Cohomology of the minimal nilpotent orbit

We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo l reduction of the Springer correspondent representation involves the sign representation exactly when l divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes. Introduction Let G be a quasi-simple complex Lie group, with Lie algebra g. We denote by N the nilpotent variety of g. The group G acts on N by the adjoint action, with finitely many orbits. If O and O are two orbits, we write O 6 O if O⊂O. This defines a partial order on the adjoint orbits. It is well known that there is a unique minimal non-zero orbit Omin (see for example [CM93], and the introduction of [KP82]). The aim of this article is to compute the integral cohomology of Omin. The nilpotent variety N is a cone in g: it is closed under multiplication by a scalar. Let us consider its image P(N ) in P(g). It is a closed subvariety of this projective space, so it is a projective variety. Now G acts on P(N ), and the orbits are the P(O), where O is a non-trivial adjoint orbit in N . The orbits of G in P(N ) are ordered in the same way as the non-trivial orbits in N . Thus P(Omin) is the minimal orbit in P(N ), and therefore it is closed: we deduce that it is a projective variety. Let xmin ∈ Omin, and let P = NG(Cxmin) (the letter N stands for normalizer, or setwise stabilizer). Then G/P can be identified to P(Omin), which is a projective variety. Thus P is a parabolic subgroup of G. Now we have a resolution of singularities (see section 2) G×P Cxmin −→ Omin = Omin ∪ {0} which restricts to an isomorphism G×P C ∗xmin −→ Omin. From this isomorphism, one can already deduce that the dimension of Omin is equal to one plus the dimension of G/P . If we fix a maximal torus T in G and a Borel subgroup B containing it, we can take for xmin a highest weight vector for the adjoint action on g. Then P is the standard parabolic subgroup corresponding to the simple roots orthogonal to the highest root, and the dimension of G/P is the number of positive roots not orthogonal to the highest root, which is 2h − 3 in the simply-laced types, where h is the Coxeter number (see [Bou68, chap. VI, §1.11, prop. 32]). So the dimension of Omin is 2h − 2 is that case. In [Wan99], Wang shows that this formula is still valid if we replace h by the dual Coxeter number h (which is equal to h only in the simply-laced types). ∗UFR de mathématiques, Université Denis Diderot Paris 7 1 We found a similar generalization of a result of Carter (see [Car70]), relating the height of a long root to the length of an element of minimal length taking the highest root to that given long root, in the simply-laced case: the result extends to all types, if we take the height of the corresponding coroot instead (see Section 1, and Theorem 1.14). To compute the cohomology of Omin, we will use the Leray-Serre spectral sequence associated to the C-fibration G×P Cxmin −→ G/P . The Pieri formula of Schubert calculus gives an answer in terms of the Bruhat order (see section 2). Thanks to the results of section 1, we translate this in terms of the combinatorics of the root system (see Theorem 2.1). As a consequence, we obtain the following results (see Theorem 2.2): Theorem (i) The middle cohomology of Omin is given by H ∨ (Omin,Z) ≃ P (Φ)/Q(Φ) where Φ is the sub-root system of Φ generated by the long simple roots, and P(Φ) (resp. Q(Φ)) is its coweight lattice (resp. its coroot lattice). (ii) If l is a good prime for G, then there is no l-torsion in the rest of the cohomology of Omin. Part (i) is obtained by a general argument, while (ii) is obtained by a case-by-case analysis (see section 3, where we give tables for each type). In section 4, we explain a second method for the type An−1, based on another resolution of singularities: this time, it is a cotangent bundle on a projective space (which is also a generalized flag variety). This cannot be applied to other types, because the minimal class is a Richardson class only in type A. The motivation for this calculation is the modular representation theory of the Weyl group W . To each rational irreducible representation ofW , one can associate, via the Springer correspondence (see for example [Spr76, Spr78, BM81, KL80, Slo80, Lus84, Sho88]), a pair consisting in a nilpotent orbit and a G-equivariant local system on it (or, equivalently, a pair (x, χ) where x is a nilpotent element of g, and χ is an irreducible character of the finite group AG(x) = CG(x)/C 0 G(x), up to G-conjugation). Note that Springer’s construction differs from the others by the sign character. All the pairs consisting of a nilpotent orbit and the constant sheaf on this orbit arise in this way. In the simply-laced types, the irreducible representation of W corresponding to the pair (Omin,Q) is the natural representation tensored with the sign representation. In the other types, we have a surjection from W to the reflection group W ′ corresponding to the subdiagram of the Dynkin diagram of W consisting in the long simple roots. The Springer correspondent representation is then the natural representation of W ′ lifted to W , tensored with the sign representation. We believe that the decomposition matrix of the Weyl group (and, in fact, of an associated Schur algebra) can be deduced from the decomposition matrix of G-equivariant perverse sheaves on the nilpotent variety N . In [Jut], we will use Theorem 2.2 to determine some decomposition numbers for perverse sheaves (which give some evidence for this conjecture). Note that we are really interested in the torsion. The rational cohomology must already be known to the experts, as will be explained in [Jut]. All the results and proofs of this article remain valid for G a quasi-simple reductive group over Fp, with p good for G, using the étale topology. In this context, one has to take Ql and Zl coefficients, where l is a prime different from p, instead of Q and Z, 1 Long roots and distinguished coset representatives The Weyl group W of an irreducible and reduced root system Φ acts transitively on the set Φlg of long roots in Φ, hence if α is an element of Φlg, then the long roots are in bijection with W/Wα, where Wα is the stabilizer of α in W (a parabolic subgroup). Now, if we fix a basis ∆ of Φ, and if we choose for α the highest root α̃, we find a relation between the partial orders on W and 2 Φlg defined by ∆, and between the length of a distinguished coset representative and the (dual) height of the corresponding long root. After this section was written, I realized that the result was already proved by Carter in the simply-laced types in [Car70] (actually, this result is quoted in [Spr76]). We extend it to any type and study more precisely the order relations involved. I also came across [BB05, §4.6], where the depth of a positive root β is defined as the minimal integer k such that there is an element w in W of length k such that w(β) < 0. By the results of this section, the depth of a long root is nothing but the height of the corresponding coroot (and the depth of a short root is equal to its height). For the classical results about root systems that are used throughout this section, the reader may refer to [Bou68, Chapter VI, §1]. It is now available in English [Bou02]. 1.1 Root systems Let V be a finite dimensional R-vector space and Φ a root system in V . We note V ∗ = Hom(V,R) and, if α ∈ Φ, we denote by α the corresponding coroot and by sα the reflexion sα,α∨ of [Bou68, chap. VI, §1.1, déf. 1, (SRII)]. Let W be the Weyl group of Φ. The perfect pairing between V and V ∗ will be denoted by 〈, 〉. Let Φ = {α | α ∈ Φ}. In all this section, we will assume that Φ is irreducible and reduced. Let us fix a scalar product ( | ) on V , invariant under W , such that min α∈Φ (α|α) = 1. We then define the integer r = max α∈Φ (α|α). Let us recall that, since Φ is irreducible, we have r ∈ {1, 2, 3} and (α|α) ∈ {1, r} if α ∈ Φ (see [Bou68, chap. VI, §1.4, prop. 12]). We define Φlg = {α ∈ Φ | (α|α) = r} and Φsh = {α ∈ Φ | (α|α) < r} = Φ \ Φlg. If α and β are two roots, then (1) 〈α, β〉 = 2(α|β) (β|β) . In particular, if α and β belong to Φ, then (2) 2(α|β) ∈ Z and, if α or β belongs to Φlg, then (3) 2(α|β) ∈ rZ The following classical result says that Φlg is a closed subset of Φ. Lemma 1.1 If α, β ∈ Φlg are such that α+ β ∈ Φ, then α+ β ∈ Φlg. Proof : We have (α+β | α+β) = (α|α)+(β|β)+2(α|β). Thus, by (3), we have (α+β | α+β) ∈ rZ, which implies the desired result. 2