THE LIE ALGEBRA OF TYPE G_{2} IS RATIONAL OVER ITS QUOTIENT BY THE ADJOINT ACTION

DAVE ANDERSON, MATHIEU FLORENCE, AND ZINOVY REICHSTEIN

Abstract. Let G be a split simple group of type G_{2} over a field k, and let 𝔤 be its Lie algebra. Answering a question of J.-L. Colliot-Thélène, B. Kunyavskiĭ, V. L. Popov, and Z. Reichstein, we show that the function field k(𝔤) is generated by algebraically independent elements over the field of adjoint invariants k(𝔤)^{G}.

Résumé. Soit G un groupe algébrique simple et déployé de type G_{2} sur un corps k. Soit 𝔤 son algèbre de Lie. On démontre que le corps des fonctions k(𝔤) est transcendant pur sur le corps k(𝔤)^{G} des invariants adjoints. Ceci répond par l’affirmative à une question posée par J.-L. Colliot-Thélène, B. Kunyavskiĭ, V. L. Popov et Z. Reichstein.

I. Introduction. Let G be a split connected reductive group over a field k and let 𝔤 be the Lie algebra of G. We will be interested in the following natural question:

Question 1. Is the function field k(𝔤) purely transcendental over the field of invariants k(𝔤)^{G} for the adjoint action of G on 𝔤? That is, can k(𝔤) be generated over k(𝔤)^{G} by algebraically independent elements?

In [5], the authors reduce this question to the case where G is simple, and show that in the case of simple groups, the answer is affirmative for split groups of types A_{n} and C_{n}, and negative for all other types except possibly for G_{2}. (The standing assumption in [5] is that char(k) = 0, but here we work in arbitrary characteristic.)

The purpose of this note is to settle Question 1 for the remaining case G = G_{2}.

Theorem 2. Let k be an arbitrary field and G be the simple split k-group of type G_{2}. Then k(𝔤) is purely transcendental over k(𝔤)^{G}.

Apart from settling the last case left open in [5], we were motivated by the (still mysterious) connection between Question 1 and the Gelfand-Kirillov (GK) conjecture [9]. Here char(k) = 0. A. Premet [11] recently showed that the GK conjecture fails for simple Lie algebras of any type other than A_{n}, C_{n} and G_{2}. His paper relies on the negative results of [5] and their characteristic p analogues (proved in [11]). It is not known whether a positive answer to Question 1 for 𝔤 implies the GK conjecture for 𝔤. The GK conjecture has been proved for algebras of type A_{n} (see [9]), but remains open for types C_{n} and G_{2}. While Theorem 2 does not settle the GK conjecture for type G_{2}, it puts the remaining two open cases—for algebras of type C_{n} and G_{2}—on equal footing vis-à-vis Question 1.

II. Twisting. Temporarily, let W be a linear algebraic group over k.

Let X be a quasi-projective variety with a (right) W -action defined over k, and let ζ be a (left) W -torsor over k. The diagonal left action of W on X ×_{Spec(k)} ζ (by g.(x, z) = (xg^{−1}, gz)) makes X ×_{Spec(k)} ζ into the total space of a W -torsor X ×_{Spec(k)} ζ → B. The base space B of this torsor is usually called the twist of X by ζ. We denote it by ^{ζ}X.

Date: August 26, 2013.

D.A. was partially supported by NSF Grant DMS-0902967. Z.R. was partially supported by National Sciences and Engineering Research Council of Canada grant No. 250217-2012.

It is easy to see that if ζ is trivial then ^{ζ}X is k-isomorphic to X. Hence, ^{ζ}X is a k-form of X, i.e., X and ^{ζ}X become isomorphic over an algebraic closure of k.

The twisting construction is functorial in X: a W -equivariant morphism X → Y (or rational map X −−→ Y) induces a k-morphism ^{ζ}X → ^{ζ}Y (resp., rational map ^{ζ}X −−→ ^{ζ}Y). For details, see [7, Section 3], [8, Section 2], or [5, Section 2].

III. The split group of type G_{2}. We fix notation and briefly review the basic facts, referring to [13], [1], or [2] for more details. Over any field k, a simple split group G of type G_{2} has a faithful seven-dimensional representation V. Following [2, (3.11)], one can fix a basis f_{1}, …, f_{7}, with dual basis X_{1}, …, X_{7}, so that G preserves the nonsingular quadratic norm N = X_{1}X_{7} + X_{2}X_{6} + X_{3}X_{5} + X_{4}^{2}. (See [1, §6.1] for the case char(k) = 2. In this case V is not irreducible, since the subspace spanned by f_{4} is invariant; the quotient V/(k · f_{4}) is the minimal irreducible representation. However, irreducibility will not be necessary in our context.) The corresponding embedding G ↪ GL_{7} yields a split maximal torus and Borel subgroup T ⊂ B ⊂ G, by intersecting with diagonal and upper-triangular matrices. Explicitly, the maximal torus is

(1)  T = diag(t_{1}, t_{2}, t_{1}t_{2}^{−1}, 1, t_{1}^{−1}t_{2}, t_{2}^{−1}, t_{1}^{−1});

cf. [2, Lemma 3.13].

The Weyl group W = N(T)/T is isomorphic to the dihedral group with 12 elements, and the surjection N(T) → W splits. The inclusion G ↪ GL_{7} thus gives rise to an inclusion N(T) = T ⋊ W ↪ D ⋊ S_{7}, where D ⊂ GL_{7} is the subgroup of diagonal matrices. On the level of the dual basis X_{1}, …, X_{7}, we obtain an isomorphism W ≅ S_{3} × S_{2} realized as follows: S_{3} permutes the pairs (X_{1}, X_{7}), (X_{2}, X_{6}) and (X_{3}, X_{5}), and S_{2} exchanges the triples (X_{1}, X_{5}, X_{6}) and (X_{2}, X_{3}, X_{7}). The variable X_{4} is fixed by W. For details, see [2, §A.3].

The subgroup P ⊂ G stabilizing the isotropic line spanned by f_{1} is a maximal standard parabolic, and the corresponding homogeneous space P \G is isomorphic to the five-dimensional quadric Q ⊂ ℙ(V) defined by the vanishing of the norm, i.e., by the equation

(2)  X_{1}X_{7} + X_{2}X_{6} + X_{3}X_{5} + X_{4}^{2} = 0.

An easy tangent space computation shows that P is smooth regardless of the characteristic of k.

Lemma 3. The group P is special, i.e., H^{1}(l, P) = {1} for every field extension l/k. Moreover, P is rational, as a variety over k.

Proof. Since the split group of type G_{2} is defined over the prime field, we may replace k by the prime field for the purpose of proving this lemma, and in particular, we can assume k is perfect. We begin by briefly recalling a construction of Chevalley [4]. The isotropic line E_{1} ⊂ V stabilized by P is spanned by f_{1}, and P also preserves an isotropic 3-space E_{3} spanned by f_{1}, f_{2}, f_{3}; see, e.g., [2, §2.2]. There is a corresponding map P → GL(E_{3}/E_{1}) ≅ GL_{2}, which is a split surjection thanks to the block matrix described in [10, p. 13] as the image of “B” in GL_{7}. The kernel is unipotent, and we have a split exact sequence corresponding to the Levi decomposition:

(3)  1 → R_{u}(P) → P → GL_{2} → 1.

Combining the exact sequence in cohomology induced by (3) with the fact that both R_{u}(P) and GL_{2} are special (see [12, pp. 122 and 128]), shows that P is special.

Since P is isomorphic to R_{u}(P) × GL_{2} as a variety over k, and P is smooth, so is R_{u}(P). A smooth connected unipotent group over a perfect field is rational [6, IV, §2(3.10)]; therefore R_{u}(P) is k-rational, and so is P. □

IV. Proof of Theorem 2. Let G be the split simple group of type G_{2} over k, T ⊂ G be a split maximal torus and W = N(T)/T be the Weyl group. We begin by reducing Theorem 2 to a statement about rationality of a twisted quotient of the quadric Q.

Proposition 4. Consider the following assertions:

(a) The twisted variety ^{ζ}(G_{K}/T_{K}) is rational over K, for any W -torsor ζ over any field K/k.

(b) The twisted variety ^{ζ}(Q_{K}/T_{K}) is rational over K, for any W -torsor ζ over any field K/k. Then (b) ⟹ (a) ⟹ Theorem 2.

A dominant rational map Q −−→ Y induced by the inclusion of fields k(Q)^{T} ↪ k(Q) is called the rational quotient map for the T -action on Q. After replacing Y by a dense open subset, we may assume that the W -action on Q descends to Y. The resulting variety Y is unique up to a W -equivariant birational isomorphism; this is the W -variety Q/T in the statement of part (b) (and similarly for Q_{K}/T_{K}). We will construct an explicit birational model for Q/T below.

Proof. (a) ⟹ Theorem 2: Let 𝔤_{reg} and 𝔱_{reg} denote the open subsets of regular semisimple elements in the Lie algebras of G and T, respectively. The following diagram commutes:

\xymatrix{G/T ×_{Spec(k)} 𝔱_{reg} \ar[r] \ar[d] & 𝔤_{reg} \ar[d]^{π} \\ 𝔱_{reg} \ar[r]^{π} & 𝔤_{reg}//G.}

Here π is the categorical quotient map, and the top horizontal map, given by (g, t) ↦ ad(g) · t, is G-equivariant. The Weyl group acts on 𝔱_{reg} and G/T (on the right), and diagonally on G/T × 𝔱_{reg}. The horizontal maps are W -torsors; see [5, Proposition 2.9]. Thus we have the following diagram of inclusions of fields:

[figure: k(G/T ×_{Spec(k)} 𝔱_{reg})^{W}; k(𝔤); ∼; k(𝔱)^{W}; k(𝔤)^{G}.; ∼]

Setting L = k(𝔱) and K = k(𝔱)^{W} and noting that

k(G/T ×_{Spec(k)} 𝔱_{reg})^{W} = K((G/T)_{K} ×_{Spec(K)} Spec L),

the field extension on the left can be rewritten as K(^{ζ}(G/T))/K, where ζ is the W -torsor Spec L. By part (a) this field extension is purely transcendental. Hence, so is the vertical extension on the right side of the diagram, i.e., Theorem 2 holds.

(b) ⟹ (a): For the purpose of proving this implication, we may we may view K as a new base field and replace it with k.

We claim that the left action of P on G/T is generically free. By the (characteristic-free) argument at the beginning of the proof of [5, Lemma 9.1], in order to establish this claim it suffices to show that the right T -action on Q = P \G is generically free. The latter action, given by restricting the linear action (1) of T on ℙ^{6} to the quadric Q given by (2), is clearly generically free.

The W -equivariant rational map G/T −−→ Q/T induced by the projection G → P \G = Q is the rational quotient map for the left P -action on G/T; cf. [5, p. 458]. Since the P -action is generically free, this map is a P -torsor over the generic point of Q/T; see [3, Theorem 4.7]. By the functoriality of the twisting operation, after twisting by a W -torsor ζ, we obtain a rational map ζ(G/T) −−→ ^{ζ}Q/T, which is a P -torsor over the generic point of ^{ζ}Q/T. This torsor has a rational section, since P is special; in particular, ^{ζ}(G/T) is k-birationally isomorphic to P ×^{ζ}(Q/T). Since P is k-rational, ^{ζ}(G/T) is rational over ^{ζ}(Q/T), and we conclude that (b) ⟹ (a), as desired. □

It remains to show that the assertion of Proposition 4(b) always holds. As before, we may replace the field K with k. The following lemma completes the proof of Theorem 2.

Lemma 5. The twisted variety ^{ζ}(Q/T) is rational over k, for any W -torsor ζ over k.

Proof. We begin by constructing an explicit birational model for the W -variety Q/T. The affine open subset Q^{aff} = {x_{1}x_{7} + x_{2}x_{6} + x_{3}x_{5} + 1 = 0} ⊂ 𝔸^{6} (where X_{4} ≠ 0) is N(T)-invariant. Here the affine coordinates on 𝔸^{6} are x_{i} := X_{i}/X_{4}, for i ≠ 4. The field of invariant rational functions for the T -action on Q^{aff} is k(y_{1}, y_{2}, y_{3}, z_{1}, z_{2}), where the variables

y_{1} = x_{1}x_{7}, y_{2} = x_{2}x_{6}, y_{3} = x_{3}x_{5}, z_{1} = x_{1}x_{5}x_{6}, and z_{2} = x_{2}x_{3}x_{7}

are subject to the relations y_{1} + y_{2} + y_{3} + 1 = 0 and y_{1}y_{2}y_{3} = z_{1}z_{2}. In other words, the rational quotient Q^{aff}/T (or equivalently, Q/T) is W -equivariantly birationally isomorphic to the affine subvariety Λ_{1} of 𝔸^{5} given by these two equations, where W = S_{2} × S_{3} acts on the coordinates as follows: S_{2} permutes z_{1}, z_{2}, and S_{3} permutes y_{1}, y_{2}, y_{3}. We claim that Λ_{1} is W -equivariantly birationally isomorphic to

Λ_{2} = {(Y_{1} : Y_{2} : Y_{3} : Z_{0} : Z_{1} : Z_{2}) : Y_{1} + Y_{2} + Y_{3} + Z_{0} = 0 and Y_{1}Y_{2}Y_{3} = Z_{1}Z_{2}Z_{0}} ⊂ ℙ^{5},
Λ_{3} = {(Y_{1} : Y_{2} : Y_{3} : Z_{1} : Z_{2}) : Y_{1}Y_{2}Y_{3} + (Y_{1} + Y_{2} + Y_{3})Z_{1}Z_{2} = 0} ⊂ ℙ^{4}, and
Λ_{4} = {(Y_{1} : Y_{2} : Y_{3} : Z_{1} : Z_{2}) : Z_{1}Z_{2} + Y_{2}Y_{3} + Y_{1}Y_{3} + Y_{1}Y_{2} = 0} ⊂ ℙ^{4} ,

where W acts on the projective coordinates Y_{1}, Y_{2}, Y_{3}, Z_{1}, Z_{2}, Z_{0} as follows: S_{2} permutes Z_{1}, Z_{2}, S_{3} permutes Y_{1}, Y_{2}, Y_{3}, and every element of W fixes Z_{0}. Note that Λ_{2} ⊂ ℙ^{5} is the projective closure of Λ_{1} ⊂ 𝔸^{5}; hence, using ≃ to denote W -equivariant birational equivalence, we have Λ_{1} ≃ Λ_{2}. The isomorphism Λ_{2} ≃ Λ_{3} is obtained by eliminating Z_{0} from the system of equations defining Λ_{2}. Finally, the isomorphism Λ_{3} ≃ Λ_{4} comes from the Cremona transformation ℙ^{4} −−→ ℙ^{4} given by Y_{i} → 1/Y_{i} and Z_{j} → 1/Z_{j} for i = 1, 2, 3 and j = 1, 2.

The resulting W -equivariant birational isomorphism Q/T ≃ Λ_{4} gives rise to a birational isomorphism ^{ζ}(Q/T) ≃ ^{ζ}Λ_{4} of k-varieties, for any W -torsor ζ over k. Since Λ_{4} is a W -equivariant quadric hypersurface in ℙ^{4}, and the W -action on ℙ^{4} is induced by a linear representation W → GL_{5}, Hilbert’s Theorem 90 tells us that ^{ζ}ℙ_{4} is k-isomorphic to ℙ^{4}, and ^{ζ}Λ_{4} is isomorphic to a quadric hypersurface in ℙ^{4} defined over k; see [7, Lemma 10.1]. It is easily checked that Λ_{4} is smooth over k, and therefore so is ^{ζ}Λ_{4}. The zero-cycle of degree 3 given by (1 : 0 : 0 : 0 : 0) + (0 : 1 : 0 : 0 : 0)+(0 : 0 : 1 : 0 : 0) in Λ_{4} is W -invariant, so it defines a zero-cycle of degree 3 in ^{ζ}Λ_{4}. By Springer’s theorem, the smooth quadric ^{ζ}Λ_{4} has a k-rational point, hence is k-rational. □

Acknowledgement. We are grateful to J.-L. Colliot-Thélène for stimulating conversations.

References

[1] D. Anderson, Degeneracy loci and G_{2} flags, Ph. D. thesis, University of Michigan, 2009.

[2] D. Anderson, “Chern class formulas for G_{2} Schubert loci,” Trans. Amer. Math. Soc. 363 (2011), no. 12, 6615– 6646.

[3] G. Berhuy, G. Favi, Essential Dimension: A Functorial Point of View (after A. Merkurjev), Documenta Math. 8 (2003), 279–330.

[4] C. Chevalley, “Les Groupes de type G_{2},” Séminaire Claude Chevalley, tome 2 (19561958), reprinted in Classification des Groupes Algébriques Semi-Simples, Springer, 2005.

[5] J.-L. Colliot-Thélène, B. Kunyavskiĭ, V. L. Popov, and Z. Reichstein, “Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?” Compositio Math. 147 (2011), 428–466.

[6] M. Demazure and P. Gabriel, Groupes algébriques, Tome I, Masson & Cie, Paris, 1970.

[7] A. Duncan, Z. Reichstein, “Versality of algebraic group actions and rational points on twisted varieties,” J. Alg. Geom., to appear, arXiv:1109.6093

[8] M. Florence, “On the essential dimension of cyclic p-groups,” Invent. Math. 171 (2008), no. 1, 175–189.

[9] I. M. Gelfand, A. A. Kirillov, “Sur les corps liés aux algèbres enveloppantes des algèbres de Lie,” IHES Publ. Math. 31 (1966), 5–19.

[10] J. Heinloth, “Bounds for Behrend’s conjecture on the canonical reduction,” Int. Math. Res. Not. IMRN (2008), no. 14, Art. ID rnn045, 17 pp.

[11] A. Premet, “Modular Lie algebras and the Gelfand-Kirillov conjecture,” Invent. Math. 181 (2010), no. 2, 395– 420.

[12] J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997.

[13] T. A. Springer and F. D. Veldkamp, Octonions, Jordan Algebras, and Exceptional Groups, Springer, 2000.

Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ 22460-320 Brasil E-mail address: dave@impa.br

Institut de Mathématiques de Jussieu, Université Paris 6, place Jussieu, 75005 Paris, France

E-mail address: florence@math.jussieu.fr

Department of Mathematics, University of British Columbia, BC, Canada V6T 1Z2

E-mail address: reichst@math.ubc.ca