 5.4. (G_{0}, ψ) is the quotient for the action of W on G/T × T_{0}.

Proof. By [Bor, Prop. II.6.6], as G_{0} is normal and ψ is surjective and separable, it suffices to prove that the fibers of ψ are W -orbits.

Using (1) and (38) one immediately verifies that the fibers of ψ are W -stable. On the other hand, let ψ(g_{1}T, t_{1}) = ψ(g_{2}T, t_{2}). By (38) this equality is equivalent to (g_{1}^{−1}g_{2})t_{2}(g_{1}^{−1}g_{2})^{−1}= t_{1}. By [Ste_{1}, 6.1] the latter, in turn, implies that there is an element w ∈ W such that

w^{.}t_{2}w^{.}^{−1} = (g_{1}^{−1}g_{2})t_{2}(g_{1}^{−1}g_{2})^{−1}.

Hence g_{1}^{−1}g_{2} = w^{.}z for z ∈ G_{t_{2}}. As t_{2} ∈ T is strongly regular, this yields that z ∈ T. Therefore,

(g_{2}T, t_{2}) = (g_{1}w^{.}T, w^{.}^{−1}t_{1}w^{.}) = w^{−1} · (g_{1}T, t_{1}).

Thus, (g_{1}T, t_{1}) and (g_{2}T, t_{2}) are in the same W -orbit. This completes the proof. □

Let π_{2} : G/T × T_{0} → T_{0} be the second projection. Clearly, (T_{0}, π_{2}) is the geometric quotient for the action of G on G/T × T_{0}. As ψ is G-equivariant, this implies that there is a morphism ϕ: T_{0} → G//G such that the following diagram is commutative:

\xymatrix{G/T × T_{0} \ar[d]_{π_{2}} \ar[r]^{ψ} & G_{0} \ar[d]^{π_{G}} \\ T_{0} \ar[r]^{ϕ} & (G//G)_{0}} .  (40)

Lemma 5.5.

(i) ϕ = π_{G}|_{T_{0}}.

(ii) For every point t ∈ T_{0}, the restriction of ψ to π_{2}^{−1}(t) is a G-equivariant isomorphism π_{2}^{−1}(t) → π_{G}^{−1}(ϕ(t)).

Proof. Take a point t ∈ T_{0}. Commutativity of diagram (40) and formula (38) yield that π_{G}(t) = π_{G}(ψ(eT, t)) = ϕ(π_{2}(eT, t)) = ϕ(t). This proves (i).

Commutativity of diagram (40) implies that the restriction of ψ to π_{2}^{−1}(t) is a G-equivariant morphism π_{2}^{−1}(t) → π_{G}^{−1}(ϕ(t)). As both π_{2}^{−1}(t) and π_{G}^{−1}(ϕ(t)) are the G-orbits and the stabilizers of their points are conjugate to T, this morphism is bijective. By Lemma 5.2 it is separable. Then, as π_{G}^{−1}(ϕ(t)) is normal, it is an isomorphism. This proves (ii). □

Proof of Theorem 1.5. Assume that (i) holds. Let σ: G//G −−→ G be a rational section of π_{G}, i.e., a section of π_{G} over a dense open subset U of (G//G)_{0}. Let S be the closure of σ(U). Put ρ := π_{G}|_{S}: S → (G//G)_{0}. Since π_{G} ∘ σ = id, shrinking U if necessary, we may assume that, for every point x ∈ U, the following properties hold:

(a) S ∩ π_{G}^{−1}(x) is a single point s;

(b) dρ_{s} is an isomorphism.

Since ψ is an isomorphism on the fibers of π_{2}, property (a) implies that, for every point t ∈ ϕ^{−1}(U), the W -stable closed set ψ^{−1}(S) intersects π_{2}^{−1}(t) at a single point. From this we infer that ψ^{−1}(S) has a unique irreducible component S̃ whose image under π_{2} is dense in T_{0} — the urgument is the same as that in the proof of Claim 2(i) in Section 2. Due to the uniqueness, this S̃ is W -stable.

Let V ⊆ π_{2}(S̃) ∩ ϕ^{−1}(U) be an open subset of T_{0}. Replacing it, if necessary, by ⋂_{w∈W} w(V), we may assume that V is W -stable. Let ρ̃: π_{2}^{−1}(V) ∩ S̃ → V be the restriction of π_{2} to π_{2}^{−1}(V) ∩ S̃. Then ρ̃ is a bijective W -equivariant morphism. We claim that it is separable and hence, by Zariski’s Main Theorem, an isomorphism (as V is normal). Indeed, take a point s̃ ∈ π_{2}^{−1}(V) ∩ S̃ and put π_{2}(s̃) = t. Then property (b), Lemma 5.2, and commutativity of diagram (40) imply that dρ̃_{s}: T_{s̃,S̃} → T_{t,V} is an isomorphism; whence the claim by [Bor, AG.17.3].

Thus, ρ^{−1} : V → π_{2}^{−1}(V) ∩ S̃ is a rational W -equivariant section of π_{2}. Its composition with the first projection G/T × T_{0} → G/T is then a W -equivariant rational map T −−→ G/T. This proves (i)⇒(ii).

Conversely, assume that (ii) holds. Let γ: T −−→ G/T be a W -equivariant rational map. Then ς := (γ, id): T_{0} −−→ G/T × T_{0} is a W -equivariant rational section of π_{2}, i.e., a section of π_{2} over a dense open subset V of T_{0}. We may assume that ς(V) and S := ψ(ς(V)) are open in their closures, ς: V → ς(V) is an isomorphism, and the subsets ϕ(V), π_{G}(S) of G//G are open and coincide. As above, we may also assume that V is W -stable.

Taking into account that ς is W -equivariant, ς(V) ∩ π_{2}^{−1}(t) is a single point for every t ∈ V, and ψ is an isomorphism on the fibers of π_{2}, we conclude that property (a) holds for every x ∈ ς(V). Thus, ρ := π_{G}|_{S}: S → ϕ(V) is a bijection.

We claim that ρ is separable, hence an isomorphism as ϕ(V) is normal by Lemma 5.1(i). Indeed, dϕ_{t} is an isomorphism by Lemma 5.5(i) and Lemma 5.1(ii). Let s = ψ(ς(t)) ∈ S. Since the restriction of (dπ_{2})_{ς(t)} to T_{ς(t),ς(V)} is an isomorphism with T_{t,V}, commutativity of diagram (40) and Lemma 5.2 imply that property (b) holds; whence the claim.

Thus, the composition of ρ^{−1} : ϕ(V) → S and the identical embedding S ↪ G is a rational section of π_{G}. This proves (ii)⇒(i) and completes the proof of the theorem. □

Recall some definitions from [CTKPR, Sects. 2.2, 2.3, and 3].

Let P be a linear algebraic group acting on a variety X and let Q be its closed subgroup. X is called a (P, Q)-variety if in X there is a dense open P -stable subset U, called a friendly subset, such that a geometric quotient π_{U}: U → U/P exists and π_{U} becomes the second projection P/Q × { \atop U/P} → { \atop U/P} after a surjective étale base change { \atop U/P} → U/P. If there is a rational section of π_{U}, one says that X admits a rational section. X is called a versal (P, Q)-variety if U/P is irreducible and, for every its dense open subset (U/P)_{0} and (P, Q)-variety Y, there is a friendly subset V of Y such that π_{V} is induced from π_{U} by a base change V → (U/H)_{0}.

Now we shall give the characteristic free proofs of the following two statements proved in [CTKPR] for char = 0.

Lemma 5.6. Let X be an irreducible variety endowed with a faithful action of a finite algebraic group H. Then

(i) X is an (H, {e})-variety; (ii) X is a versal (H, {e})-variety in each of the following cases:

(a) X is a free H-module;

(b) X is a linear algebraic torus and H acts by its automorphisms.

Proof. (i) Replacing X by its smooth locus, we may assume that X is smooth.

As H is finite, for any nonempty open affine subset U of X, the set ⋂_{h∈H} h(U) is H-stable, affine, and open in X. So, replacing X by it, we may assume that X is affine. Then, as is well known, for the action of H on X there is a geometric quotient π: X → X/H (see, e.g., [Bor, Prop. 6.15]). As X is normal, X/H is normal as well.

Since H is finite and the action is faithful, the points with trivial stabilizer form an open subset of X. Replacing X by it, we may also assume that the action is settheoretically free, i.e., the H-stabilizer of every point of X is trivial. As X and X/G are normal, by [G_{3}, Exp. I, Théorème 9.5(ii)] and [Bou_{1}, V.2.3, Cor. 4] this property implies that π is étale and hence X/H is smooth.

For every base change β: Y → X/H of π, the group H acts on X ×_{X/H} Y via X. As the action of H on X is set-theoretically free, taking Y = X and β = π, we obtain

X ×_{X/H} X = ⨆_{h∈H} h(D) where D := {(x, x) | x ∈ X}.

¿From this we deduce that in the commutative diagram

\xymatrix{H × X \ar[rr]^{α} \ar@{-->}[dr] & & X ×_{X/H} X \ar@{-->}[dl] \\ & X} ,

where α(h, x) := (h(x), x) and two other maps are the second projections, α is an H-equivariant isomorphism. This proves (i).

The proofs of (ii)(a) and (ii)(b) are the same as that of (b) and (d) in [CTKPR, Lemma 3.3] if one replaces in them the references to [CTKPR, Theorem 2.12] (whose proof is based on the assumption char k = 0) by the references to statement (i) of the present lemma. □

Remark 5.7. The proof of (i) shows that, for finite group actions, set-theoretical freeness coincides with that in the sense of GIT, [MF, Def. 0.8].

Lemma 5.8. G is a versal (G, T)-variety.

Proof. First we shall give a characteristic free proof of the fact that G is a (G, T)- variety (the proof given in [CTKPR] is based on the assumption char k = 0). By Lemma 5.1(iii) this is equivalent to proving the existence of a dense open subset U of (G//G)_{0} such that after a surjective étale base change U′ → U morphism (37) becomes the second projection G/T × U′ → U′.

Consider the base change of π_{G} in (40) by means of ϕ. Lemma 5.5(i) implies that

F := G_{0} ×_{(G//G)_{0}} T_{0} = {(g, t) ∈ G_{0} × T_{0} | G(g) = G(t)}  (41)

(see (5)). We have the canonical map corresponding to commutative diagram (40):

γ := ψ × id: G/T × T_{0} ⟶ F,  (gT, t) ↦ (gtg^{−1}, t).  (42)

It follows from (41) that γ is surjective; whence F is irreducible. But if for t ∈ T_{0} and g_{1}, g_{2} ∈ G we have g_{1}tg_{1}^{−1} = g_{2}tg_{2}^{−1}, then g_{1}T = g_{2}T since G_{t} = T. Therefore, γ is bijective. Lemma 5.2 and (42) show that dγ_{x} is injective for every x ∈ G/T × T_{0}. Hence if γ(x) lies in the smooth locus F_{sm} of F, then dγ_{x} is the isomorphism. This implies that γ is separable and then, by Zariski’a Main Theorem, that the restriction of γ to γ^{−1}(F_{sm}) is an isomorphism γ^{−1}(F_{sm}) → F_{sm}.

As F_{sm} is G-stable and γ is G-equivariant, γ^{−1}(F_{sm}) is a G-stable open subset of G/T ×T_{0}. Hence it is of the form G/T ×U′ for an open subset U′ of T_{0}. But Lemmas 5(ii) and 5.5(i) imply that U := ϕ(U′) is open in (G//G)_{0} and ϕ|_{U′}: U′ → U is étale. This proves that after the étale base change ϕ|_{U′}: U′ → U morphism (37) becomes the second projection G/T × U′ → U′. Hence G is a (G, T)-variety.

By Lemma 5.6(b), T is a versal (W, {e})-variety. The characteristic free arguments from [CTKPR, proof of Prop. 4.3(c)] then show that this fact implies versality of the (G, T)-variety G. This completes the proof of the lemma. □

Proof of Theorem 1.6. Let us first show how to deduce (i) from (ii). Consider commutative diagram (9). As μ is separable, it is central. Therefore, if (ii) holds, there is a rational section of πG̃; whence there is a rational cross-section in G̃. Then (i) follows from Corollary 2.4.

Now we shall prove (ii). Assume that τ is central. Then the natural morphism Ĝ/T̂ → G/T is an isomorphism by [Bor, Props. 6.13, 22.4].

Using τ, every action of G naturally lifts to an action of Ĝ on the same variety. In particular, G is endowed with an action of Ĝ. But G is a (G, T)-variety by Lemma 5.8(i). As Ĝ/T̂ and G/T are isomorphic, this means that G is a (Ĝ, T̂)-variety. But Ĝ is a versal (Ĝ, T̂)-variety (by Lemma 5.8(i)) that admits a rational section (by Lemma 5.1(iii) and [Ste_{1}]). Hence by [CTKPR, Theorem 3.6(a)] (the proof of this result is characteristic free) every (Ĝ, T̂)-variety admits a rational section. In particular, this is so for G. This proves (ii) and completes the proof of the theorem. □

6. Remarks

1. Cross-sections versus sections. If there is a section σ: G//G → G of π_{G}, then σ(G//G) is a cross-section in G. Indeed, as id_{k[G//G]} is the composition of the homomorphisms

k[G//G] ⟶^{π_{G}^{∗}} k[G] ⟶^{σ^{∗}} k[G//G],

π_{G}^{∗} is surjective; by [G_{2}, Cor. 4.2.3] this means that σ is a closed embedding.

The cross-section σ(G//G) has the property that the restriction of π_{G} to σ(G//G) is an isomorphism σ(G//G) → G//G. Conversely, let S be a cross-section in G. If π_{G}|_{S}: S → G//G is separable, then, since π_{G}|_{S} is bijective and G//G is normal, Zariski’s Main Theorem implies that π_{G}|_{S} is an isomorphism (cf. [Bor, AG 18.2]). So in this case the composition of (π_{G}|_{S})^{−1} with the identity embedding S ↪ G is a section of π_{G} whose image is S. In particular, if char k = 0, then every cross-section in G is the image of a section of π_{G}. If char k > 0, then in the general case this is not true.

Example 6.1. Let G = SL_{3} and char k = p > 0. Then for every integer d > 0,

S := \{s(a_{1}, a_{2}) | a_{1}, a_{2} ∈ k\}, where s(a_{1}, a_{2}) := (\begin{matrix}a_{1} & a_{2} & 1 \\ 1 & a_{1}^{p^{d}} − a_{1} & 0 \\ 0 & 1 & 0 \end{matrix}) ,

is a cross-section in G such that ρ is not separable. Indeed, as χ_{ϖ_{i}}(g) is the sum of principal i-minors of g ∈ G, we have (see Lemma 2.1(ii))

(λ ∘ ρ)(s(a_{1}, a_{2})) = (a_{1}^{p^{d}}, a_{1}(a_{1}^{p^{d}} − a_{1}) − a_{2}). □

Similarly, if σ: G//G −−→ G is a rational section of π_{G} and S the closure S of its image, then S is a rational cross-section in G such that the restriction of π_{G} to it is a birational isomorphism with G//G.

2. Group action on the set of cross-sections. Let Mor(G//G, G) be the group of morphisms G//G → G. If S is a cross-section in G and γ ∈ Mor(G//G, G), then

γ(S) := {γ(s)sγ(s)^{−1} | s ∈ S}

is a cross-section in G. This defines an action of Mor(G//G, G) on the set of crosssections in G. If char k = 0, then by [FM] this action is transitive. If char k > 0, then in the general case this is not true: in Example 6.1, Steinberg’s section and S are not in the same Mor(G//G, G)-orbit since, for the former, the restriction of π_{G} is separable [Ste_{1}, Theorem 1.5], but, for the latter, it is not.

3. Lifting T -action. By Theorem 3.5 there is an action of T on T/W determining a structure of a toric variety. This action cannot be lifted to T making π_{T} : T → T/W equivariant. This follows from the fact that the automorphism group of the underlying variety of T is GL_{r}(Z) ⋉ T.

4. Questions. Given Theorem 1.5 and Corollary 1.7, it would be interesting to construct explicitly an example of a W -equivariant rational map T −−→ G/T for central τ.

Is there such a map defined on T_{0}?

Is there a rational section of π_{G} defined on (G//G)_{0}?

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Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow, 119991, Russia

   E-mail address: popovvl@orc.ru