RATIONAL SMOOTHNESS OF VARIETIES OF REPRESENTATIONS FOR QUIVERS OF DYNKIN TYPE

PHILIPPE CALDERO AND RALF SCHIFFLER

Abstract. Let U^{+} be the positive part of the quantized enveloping algebra U of type A, D or E. The change of basis between canonical, and PBW-basis of U^{+} has a geometric interpretation in terms of local intersection cohomology of some affine algebraic varieties, namely the Zariski closures of orbits of representations of a quiver of type A, D or E. In this paper we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth. This provides an analogue of theorems of V. Deodhar, and J. Carrell and D. Peterson on Schubert varieties.

1. Introduction

Let 𝔽 be an algebraically closed field, d = (d_{1}, d_{2}, …, d_{n}) ∈ ℕ^{n} and G_{d} = ∏_{i=1}^{n} GL_{d_{i}}(𝔽). Let Q be a fixed quiver whose underlying graph Δ is the Dynkin graph of type A_{n}, D_{n} or E_{n}. G_{d} acts on E_{d} = ⊕_{i→j∈Q}Hom_{𝔽}(𝔽^{d_{i}}, 𝔽^{d_{j}}), by conjugation. Let O be a G_{d}-orbit and \overline{O} its Zariski closure. In [BS] the complete list of rationally smooth orbit closures of type A_{n} was obtained. As a consequence it was shown that, in type A_{n}, \overline{O} is rationally smooth if and only if \overline{O} is smooth. In this paper we will generalize these results to the types D_{n} and E_{n}, see theorem 3.6 and corollary 3.7.

Rational smoothness is a topological property, which is defined using local intersection cohomology, and has been extensively studied for Schubert varieties, [Deo85], [Car94]. For a survey of some of these results, see [BL00] and [Bri98].

Let U^{+} be the positive part of the quantized enveloping algebra U over ℚ(v) associated by Drinfeld and Jimbo to the root system of type Δ. Kashiwara and Lusztig have constructed independently of each other a unique canonical basis B of U^{+} in [Kas91] and [Lus90a]. For each reduced expression i of the longest element w_{0} of the Weyl group W of type Δ, there is also a PBW-basis B_{i}. Some of the reduced expressions are adapted to the quiver Q. In this case, Lusztig has shown in [Lus90a] that the entries of the transition matrix between the bases B and B_{i} have a description in terms of local intersection cohomology of orbit closures. We use this approach to study rational smoothness of orbit closures. One important ingredient therefore is the action of the bar involution of U on the elements of the PBW-basis.

This paper is organized as follows. In section 2 we fix notations and recall some results that we will need at a later stage. In particular, we recall the Hall algebra realization of U^{+}. We then present two different approaches to study rational smoothness of orbit closures: one algebraic and the other geometric.

2000 Mathematics Subject Classification. Primary 17B37.

The second author was supported in part by FCAR Grant.

Section 3 contains the algebraic approach. Here we consider the bar of a PBWbasis element as a linear combination in the PBW-basis, and study the coefficients of this expansion; to be more precise, we calculate the derivative at v = 1 of these coefficients. The method used to calculate the coefficients differs from [BS]. In fact, we use the dual approach, with the help of the canonical form on U^{+}. Indeed, the dual PBW-basis is known in the Hall algebra context by [Gre95]. Moreover, the adjoint σ of the bar involution is a ℚ-antiautomorphism (up to a power of q) given by Lusztig, [Lus93]. The reason why things are more convenient when we dualize is that the dual of an irreducible element of the PBW-basis is an element of the dual canonical basis, [Cal], and so is stable by σ. Then, the image by σ of a general dual PBW-basis element can be easily calculated by the antiautomorphism property. This enables us to realize the desired coefficients in terms of generalized Hall polynomials, see Proposition 3.1. Using the same methods as in [BS], we then obtain the complete list of rationally smooth orbit closures. It is easy to see that each orbit closure in this list is smooth. In section 4 we present another proof of this characterization using a geometric approach. According to an idea of Michel Brion, we calculate the Euler-Poincaré characteristic of the projectivization of the orbit closures \overline{O}. Indeed, it is known that, for rationally smooth cones, the Euler-Poincaré characteristic of the projectivization of the cone equals the dimension of the cone. Thanks to a theorem of Deligne on the Weil conjecture, the characteristic can be calculated by counting the number of 𝔽_{q}-rational points of an orbit closure viewed as a variety on an algebraic closure \overline{𝔽_{q}} of 𝔽_{q}, and specializing at q = 1. This method provides a geometric interpretation of the proof given in [BS], and followed in 3. It also provides an interesting homological realization of algebraic elements, namely the derivative at q = 1 of the coefficients of the bar involution in the PBW-basis.

Our main result, Theorem 3.3, generalizes [BS, Theorem 5.4]. Nevertheless, the E-case slightly differs from the A-D case. Let’s explain how. First of all, by Ringel’s Hall algebra approach of quantum groups, the coefficients of the bar automorphism discussed above are parameterized by couples (O, O′) of G_{d} orbits, and are non zero if and only if O′ ⪯ O for the so-called degeneration ordering. Now, Klaus Bongartz gave, [Bon95], a description of this ordering in the representation theory of the quiver Q. In particular, we can define “elementary degenerations” corresponding to certain non-split exact sequences, called elementary operations, in the Auslander-Reiten quiver of Q, and O′ ⪯ O if and only if there exists a chain of elementary degenerations relying O′ and O. The results in [BS] proves that the coefficient parameterized by (O, O′) ”sees” the minimal length of such a chain, and that the derivative at q = 1 of the coefficient is non zero if and only if the degeneration corresponding to (O, O′) is elementary. In order to generalize this result, we have to replace elementary degenerations by a finer notion, see 3.2.

2. Notations and recollections

2.1. The quantized enveloping algebra U. Let Δ be a Dynkin diagram of type A, D or E, let n be the number of vertices of Δ and let (a_{ij}) be the corresponding Cartan matrix. Thus

a_{ij} = \{ \begin{matrix}2 & if i = j \\ −1 & if i j is an edge in Δ \\ 0 & otherwise. \end{matrix}

Let v be an indeterminate and U the quantized enveloping algebra of Drinfeld-Jimbo of type Δ over the field ℚ(v) of rational functions. U is a ℚ(v)-algebra with generators: E_{i}, F_{i}, K_{i}, K_{i}^{−1}(1 ≤ i ≤ n) and relations:

(r.1)  K_{i}K_{i}^{−1} = K_{i}^{−1}K_{i} = 1, K_{i}K_{j} = K_{j}K_{i};
(r.2)  K_{i}E_{j} = v^{a_{ij}}E_{j}K_{i}
(r.3)  K_{i}F_{j} = v^{−a_{ij}}
(r.4)  E_{i}F_{j} − F_{j}E_{i} = δ_{ij} \frac{(K_{i} − K_{i}^{−1})}{(v − v^{−1})} where δ_{ij} = \{ \begin{matrix}1 & if i = j \\ 0 & if i ≠ j. \end{matrix}

(r.5)  \{ \begin{matrix}E_{i}^{2}E_{j} − (v + v^{−1})E_{i}E_{j}E_{i} + E_{j}E_{i}^{2} = 0 & if a_{ij} = −1 \\ E_{i}E_{j} − E_{j}E_{i} = 0 & if a_{ij} ≠ −1. \end{matrix}

(r.6)  \{ \begin{matrix}F_{i}^{2}F_{j} − (v + v^{−1})F_{i}F_{j}F_{i} + F_{j}F_{i}^{2} = 0 & if a_{ij} = −1 \\ F_{i}F_{j} − F_{j}F_{i} = 0, & if a_{ij} ≠ −1 \end{matrix}

Let U^{+} be the ℚ(v)-subalgebra generated by the E_{i} (1 ≤ i ≤ n). Let \overline{( )} : U → U be the involution of ℚ-algebras defined by

E_{i} ↦ E_{i}, F_{i} ↦ F_{i}, K_{i} ↦ K_{i}^{−1} for all 1 ≤ i ≤ n and v ↦ v^{−1}.

Note that \overline{U^{+}} = U^{+}.

Let Q, resp. Q^{+}, be the free abelian group, resp. semigroup, with basis {α_{1}, α_{2}, …, α_{n}}. Define an inner product ( , )_{Q} on Q by (α_{i}, α_{j})_{Q} = a_{ij}. Let R = {α ∈ Q | (α, α)_{Q} = 2}. R is a root system of type Δ whose set of simple roots is {α_{1}, α_{2}, …, α_{n}}. Let R^{+} = {α ∈ R | α = ∑_{j} c_{j}α_{j} with c_{j} ∈ ℕ} be the subset of positive roots.

Each α ∈ R defines a reflection s_{α} : Q → Q, z ↦ z − (z, α)_{Q} α. We will write s_{i} instead of s_{α_{i}}. Let W be the Weyl group of R. This is the subgroup of Aut(Q) generated by the reflections s_{i}, (1 ≤ i ≤ n). Let ℓ(w) be the length of w with respect to the generators {s_{1}, s_{2}, …, s_{n}} and denote by w_{0} the unique element of W of maximal length. It is known that ℓ(w_{0}) = ν = #(R^{+}). Let Tr be the linear form on Q^{+} such that Tr(α_{i}) = 1, 1 ≤ i ≤ n.

We shall use the Q^{+}-grading wt of U^{+} defined by wt(E_{i}) = α_{i}.

Lusztig has defined an action of the braid group on U [Lus90a] and used it to define bases of PBW type of U^{+}. We now recall these definitions.

For i ∈ {1, …, n}, let T̃_{i} : U → U be the automorphism of ℚ(v)-algebras defined by

E_{i} ↦ −K_{i}^{−1}F_{i} ,  F_{i} ↦ −E_{i}K_{i} ,  K_{i} ↦ K_{i}^{−1} ,
E_{j} ↦ E_{j},  F_{j} ↦ F_{j},  K_{j} ↦ K_{j},  if a_{ij} = 0,
E_{j} ↦ (E_{j}E_{i} − v^{−1}E_{i}E_{j}), F_{j} ↦ (F_{i}F_{j} − vF_{j}F_{i}), K_{j} ↦ K_{i}K_{j} , if a_{ij} = −1.

We have T̃_{i}T̃_{j}T̃_{i} = T̃_{j}T̃_{i}T̃_{j} if a_{ij} = −1 and T̃_{i}T̃_{j} = T̃_{j}T̃_{i} if a_{ij} ≠ −1. This gives us a braid group action. Moreover T̃_{i}(E_{j}) = T̃_{j}^{−1}(E_{i}) if a_{ij} = −1.

Given integers M, N ≥ 0, we define

[N]! = ∏_{h=1}^{N} \frac{(v^{h} − v^{−h})}{(v − v^{−1})} ∈ ℤ[v, v^{−1}], \left[{M + N \atop N}\right] = \frac{[M + N]!}{[M]![N]!} ∈ ℤ[v, v^{−1}]

and E_{i}^{(N)} = \frac{E_{i}^{N}}{[N]!} for 1 ≤ i ≤ n.

Let I be the set of sequences i = (i_{1}, …, i_{ν}) of elements in {1, …, n} such that s_{i_{1}} … s_{i_{ν}} is a reduced expression of w_{0}. Each i ∈ I gives rise to a total order on R^{+} = {α^{1}, …, α^{ν}}, where α^{t} = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{t−1}}(α_{i_{t}}) for t = 1, …, ν. We say that an element c = (c_{1}, …, c_{ν}) ∈ ℕ^{ν} is of i-homogeneity d = (d_{1}, …, d_{n}) ∈ ℕ^{n} if

∑_{t=1}^{ν} c_{t} α^{t} = ∑_{k=1}^{n} d_{k} α_{k} .

Let b(t) be the vector (0, …, 0, 1, 0, …, 0) ∈ ℕ^{ν} whose only nonzero component is in the t^{th} column and is 1, where 1 ≤ t ≤ ν.

For i = (i_{1}, …, i_{ν}) ∈ I and c = (c_{1}, …, c_{ν}) ∈ ℕ^{ν}, define

E_{i}^{c} = E_{i_{1}}^{(c_{1})} T̃_{i_{1}} (E_{i_{2}}^{(c_{2})}) T̃_{i_{1}} T̃_{i_{2}} (E_{i_{3}}^{(c_{3})}) … T̃_{i_{1}} T̃_{i_{2}} ⋯ T̃_{i_{(ν−1)}} (E_{i_{ν}}^{(c_{ν})}) = ∏_{t=1}^{ν} E_{i}^{c_{t}b(t)}.

Note that if c is of i-homogeneity d then Tr(wt(E_{i}^{c})) = ∑_{k=1}^{n} d_{k}

Proposition 2.1. Let i ∈ I. Then B_{i} = {E_{i}^{c} | c ∈ ℕ^{ν}} is a ℚ(v)-basis of U^{+}. We say that B_{i} is a basis of PBW type.

Proof. [Lus90b, sect. 1.8 and 1.13] □

We now recall Lusztig’s construction of the canonical basis of U^{+}.

Theorem 2.2. Let i ∈ I and L_{i} the ℤ[v^{−1}]-submodule of U^{+} generated by B_{i}.

(i) L_{i} is independent of i. We denote L_{i} by L.

(ii) π(B_{i}) is a ℤ-basis of L/v^{−1}L independent of i. Here π : L → L/v^{−1}L is

the canonical projection. We denote π(B_{i}) by B. (iii) The restriction of π : L → L/v^{−1}L defines an isomorphism of ℤ-modules

π′ : L ∩ \overline{L} → L/v^{−1}L where \overline{L} is the image of L under \overline{( )}. In particular, B = π′^{−1}(B) is a ℤ-basis of L ∩ \overline{L}.

(iv) B is a ℤ[v^{−1}]-basis of L and a ℚ(v)-basis of U^{+}. B is said to be the

canonical basis of U^{+}.

(v) Each element of B is fixed by \overline{( )} : U^{+} → U^{+}.

Proof. [Lus90a] □

2.2. Specialization at v = 1. Let A = ℚ[v]_{(v−1)} denote the subring of ℚ(v) consisting of functions regular at v = 1. Define the A-form U_{A} of U^{+} to be the A-subalgebra of U^{+} generated by E_{i}, 1 ≤ i ≤ n. Denote by U_{1}^{+} = U_{A}/(v − 1)U_{A} the specialization of U^{+} at v = 1. This is the positive part of the classical universal enveloping algebra of type Δ, with generators E_{1}, …, E_{n} and classical Serre relations

\{ \begin{matrix}E_{i}^{2}E_{j} − 2 E_{i}E_{j}E_{i} + E_{j}E_{i}^{2} = 0 & if a_{ij} = −1 \\ E_{i}E_{j} − E_{j}E_{i} = 0 & if a_{ij} ≠ −1. \end{matrix}

Note that in U_{1}, T̃_{i}(E_{j}) becomes the usual bracket E_{j}E_{i} − E_{i}E_{j} and the specialization B_{i}(1) = {E_{i}^{c}(1) | c ∈ ℕ^{ν}} of the PBW-basis is a PBW-basis of U_{1}.

2.3. Quiver modules. Let Q = (Q^{0}, Q^{1}) be a quiver whose underlying graph is Δ, i.e. for each edge {i, j} of Δ we fix an orientation. (We use the notation Q^{0} for the set of vertices of the quiver Q and Q^{1} for the set of arrows.) A vertex i ∈ Q^{0} is a sink (respectively a source) of Q if there is no arrow i → j (respectively i ← j) ∈ Q^{1}. An element i = (i_{1}, …, i_{ν}) ∈ I is adapted to the quiver Q if i_{1} is a sink of Q_{1} = Q and i_{k} is a sink of the quiver Q_{k} = s_{i_{k−1}}(Q_{k−1}) obtained from Q_{k−1} by reversing the orientation of all arrows ending at i_{k−1}, where 2 ≤ k ≤ ν. It is easy to see that there is an element i ∈ I adapted to Q and that an element i of I can be adapted to at most one quiver.

For the rest of this paper, let i be adapted to the quiver Q.

Let F be any field. A module (or representation) V = (V_{i}, f_{ij}) of Q is a collection of n finite dimensional F -vector spaces V_{i}, (1 ≤ i ≤ n) and of (n−1) F -linear maps f_{ij} : V_{i} → V_{j}, (i → j ∈ Q^{1}). A morphism from the module V = (V_{i}, f_{ij}) to the module V′ = (V′_{i}, f′_{ij}) is a collection of F -linear maps g_{i} : V_{i} → V′_{i}, 1 ≤ i ≤ n such that f′_{ij} ∘ g_{i} = g_{j} ∘ f_{ij} for each i → j ∈ Q^{1}. These modules and morphisms form an abelian category Mod(Q). If V is a module of Q, denote by [V] its isomorphism class in Mod(Q).

The dimension of the module V = (V_{i}, f_{ij}) is the n-tuple

dim(V) = (dim_{F} (V_{1}), dim_{F} (V_{2}), …, dim_{F} (V_{n})) ∈ ℕ^{n}.

A module V of Q is indecomposable if V cannot be written as the direct sum of proper submodules.

Theorem 2.3. (i) For all α ∈ R^{+}, there is a unique indecomposable module (up to isomorphism), denoted e_{α} ∈ Mod(Q), such that dim(e_{α}) = (d_{1}, …, d_{n}) and α = ∑_{i=1}^{n} d_{i}α_{i}; any indecomposable module is isomorphic to e_{α} for a unique α. This is Gabriel’s theorem.

(ii) There exists an ordering α^{t}, 1 ≤ t ≤ ν of the positive roots such that

Hom_{Q}(e_{α^{t}}, e_{α^{s}}) = 0 if s < t. (iii) There exists a bijection c = (c_{1}, c_{2}, …, c_{ν}) ↦ [e(c)] between ℕ^{ν} and the set

of isomorphism classes of modules of Q, where e(c) = ⊕_{t=1}^{ν}c_{t} e_{α^{t}}. In this case, dim(e(c)) = (d_{1}, …, d_{n}), where ∑_{t=1}^{ν} c_{t} α^{t} = ∑_{i=1} d_{i} α_{i}, i.e. c is of i-homogeneity d.

In particular, the classification of indecomposable modules is independent of the ground field.

Proof. [Lus90a, sect. 4.12 – 4.15] □

Set [V, V′] = dim_{F} Hom_{Q}(V, V′) and [V, V′]^{1} = dim_{F} Ext_{Q}^{1}(V, V′). Note that Hom_{Q}(V, V′) is the F -vector space of morphisms g : V → V′ in Mod(Q) and Ext_{Q}^{1}(V, V′) is the F -vector space of extensions 0 → V′ → E → V → 0 in Mod(Q).

For d = (d_{1}, …, d_{n}) ∈ ℕ^{n}, define

E_{d} = ⨁_{i→j∈Q^{1}} Hom_{F} (F^{d_{i}}, F^{d_{j}}) and G_{d} = ∏_{i=1}^{n} GL_{d_{i}}(F).

The group G_{d} acts on E_{d} by (g · f)_{i→j} = (g_{j} f_{ij} g_{i}^{−1})_{i→j}. An element of E_{d} can be seen as a module in Mod(Q) of dimension d. Two elements of E_{d} define isomorphic modules if and only if they are in the same G_{d}-orbit. By theorem 2.3, there exists a bijection between the set of ν-tuples c = (c_{1}, …, c_{ν}) of i-homogeneity d and the set of G_{d}-orbits in E_{d}, where c = (c_{1}, …, c_{ν}) corresponds to the orbit O_{c} whose elements are isomorphic to e(c).

There is a partial order on ℕ^{ν} given by c′ ⪯ c if c′ and c have the same ihomogeneity and the orbit O_{c′} is contained in the Zariski closure \overline{O}_{c} of O_{c}. This is the so-called degeneration ordering.

Let S be the set of non-split short exact sequences of modules of Q and let Op be the subset of S consisting of all sequences for which the first and the last module are indecomposable. Hence if Υ ∈ Op then

Υ : 0 → e_{α^{s}} → V → e_{α^{t}} → 0

for some s, t ∈ {1, …, ν} and some module V. The elements of Op are called elementary operations. For Υ : 0 → e_{α^{s}} → V → e_{α^{t}} → 0 ∈ Op define in(Υ) = s and out(Υ) = t and denote by op^{Υ} the vector (op_{1}^{Υ}, …, op_{ν}^{Υ}) ∈ ℤ^{ν} given by

op_{r}^{Υ} = \{ \begin{matrix}−1 & if r = s, t \\ 1 & if e_{α^{r}} is a direct summand of V \\ 0 & otherwise. \end{matrix}

For all c ∈ ℕ^{ν} define Op(c) = {Υ ∈ Op | c + op^{Υ} ∈ ℕ^{ν}}. Thus an elementary operation Υ ∈ Op(c) allows us to go from one orbit O_{c} to another orbit O_{c+op^{Υ}}. As we will see in Theorem 2.4 below, elementary operations do not only preserve the i-homogeneity but they are also compatible with the partial ordering ⪯.

The following theorem is shown in [Bon95].

Theorem 2.4. Let c, c′ ∈ ℕ^{ν}. Then the following four statements are equivalent: (i) c′ ⪯ c (ii) There is a sequence of elementary operations Υ_{1}, Υ_{2}, …, Υ_{k} such that Υ_{l} ∈

Op(c′ + ∑_{i=1}^{l−1} op^{Υ_{i}}) and c′ + ∑_{i=1}^{k} op^{Υ_{i}} = c (iii) [e_{α}, e(c′)] ≥ [e_{α}, e(c)] for all indecomposable modules e_{α}. (iv) [e(c′), e_{α}] ≥ [e(c), e_{α}] for all indecomposable modules e_{α}.

2.4. Hall algebras. Fix a quiver Q and a ν-tuple i adapted to Q. Let H_{Q} = H be the twisted Hall algebra associated to the quiver Q. H is the free ℚ(v)-module with basis B the set of isomorphism classes of representations of the quiver Q, with multiplication

(2.1)  [e(c′)] · [e(c′′)] = v^{<e(c′),e(c′′)>} ∑_{c∈ℕ^{ν}} F_{c′,c′′}^{c} [e(c)]

where < e(c′), e(c′′) > = dim_{𝔽_{v^{2}}} Hom (e(c′), e(c′′)) − dim_{𝔽_{v^{2}}} Ext (e(c′), e(c′′)) and F_{c′,c′′}^{c} is the number of submodules of e(c) that are isomorphic to e(c′′) and are such that the corresponding quotient module is isomorphic to e(c′), as representations over 𝔽_{v^{2}}. This defines polynomials F_{c′,c′′}^{c} which are called Hall polynomials.

Theorem 2.5. There exists an isomorphism η : U^{+} → H of ℤ^{n}-graded ℚ(v)- algebras such that η(E_{i}) = [e_{α_{i}}]. It maps E_{i}^{c} to v^{[e(c),e(c)]−Tr(wt(E_{i}^{c}))}[e(c)].

Proof. [Rin93][Gre95] □

We shall need the following corollary, see [Cal] :

Corollary 2.6. Up to a power of v, the E_{i}^{c′}-coefficient of ∏_{i=1}^{k} E_{i}^{c_{i}} is F_{c_{1},...,c_{k}}^{c′}(v^{2}), where the polynomial F_{c_{1},...,c_{k}}^{c′}(q) denotes the number of filtrations of e(c′) with successive quotients isomorphic to e(c_{1}), …, e(c_{k}) over 𝔽_{q}.

2.5. Coalgebra structure. Define an algebra structure on U^{+} ⊗_{ℚ(v)} U^{+} by

(x ⊗ x′).(y ⊗ y′) = v^{(wt(x′),wt(y))}(xy) ⊗ (x′y′)

on homogeneous elements. Let δ : U^{+} → U^{+} ⊗_{ℚ(v)} U^{+} be the ℚ(v)-algebra map given by δ(E_{i}) = E_{i} ⊗ 1 + 1 ⊗ E_{i}. Now, there exists a unique ℚ(v)-bilinear form ( , ) on U^{+} such that (E_{i}, E_{j}) = δ_{i,j}(1 − v^{−2})^{−1} and (x, yy′) = (δ(x), y ⊗ y′), where ( , ) is extended to U^{+} ⊗ U^{+} by the rule (x ⊗ x′, y ⊗ y′) = (x, y).(x′, y′).

The Hall algebra approach of the quantum algebra U^{+} gives a nice description of its coalgebra structure, [Gre95]. For every representation e(c) of the quiver Q, let a_{c}(v^{2}) be the number of automorphisms of e(c) as a representation over 𝔽_{v^{2}}. Recall [Rin90] that

(2.2)  a_{c}(v^{2}) = v^{2 ∑_{s<t} c_{s}c_{t}[e_{α^{s}} ,e_{α^{t}}]} ∏_{s=1}^{ν} | GL_{c_{s}}(𝔽_{v^{2}})) |

We have [Rei99]:

(2.3)  (E_{i}^{c}, E_{i}^{c′}) = δ_{c,c′} v^{2[e(c),e(c)]} a_{c}(v^{2})^{−1}.

This proves that the basis B_{i} is orthogonal. We can define the dual basis for this pairing B_{i}^{∗} = {E_{i}^{c∗}, c ∈ ℕ^{ν}}, with

(2.4)  E_{i}^{c∗} = v^{−2[e(c),e(c)]} a_{c}(v^{2} )E_{i}^{c}.

This implies that the algebra U^{+} is generated by E_{i}^{b(s)∗}, 1 ≤ s ≤ ν, with straightening relations :

(2.5)  E_{i}^{b(t)∗}E_{i}^{b(s)∗} = ∑_{b(t)+b(s)⪯c} G_{b(t),b(s)}^{c}E_{i}^{c∗}.

In order to understand those straightening relations, we have to define another polynomial. Let 0 → e(c′) → e(c) → e(c′′) → 0 be a non-split exact sequence of representations of Q. Then, the set of points C_{c′′,c′}^{c} in the 𝔽_{v^{2}}-space Ext^{1}(e(c′′), e(c′)) corresponding to this extension is a cone (minus the zero point). Set

E_{c′′,c′}^{c}(v^{2}) := | C_{c′′,c′}^{c} | .

It is known that E_{c′′,c′}^{c} is a polynomial.

Proposition 2.7. Suppose 1 ≤ s < t ≤ ν. Then, for all c in ℕ^{ν} and up to a power of v, we have

G_{b(t),b(s)}^{c} = \{ \begin{matrix}1 & if c = b(s) + b(t), \\ E_{b(t),b(s)}^{c}(v^{2}) & if c = b(s) + b(t) + op^{Υ} for some Υ ∈ Op, \\ 0 & otherwise. \end{matrix}

Proof. This is a direct consequence of (2.4) and the Riedtmann formula which states

(2.6)  F_{b(t),b(s)}^{c}(v^{2}) a_{b(t)}(v^{2}) a_{b(s)}(v^{2}) a_{c}(v^{2})^{−1} = E_{b(t),b(s)}^{c}(v^{2}).

□

Remark 2.8. As the set C_{c′′,c′}^{c} ∪{0} is a cone, we see that the polynomial E_{c′′,c′}^{c}(v^{2}) can be factorized by v^{2} − 1. So, the proposition shows that the A-space generated by the E_{i}^{c∗} is an algebra, and its specialization at v = 1 is a (commutative) polynomial algebra. Indeed, it is known that this specialization is the algebra of regular functions on the maximal unipotent group.

Now, let σ be the ℚ-antiautomorphism of U^{+} such that σ(E_{i}) = E_{i}, 1 ≤ i ≤ n, σ(v) = v^{−1}.

Lemma 2.9. Let x, y be homogeneous elements in U^{+}, and let t ∈ {1, …, ν}. We have

(i)  (\overline{x}, y) = (−v)^{Tr(wt(x))} v^{\frac{1}{2} (wt(x),wt(x))_{Q}} \overline{(x, σ(y))}.
(ii) σ(E_{i}^{b(t)}) = (−1)^{Tr(α^{t})−1} E_{i}^{b(t)}, up to a power of v.

Proof. (i) is a direct consequence of [Rei99, 4.3] and (iii) is proved in [Cal, prop. 2.1]. □

2.6. Local intersection cohomology of orbit closures. In this subsection, let 𝔽 be an algebraic closure of a finite field 𝔽_{q} with q = p^{e} elements, where p is a prime number, and let d = (d_{1}, …, d_{n}) ∈ ℕ^{n}. We will write the dimension dim(O_{c}) of the orbit O_{c} by d(c).

The results of this subsection have been proved in [Lus90a, chap. 9 – 10].

Proposition 2.10. Let c ∈ ℕ^{ν} be of i-homogeneity d. Then for each c′ ⪯ c, there exists ω_{c′}^{c} ∈ ℤ[v, v^{−1}] such that

\overline{E_{i}^{c}} = ∑_{c′⪯c} ω_{c′}^{c} E_{i}^{c′}.

Moreover ω_{c}^{c} = 1 and for all c′ ⪯ c, Ω_{c′}^{c} =^{def} v^{d(c)−d(c′)}ω_{c′}^{c} is an element of ℤ[v^{2}, v^{−2}].

Theorem 2.11. Let c ∈ ℕ^{ν} be of i-homogeneity d and let E^{c} ∈ B be the unique canonical basis element such that π(E^{c}) = π(E_{i}^{c}). Then

(i) E^{c} = ∑_{c′} ζ_{c′}^{c} E_{i}^{c′}, where c′ runs over the set of elements of ℕ^{ν} of ihomogeneity d, ζ_{c}^{c} = 1 and ζ_{c′}^{c} ∈ v^{−1}ℤ[v^{−1}] for c′ ≠ c.

(ii) If c′ ⪯̸ c, then ζ_{c′}^{c} = 0.

(iii) If \overline{( )} is the ℤ-linear involution of ℤ[v, v^{−1}] sending v to v^{−1}, then

ζ_{c′}^{c} = ∑_{{c′′ \atop c′⪯c′′⪯c}} w_{c′}^{c′′} \overline{ζ_{c′′}^{c}}.

(iv) If c′ ⪯ c, f is a 𝔽_{q}-rational point of the orbit O_{c′} in E_{d} and H_{f}^{a} is the stalk at f of the a-th cohomology sheaf of the intersection cohomology complex of the Zariski closure \overline{O}_{c} of O_{c} with coefficients in \overline{ℚ_{ℓ}} (extended by zero on the complement of that closure), where ℓ is a prime number ≠ p, and with the 𝔽_{q}-structure such that the Frobenius map acts as identity on the stalks of its 0-th cohomology sheaf at the rational points of the orbit O_{c}, then

H_{f}^{2a+1} = 0 for all a and v^{d(c)−d(c′)}ζ_{c′}^{c} = ∑_{a} dim(H_{f}^{2a}) v^{2a}.

In particular, v^{d(c)−d(c′)}ζ_{c′}^{c} is a polynomial in v^{2} with coefficients in ℕ.

Definition 2.12. We say that the orbit closure \overline{O}_{c} is rationally smooth at O_{c′} ⊂ \overline{O}_{c} if for all c′′ such that c′ ⪯ c′′ ⪯ c we have ∑_{a} dim(H_{f}^{2a})v^{2a} = 1 for a 𝔽_{q}-rational point f ∈ O_{c′′}, i.e. if ζ_{c′′}^{c} = v^{d(c′′)−d(c)} for all c′′ such that c′ ⪯ c′′ ⪯ c. The orbit closure \overline{O}_{c} is rationally smooth if it is rationally smooth at each O_{c′} ⊂ \overline{O}_{c}, i.e. if ζ_{c′}^{c} = v^{d(c′)−d(c)} for all c′ ⪯ c.

Remark 2.13. By a result of [BM], \overline{O}_{c} is rationally smooth iff for each point x ∈ \overline{O}_{c} we have

H^{i}(\overline{O}_{c}, \overline{O}_{c} \ \{x\}; ℚ) = \{ \begin{matrix}ℚ & if i = 2 dim \overline{O}_{c} \\ 0 & otherwise \end{matrix}

where H^{i} denotes ordinary cohomology.

3. Algebraic approach

This section gives a generalization of results in [BS]. We first prove some formulas on the coefficients Ω_{c′}^{c}.

3.1. A formula for Ω_{c′}^{c}. We fix a quiver Q and a ν-tuple i adapted to Q as in the previous section. For all c in ℕ^{ν}, set c^{t} := c_{t}b(t), 1 ≤ t ≤ ν. An explicit formula for Ω_{c′}^{c} can be calculated :

Proposition 3.1. For every c, c′ in ℕ^{ν}, we have :

(3.1)  Ω_{c′}^{c} = F_{c′^{ν},...,c′^{1}}^{c}(v^{−2}) (∏_{t=1}^{ν} a_{c′}^{t}(v^{−2})) a_{c}(v^{−2})^{−1}

up to a power of v.

Proof. In the following proof, equalities are up to a power of v. We first write Ω_{c′}^{c} = (\overline{E_{i}^{c}}, E_{i}^{c′∗}). Hence, by Lemma 2.9 (i), we have \overline{Ω_{c′}^{c}} = (−1)^{Tr(wt(E_{i}^{c′}))}(E_{i}^{c}, σ(E_{i}^{c′∗})). So, by formula (2.3), it remains to find the coefficient of E_{i}^{c} in the decomposition of σ(E_{i}^{c′∗}) in the basis B_{i}.

It is easily seen that E_{i}^{c′∗} = ∏_{t=1}^{ν} E_{i}^{c′_{t}b(t)∗}, up to a power of v. Then, using equation (2.4), Lemma 2.9 (ii) and the fact that \overline{a_{c_{t}b(t)}(v^{2})} a_{c_{t}b(t)}(v^{2})^{−1} = (−1)^{c_{t}} up to a power of v, we get

(3.2)  \overline{Ω_{c′}^{c}} = (E_{i}^{c}, ∏_{t=ν}^{1} E_{i}^{c′_{t}b(t)∗}) up to a power of v.

Finally, using equation 2.4, Corollary 2.6 (i) and equation 2.3, we obtain :

\overline{Ω_{c′}^{c}} = F_{c′^{ν},...,c′^{1}}^{c}(v^{2}) (∏_{t=1}^{ν} a_{c′^{t}}(v^{2})) a_{c}(v^{2})^{−1}

and the proposition follows. □

There is a nice particular case of this formula when e(c′) is the sum of two indecomposable modules.

Corollary 3.2. Suppose c′ = b(s) + b(t), s < t. Then, up to a power of v,

(3.3)  Ω_{c′}^{c} = E_{b(t),b(s)}^{c}(v^{−2}).

Proof. This is a direct application of the Riedtmann formula, [Rie94]. □

3.2. Derivative of Ω_{c′}^{c} at v = 1. Set

e_{c′′,c′}^{c}(v^{2}) := \frac{E_{c′′,c′}^{c}(v^{2})}{v^{2} − 1},

which is a polynomial. Let Opr be the set of regular elementary operations defined by

Opr = {Υ : 0 → e_{α^{s}} → e(c) → e_{α^{t}} → 0, e_{b(t),b(s)}^{c}(1) ≠ 0}.

With the notation above, we set e^{Υ} := e_{b(t),b(s)}^{c}(1).

By [Nör], we know that all elementary operations are regular in case A and D. This is not true in case E, see remark below. For c in ℕ^{ν}, define Opr(c) := Opr ∩ Op(c). We can now generalize [BS, Theorem 5.4].

Theorem 3.3. Let c, c′ in ℕ^{ν} with same i-homogeneity, then

(i) Ω_{c}^{c} = 1 and Ω_{c′}^{c}(1) = 0 if c′ ≺ c.

(ii) \frac{d}{dv} Ω_{c′}^{c} |_{v=1} ≠ 0 if and only if there exists Υ in Opr(c) such that c = c′+op^{Υ}. (iii) If the condition in (ii) is verified, then \frac{d}{dv} Ω_{c′}^{c} |_{v=1} = −2 c′_{s}c′_{t} e^{Υ}, where

s = in(Υ) and t = out(Υ).

Proof. The first formula of (i) is given in Proposition 2.10. For the second formula, just remark that the map \overline{( )} specializes on identity at v = 1.

Let’s prove (ii) and (iii). By equation (3.2) \overline{Ω_{c′}^{c}} = (E_{i}^{c}, ∏_{t=ν}^{1} E_{i}^{c′_{t}b(t)∗}), up to a

power of v. Say ∑_{c′′} h_{c′}^{c′′}E_{i}^{c′′∗} = ∏_{t=ν}^{1} E_{i}^{c′_{t}b(t)∗} is the expansion of ∏_{t=ν}^{1} E_{i}^{c′_{t}b(t)∗} in the dual PBW-basis, thus \overline{Ω_{c′}^{c}} = h_{c′}^{c} up to a power of v. Then, since Ω_{c′}^{c}(1) = 0, we have \frac{d}{dv} \overline{Ω_{c′}^{c}}|_{v=1} = \frac{d}{dv}h_{c′}^{c}|_{v=1}, which is non-zero iff the multiplicity of (v − 1) in h_{c′}^{c} is one. The above expansion is obtained by performing a certain number of straightening relations (2.5). By proposition 2.7 we have

(1) h_{c′}^{c′} = 1, up to a power of v.
(2) If c = c′ + op^{Υ} for some Υ ∈ Op with s = in(Υ), t = out(Υ), then
h_{c′}^{c} = E_{b(t),b(s)}^{c}(v^{2}) = (v^{2} − 1) e_{b(t),b(s)}^{c}(v^{2})

up to a power of v. Then the multiplicity of (v − 1) in h_{c′}^{c} is at least one and is equal to one iff Υ is regular.

(3) All other h_{c′}^{c′′} have (v − 1) multiplicity at least 2.

This proves (ii). Now let c = c′ + op^{Υ} with Υ ∈ Opr and s = in(Υ), t = out(Υ). Then the only way E_{i}^{c∗} terms appear in our expansion is in the straightening relations involving both E_{i}^{b(t)∗} and E_{i}^{b(s)∗}. The number of these straightening relations is c′_{s}c′_{t}. Thus, by Proposition 2.7, h_{c′}^{c} is a sum of c′_{s}c′_{t} terms equal to (v^{2}−1) e_{b(t),b(s)}^{c}(v^{2}) up to a power of v. Hence, \frac{d}{dv}h_{c′}^{c}|_{v=1} = 2 c′_{s}c′_{t} e^{Υ}. The theorem follows since \frac{d}{dv}Ω_{c′}^{c}|_{v=1} = −\frac{d}{dv} \overline{Ω_{c′}^{c}}|_{v=1}.

□

Remark 3.4. The formula in (iii) slightly differs from the one in [BS] because of the factor e^{Υ}. This factor can be obtained from the polynomials calculated in [Nör]. In type A, it can only be 1, so this agrees with the formula in [BS]. In type D this constant can be equal to 1 or -1. In type E, it can be equal to 1, -1 and 0.

3.3. Rational Smoothness. In this section, we will characterize which orbit closures \overline{O_{c}} are rationally smooth. As a consequence, we will show that if \overline{O_{c}} is rationally smooth, then \overline{O_{c}} is smooth.

Let u = v^{2}. The next proposition has been shown in [BS, Cor 6.4]. We sketch a proof of it for completeness.

Suppose that \overline{O_{c}} is rationally smooth, then for all c′ ⪯ c we have ζ_{c′}^{c} = v^{d(c′)−d(c)}. Fix now c′ ⪯ c. We obtain by Theorem 2.11 (iii) that

∑_{c′⪯c′′⪯c} Ω_{c′}^{c′′}u^{d(c)−d(c′′)} = 1.

By taking the derivative to u evaluated at u = 1, we get by Theorem 3.3 (i) :

Proposition 3.5. \overline{O_{c}} is rationally smooth then

− ∑_{{c′′ \atop c′≺c′′⪯c}} \frac{dΩ_{c′}^{c′′}}{du} |_{u=1} = d(c) − d(c′)

for all c′ ⪯ c.

For a subset J of the set Q^{1} of arrows of the quiver Q, we define

E_{d}(J) = {(f_{ij})_{i→j} ∈ E_{d} | f_{ij} = 0 if {i, j} ∈ Q^{1} \ J}.

The following properties are easily proved and left to the reader. Let J, J′ be two subsets of the set Q^{1} of edges of Q.

(1) E_{d}(J) is a linear subspace of E_{d} of dimension dim(E_{d}(J)) = ∑_{{i,j}∈J} d_{i} d_{j}. In particular, E_{d}(J) is a smooth variety.

(2) E_{d}(J) is a G_{d}-stable closed subset of E_{d} and it is a finite union of G_{d}- orbits. As a consequence and because the field 𝔽 is algebraically closed, we get that there is a unique open dense G_{d}-orbit in E_{d}(J). We will denote this orbit by O(J).

(3) E_{d}(J) ∩ E_{d}(J′) = E_{d}(J ∩ J′) and E_{d}(J) + E_{d}(J′) = E_{d}(J ∪ J′).

(4) E_{d}(J) ⊆ E_{d}(J′) if J ⊆ J′.

(5) E_{d}(∅) = {0} and E_{d}(Q^{1}) = E_{d}.

Because of (3) and (5) we see that for each G_{d}-orbit O_{c} in E_{d}, there is a unique smallest subset J(c) of Q^{1} for which O_{c} ⊆ E_{d}(J(c)). In fact

J(c) = ⋂_{O_{c}⊆E_{d}(J)} J.

Because E_{d}(J) is closed, then O_{c} ⊆ E_{d}(J) if and only if \overline{O_{c}} ⊆ E_{d}(J). As a consequence, if c′′ ⪯ c, then we get easily that J(c′′) ⊆ J(c). Note also that we don’t necessarily have O_{c} = O(J(c)). In fact, we will prove in theorem 3.6 that we get this equality precisely when \overline{O_{c}} is rationally smooth.

Let c^{min} = (c_{1}^{min}, c_{2}^{min}, …, c_{ν}^{min}) ∈ ℕ^{ν} be defined by

c_{k}^{min} = \{ \begin{matrix}d_{i} & if α^{k} = α_{i} for some i, 1 ≤ i ≤ n \\ 0 & otherwise. \end{matrix}

O_{c^{min}} is the minimal G_{d}-orbit of E_{d} and it consists of only the identity element of E_{d}.

Theorem 3.6. (1) We have the equality

− ∑_{{c′′ \atop c^{min}≺c′′⪯c}} \frac{dΩ_{c^{min}}^{c′′}}{du} |_{u=1} = dim(E_{d}(J(c))).

(2) \overline{O_{c}} is rationally smooth if and only if \overline{O_{c}} = E_{d}(J(c)).

Proof. By theorem 3.3,

− ∑_{{c′′ \atop c^{min}≺c′′⪯c}} \frac{dΩ_{c^{min}}^{c′′}}{du} |_{u=1} = ∑_{Υ∈Opr(c^{min}) c^{min}+op^{Υ}⪯c} c_{in(Υ)}^{min} c_{out(Υ)}^{min} e^{Υ}.

Now Υ ∈ Opr(c^{min}) iff Υ ∈ Opr and c_{in(Υ)}^{min} ≠ 0 ≠ c_{out(Υ)}^{min}, i.e. α^{in(Υ)} = α_{j} and α^{out(Υ)} = α_{i} are simple roots, c_{in(Υ)}^{min} = d_{j}, c_{out(Υ)}^{min} = d_{i} and there is an arrow i → j ∈ Q^{1}. Moreover Υ : 0 → e_{α_{j}} → e_{α_{i}+α_{j}} → e_{α_{i}} → 0 and e^{Υ} = 1. Note also that in this situation c^{min} + op^{Υ} ⪯ c iff i → j ∈ J(c). Thus

− ∑_{{c′′ \atop c^{min}≺c′′⪯c}} \frac{dΩ_{c^{min}}^{c′′}}{du} |_{u=1} = ∑_{i→j∈J(c)} d_{i}d_{j} = dim(E_{d}(J(c))),

this proves (1).

(2) E_{d}(J(c)) is smooth, hence rationally smooth.

Conversely if \overline{O_{c}} is rationally smooth, then applying proposition 3.5 for c′ = c^{min} and using part (1), we get dim(E_{d}(J(c))) = d(c). Hence O_{c} is the unique dense orbit in E_{d}(J(c)), thus \overline{O_{c}} = E_{d}(J(c)). □

Corollary 3.7. An orbit closure of type A, D, E is smooth, if and only if it is rationally smooth.

4. Geometric approach.

We present in this section an independent proof for Theorem 3.6. The interest of this proof is that we give an homological realization of the constants calculated in the previous section. To be more precise, the derivative of a coefficient Ω at u = 1 can be seen as an Euler-Poincaré characteristic of a complex variety. The principle on which the alternative proof works is that if a cone X in ℂ^{n} is rationally smooth, then, the rational cohomology of the projectivization ℙ(X) in ℙ^{n−1} is the same as the rational cohomology of ℙ^{d} where d = dim ℙ(X).

4.1. Euler-Poincaré characteristic. In the following section, the varieties \overline{O} are considered on the field ℂ. We fix c ∈ ℕ^{ν} of i-homogeneity d. Set X_{c} := ℙ(\overline{O_{c}}\\{0\}), as a complex variety and let X_{c}(𝔽_{q}), resp. X_{c}(\overline{𝔽_{q}}), be the corresponding variety on the finite field 𝔽_{q}, resp \overline{𝔽_{q}}. We know that there exists a discrete valuation ring R ⊂ ℂ with residue field of characteristic p and a variety X̂_{c} defined on R such that we get the variety X_{c}(\overline{𝔽_{q}}) over \overline{𝔽_{q}} and the variety X_{c} over ℂ by the base change to Spec\overline{𝔽_{q}} and Specℂ respectively.

Let χ(X_{c}) be the Euler-Poincaré characteristic of the variety X_{c}. Then

Theorem 4.1. For every c ∈ ℕ^{ν} of i-homogeneity d we have

χ(X_{c}) = dim E_{d}(J(c)).

Proof. We first transpose the calculation of χ(X_{c}) in the context of l-adic cohomology with compact support. We have

dim_{ℚ} H^{i}(X_{c}, ℚ) = dim_{ℂ} H^{i}(X_{c}, ℂ) = dim_{ℂ} H_{c}^{i}(X_{c}, ℂ) = dim_{\overline{ℚ}_{ℓ}} H_{c}^{i}(X_{c}(\overline{𝔽_{q}}), \overline{ℚ}_{ℓ}),

where l is prime to p. Hence,

(4.1)  χ(X_{c}) = ∑_{i}(−1)^{i} dim H_{c}^{i}(X_{c}(\overline{𝔽_{q}}), \overline{ℚ}_{ℓ}).

Fix a prime number p such that l is prime to p. We know that there is an action of the Frobenius Fr on the l-adic cohomology. By the Grothendieck trace formula, [Dan96, 7.10], we know that

(4.2)  | X_{c}(𝔽_{p^{e}}) |= ∑_{i=0}^{2 dim X_{c}}(−1)^{i}tr(Fr^{e}, H_{c}^{i}(X_{c}(\overline{𝔽_{q}}), \overline{ℚ}_{ℓ})).

Now, the theorem of Deligne, [Dan96, 8.21], asserts that H_{c}^{i}(X_{c}(\overline{𝔽_{q}}), \overline{ℚ}_{ℓ}) is filtered by Fr-stable subspace W_{i,j} such that the eigenvalues of the Frobenius Fr on the successive quotients are powers of p up to a root of one. So, there exists an N such that if e is a multiple of N, the eigenvalues of Fr^{e} are powers of p^{e}. For these e, we have

(4.3)  | X_{c}(𝔽_{p^{e}}) |= ∑_{i,j}(−1)^{i}p^{ek_{i,j}} dim(W_{i,j}/W_{i,j−1}).

Now, we know that | X_{c}(𝔽_{q}) | is a polynomial P_{c}(q). So, the previous equation provides a polynomial equality which is true for an infinite number of p^{e}.

Hence, | X_{c}(𝔽_{q}) |= ∑_{i,j}(−1)^{i}q^{k_{i,j}} dim(W_{i,j}/W_{i,j−1}), which implies by (4.1) that χ(X_{c}) = P_{c}(1). The proof of the theorem relies now on the following lemma. □

Lemma 4.2. For every c ∈ ℕ^{ν} of i-homogeneity d we have

P_{c}(1) = dim(E_{d}(J(c))).

Proof. First of all, let S_{c} be the set of elements c′ of i-homogeneity d such that there exists Υ ∈ Op such that c^{min} + op^{Υ} = c′ ⪯ c. We have :

S_{c} = {c^{ij}, (i, j) ∈ J(c)},

with

c_{s}^{ij} = \{ \begin{matrix}1 & if α^{s} = α_{i} + α_{j} \\ d_{i} − 1 & if α^{s} = α_{i} \\ d_{j} − 1 & if α^{s} = α_{j} \\ 0 & otherwise. \end{matrix}

We want to calculate P_{c}(q). As \overline{O_{c}} is a disjoint union of orbits O_{c′} with c′ ⪯ c, we have to count the cardinality of the set of 𝔽_{q}-rational points O_{c′}(𝔽_{q}). This will be denoted by Q_{c′}(q), thus P_{c}(q) = \frac{1}{q−1} ∑_{c^{min}≺c′⪯c} Q_{c′}(q). Let x′ be a point of

O_{c′}(𝔽_{q}) and let G_{d,x′} be the isotropy group of x′ in G_{d}. Then, Q_{c′}(q) = \frac{|G_{d}|}{|G_{d,x′} |} and it is known that G_{d,x′} is the group of automorphism of the module e(c′). Hence, | G_{d,x′} |= a_{c′}(q) is given by (2.2). In particular, the multiplicity of (q −1) in Q_{c′}(q) is ∑_{i=1}^{n} d_{i} − ∑_{s=1}^{ν} c′_{s}.

We find that every \frac{Q_{c′}}{q−1}|_{q=1} is zero unless c′ ∈ S_{c}. Using the formula \frac{|Gl_{d_{i}} (𝔽_{q})|}{|Gl_{d_{i}−1}(𝔽_{q})|} =

(q^{d_{i}} −1)q^{d_{i}−1} and L’Hospital’s rule, we find \frac{Q_{c^{ij}}}{q−1} |_{q=1} = d_{i}d_{j}. By the decomposition P_{c}(q) = \frac{1}{q−1} ∑_{c^{min}≺c′⪯c} Q_{c′}(q), we obtain the lemma. □

4.2. Alternative proof. We now explain how the previous theorem gives a geometric version of the algebraic proof given in [BS] and in the present paper. First, as \overline{O_{c}} is a cone, we claim that the rational smoothness of \overline{O_{c}} implies that the cohomology of X_{c} is the same as the cohomology of the projective space, i.e. dim H^{i}(X_{c}, ℚ) = dim H^{i}(ℙ^{d(c)−1}, ℚ). Let’s sketch a proof for this claim, following [Bri98]. By Remark 2.13, the rational smoothness property implies that H^{i}(\overline{O_{c}}, \overline{O_{c}}\\{0\}) = 0, i ≠ 2d(c), H^{2d(c)}(\overline{O_{c}}, \overline{O_{c}}\\{0\}) = ℚ. As \overline{O_{c}} is contractible, we obtain by a long exact sequence in relative cohomology, that \overline{O_{c}}\\{0\} is a rational cohomology sphere of dimension 2d(c)−1. Now, let S^{1} ⊂ ℂ^{∗} acting naturally on the cone \overline{O_{c}}. It is clear that X_{c} and (\overline{O_{c}}\\{0\})/S^{1} have the same rational cohomology. Hence, a Gysin exact sequence gives H^{0}(X_{c}) ≃ H^{2}(X_{c}) ≃ … H^{2(d(c)−1)}(X_{c}) ≃ ℚ and H^{2k+1}(X_{c}) = 0. This proves the claim.

Now, χ(X_{c}) = χ(ℙ^{d(c)−1}) = d(c). By Theorem 4.1, this implies that dim \overline{O_{c}} = dim E_{d}(J(c)), and then \overline{O_{c}} = E_{d}(J(c)). Once again, this implies that \overline{O_{c}} is smooth.

Acknowledgements

The first author wants to thank Michel Brion for suggesting the geometric approach and Markus Reineke for interesting and helpful email conversations.

The second author thanks Robert Bédard for several discussions on the subject.

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Département de Mathématiques, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France

   E-mail address: caldero@igd.univ-lyon1.fr

School of mathematics and statistics, Carleton University, 1125 Colonel By Drive, Room 4302 Herzberg Building, Ottawa, Ontario, Canada K1S 5B6

   E-mail address: ralf@math.carleton.ca