THE BAR AUTOMORPHISM IN QUANTUM GROUPS AND GEOMETRY OF QUIVER REPRESENTATIONS

PHILIPPE CALDERO AND MARKUS REINEKE

Abstract. Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers.

1. Introduction

The canonical basis B of the positive part U_{v}(𝔤)^{+} of the quantized enveloping algebra of a semisimple Lie algebra 𝔤, constructed by G. Lusztig [5], has many favourable properties, like for example inducing bases in all the finite dimensional irreducible representations of 𝔤 simultaneously.

The basis B can be characterized algebraically by its elements being fixed under the so-called bar automorphism of U_{v}(𝔤)^{+}, and by admitting a unitriangular base change to the PBW-type bases.

The Hall algebra approach to quantum groups [12] provides a realization of certain specializations U_{q}(𝔤)^{+} via a convolution product for constructible functions on varieties R_{d} parametrizing representations of Dynkin quivers. In this realization, the elements of PBW-type bases correspond to characteristic functions of orbits in R_{d} under a natural action of an algebraic group G_{d}, whereas the elements of B correspond to constructible functions naturally associated to the intersection cohomology complexes of the closures of G_{d}-orbits.

It is therefore natural to also ask for interpretations of the bar automorphism in terms of the geometry of the varieties R_{d}, since this automorphism plays a central role in defining the canonical basis algebraically.

In the present paper, two such interpretations are given. In the geometric setup of [7], analogues of orbital varieties in the quiver context are constructed. These parametrize quiver representations fixing certain flags, and their numbers of rational points over finite fields are shown (Theorem 3.4) to give essentially the coefficients Ω_{M,N} of the bar automorphism on a PBW-type basis of U_{q}(𝔤)^{+}. The key ingredient in deriving this result in section 3 is a generalization (Corollary 2.6) of a very useful formula [9] by C. Riedtmann, relating numbers of filtrations of quiver representations over finite fields to cardinalities of orbital varieties; this generalization, together with the construction of the orbital varieties, is given in section 2. Theorem 3.4 is similar in spirit to a result in [4, Appendix], where the coefficients of an analogous bar involution in Hecke algebras are interpreted as numbers of rational points of varieties related to Schubert cells.

Date: September 7, 2005.

The second interpretation starts from a duality between constructible functions on the varieties R_{d} provided by the preprojective varieties of a quiver, also used by G. Lusztig [6]. The coefficients Ω_{M,N} are shown (Proposition 4.2) to be essentially given by a convolution operator derived from a certain twisted version of this duality, constructed in section 4.

Acknowledgments: The present work was done during a stay of both authors at the University of Antwerp in the framework of the Priority Programme ”Non-Commutative Geometry” of the European Science Foundation.

2. A generalization of Riedtmann’s formula

2.1. Let k be a field. We fix a finite quiver Q with set of vertices I and set of arrows Q_{1}, whose underlying unoriented graph is a disjoint union of Dynkin diagrams of type A_{n}, D_{n}, E_{6}, E_{7} or E_{8}. Fixing a dimension type d = ∑_{i} d_{i}i ∈ ℕI, define an I-graded vector space V_{d} := ⊕_{i∈I}k^{d_{i}}. For any subquotient W of V_{d} compatible with the grading, we set \underline{dim}(W) := ∑_{i}(dimW_{i})i ∈ ℕI, where W_{i} denotes the i-component of W.

Set R_{d} := R_{d}(Q) := ⨁_{α:i→j} Hom(k^{d_{i}}, k^{d_{j}}) and G_{d} := ∏_{i} GL_{d_{i}}(k). The algebraic group G_{d} acts linearly on the affine space R_{d} by (g.M)_{α:i→j} := g_{j}M_{α}g_{i}^{−1}.

2.2. Let ν be a positive integer, and let d = (d^{1}, …, d^{ν}) be a ν-tuple of elements in ℕI such that d = ∑_{s} d^{s}. Let F_{d} be the set of filtrations F^{∗} = (0 = F^{ν} ⊂ … ⊂ F^{1} ⊂ F^{0} = V_{d}) of the graded space V_{d} such that \underline{dim}F^{s−1}/F^{s} = d^{s}, 1 ≤ s ≤ ν. The action of the group G_{d} on V_{d} provides a transitive action of G_{d} on F_{d}.

We fix an arbitrary filtration F_{0}^{∗} in F_{d}. Choosing successive complements, we can assume that V_{d} has a direct sum decomposition V_{d} = ⨁_{s=1}^{ν} V_{d}^{s} (as I-graded k-space), such that F_{0}^{s} = ⨁_{t>s} V_{d}^{t} for s = 1, …, ν. This induces a decomposition R_{d} = ⨁_{s,t=1}^{ν} R_{d}^{s,t} by setting R_{d}^{s,t} = ⨁_{α:i→j} Hom((V_{d}^{s})_{i}, (V_{d}^{t})_{j}). Let P_{d} be the stabilizer of F_{0}^{∗} in G_{d}. This is a parabolic subgroup of G_{d}, providing an identification F_{d} ≃ G_{d}/P_{d}.

We say that an element M in R_{d} is compatible with a filtration F^{∗} in F_{d} if M_{α}(F_{i}^{s}) ⊂ F_{j}^{s} for any arrow α : i → j in Q_{1} and any s = 1, …, ν. Set

X_{d} := {(M, F^{∗}), M ∈ R_{d}, F^{∗} ∈ F_{d}, M compatible with F^{∗}} ⊂ R_{d} × F_{d}.

The diagonal action of the group G_{d} on R_{d} × F_{d} respects X_{d}, and the projections p_{1} and p_{2} on R_{d} and F_{d}, respectively, are G_{d}-equivariant.

Set Y_{d} := p_{2}^{−1}(F_{0}^{∗}), which will be identified with its image p_{1}(Y_{d}) in R_{d}. We have the identification

(2.1)  G_{d} ×^{P_{d}} Y_{d} ≃ X_{d}, \overline{(g, y)} := \{(gp, p^{−1}.y), p ∈ P_{d}\} ↦ (g.y, gF_{0}^{∗}).

Denote by U_{d} the unipotent radical of P_{d} and set X̃_{d} : G_{d} ×^{U_{d}} Y_{d}. Let π be the natural projection X̃_{d} → G_{d} ×^{P_{d}} Y_{d} ≃ X_{d}. We have a Levi decomposition P_{d} = L_{d}U_{d}, where L_{d} is the Levi of the parabolic P_{d} defined by all elements g of G_{d} fixing each V_{d}^{s}. We have a diagonal action of L_{d} on X̃_{d} defined by l.(g, y) : (gl^{−1}, l.y), and a left action of G_{d} on X̃_{d} defined by g′.(g, y) := (g′g, y). These two actions commute.

The above direct sum decomposition V_{d} = ⨁_{s} V_{d}^{s} provides a surjection ζ : R_{d} → ∏_{s} R_{d^{s}}, which restricts to a surjection ζ : Y_{d} → ∏_{s} R_{d^{s}}. This clearly defines a surjection ζ̃: G_{d} ×^{U_{d}} Y_{d} → ∏_{s} R_{d^{s}} by projecting on the second factor. We obtain the following diagram.

(2.2)  \begin{aligned}∏_{s} R_{d^{s}} & ←^{ζ̃} & X̃_{d} \\ & & ↓ π \\ R_{d} & ←^{p_{1}} & X_{d} & →^{p_{2}} & F_{d} ≃ G_{d}/P_{d} \end{aligned}

The following lemma follows immediately from the definitions:

Lemma 2.1. The morphism ζ̃ commutes with the action of L_{d}. The morphisms π, p_{1} and p_{2} commute with the action of G_{d}.

2.3. Let mod kQ be the category of finite dimensional k-representations of Q. For a representation X in mod kQ of dimension type \underline{dim}(X) = d, we denote by O_{X} the corresponding G_{d}-orbit in R_{d}. The G_{d}-orbits are in bijection with the isoclasses of representations of dimension type d in mod kQ by definition. We denote by \overline{mod}kQ this set of isoclasses, and \overline{X} will be the isoclass corresponding to the representation X.

We now suppose that ν is the number of isoclasses of indecomposable representations in modkQ (which coincides with the number of positive roots of the root system corresponding to Q by Gabriel’s Theorem), and we denote by I_{s} for 1 ≤ s ≤ ν these indecomposable representations, ordered such that

(2.3)  Hom_{Q}(I_{t}, I_{s}) = 0 for 1 ≤ s < t ≤ ν,

where Hom_{Q} denotes the space of homomorphisms in the category modkQ (such an ordering exists since the category modkQ is directed; for this and other facts on modkQ see, for example, [10]).

We fix two representations N and M in R_{d}. In the following, we define analogues of the varieties introduced in the previous section, naturally associated to N and M.

Let N = ⊕_{s}N_{s} be the unique decomposition of the representation N such that N_{s} is isomorphic to a direct sum of copies of I_{s} for 1 ≤ s ≤ ν. We can suppose without loss of generality that the spaces N_{s} are compatible with the direct sum decomposition V_{d} = ⨁_{s} V_{d}^{s}, so that N_{s} belongs to R_{d}^{s,s}.

We define X\overline{N} as the set of pairs (P, F^{∗}) in X_{d} such that the representation induced by P on F^{s−1}/F^{s} is isomorphic to N_{s} for any s = 1, …, ν. Let X\overline{M} ; \overline{N} be the subset of pairs (P, F^{∗}) in X\overline{N} such that P belongs to O_{M}. Set Y\overline{N} := ζ^{−1}(∏_{s} O_{N_{s}}) ⊂ Y_{d}. Again, the following lemma follows immediately from the definitions:

Lemma 2.2. Via the identification 2.1, we have

X\overline{N} ≃ G_{d} ×^{P_{d}} Y\overline{N} and X\overline{M} ; \overline{N} ≃ G_{d} ×^{P_{d}} (Y\overline{N} ∩ O_{M}).

We define

X̃\overline{M} ; \overline{N} := G_{d} ×^{U_{d}} (Y\overline{N} ∩ O_{M}) ⊂ X̃\overline{N} := G_{d} ×^{U_{d}} Y\overline{N} ⊂ X_{d}.

The left action of G_{d} and the diagonal action of L_{d} both stabilize these varieties.

2.4. From now on, we suppose that k is a finite field with q elements. For any representation P in R_{d}, we denote by Aut(P) ⊂ G_{d} the stabilizer of P (which coincides with the automorphisms of P as an object in mod kQ by definition). With the notation of the previous section, let Y_{N} be the fiber of ζ over the point (N_{s}), that is, Y_{N} : ζ^{−1}((N_{s})) ⊂ Y_{d}. Note that U_{d} acts on Y_{N}.

Proposition 2.3. With notation as above, we have

|p_{1}^{−1}(M) ∩ X\overline{N}| = \frac{|Aut(M)| · |Y_{N} ∩ O_{M}|}{(∏_{s} |Aut(N_{s})|) · |U_{d}|}.

Proof. First we have

π^{−1}p_{1}^{−1}(O_{M}) ∩ X̃\overline{N} = X̃\overline{M} ; \overline{N} ≃ G_{d} ×^{U_{d}} (Y\overline{N} ∩ O_{M}).

Since p_{1}π commutes with the G_{d}-action, we obtain

|π^{−1}p_{1}^{−1}(M) ∩ X̃\overline{N}| = \frac{|G_{d}| · |Y\overline{N} ∩ O_{M}|}{|O_{M}| · |U_{d}|}
= \frac{|Aut(M)| · |Y\overline{N} ∩ O_{M}|}{|U_{d}|}.

From Lemma 2.1, we conclude

|p_{1}^{−1}(M) ∩ X\overline{N}| = \frac{|Aut(M)| · |Y\overline{N} ∩ O_{M}|}{|L_{d}| · |U_{d}|}
= \frac{|Aut(M)| · (∏_{s} |O_{N_{s}}|) · |Y_{N} ∩ O_{M}|}{|L_{d}| · |U_{d}|}.

This implies the proposition. □

2.5. In this section, we give a more precise version of Proposition 2.3. Let 𝔤_{d} := ⨁_{i} 𝔤𝔩(k^{d_{i}}) be the Lie algebra of the group G_{d}. The components of an element ξ in 𝔤_{d} will be denoted by ξ_{i}. Let 𝔲_{d} ⊂ 𝔤_{d} be the Lie algebra of U_{d}. The differential of the morphism G_{d} → G_{d}.N gives rise to a morphism of vector spaces ϕ : 𝔤_{d} → R_{d} given by ϕ(ξ)_{α:i→j} = ξ_{j}N_{α} − N_{α}ξ_{i}.

Lemma 2.4. The morphism ϕ has the following properties :

(i) Ker(ϕ) = End_{Q}(N), (ii) Im(ϕ) = T_{N}, where T_{N} := T_{N}(O_{N}) is the tangent space to O_{N} at the point N, (iii) Im(ϕ) is compatible with the decomposition R_{d} = ⨁_{s,t} R_{d}^{s,t}, and it contains the subspace ⨁_{s≤t} R_{d}^{s,t}, (iv) the restriction of ϕ to 𝔲_{d} is injective.

Proof. (i) follows from the definition of ϕ and of the category modkQ. (ii) is clear. The first assertion of (iii) follows from the fact that ϕ decomposes into a direct sum ϕ = ⨁_{s,t} ϕ_{s,t}, where

(2.4)  ϕ_{s,t} : ⨁_{i} Hom_{k}((V_{d}^{s})_{i}, (V_{d}^{t})_{i}) → R_{d}^{s,t}.

The second assertion is [5, Lemma 10.4]. To prove (iv), we remark that ϕ|_{𝔲_{d}} = ⨁_{s>t} ϕ_{s,t} which implies that Kerϕ|_{𝔲_{d}} = 0 by formula 2.3 and (i). □

We now consider the affine space Y_{N} identified with its tangent vector space T_{N}(Y_{N}) by x ↦ x + N.

Proposition 2.5. The space Y_{N} contains ϕ(𝔲_{d}). Let E_{N} be any complement of ϕ(𝔲_{d}) in Y_{N} which is compatible with the decomposition ⨁ R_{d}^{s,t} of R_{d}. Then,

(i) the space E_{N} is a direct summand of T_{N}(O_{N}) in R_{d}, (ii) E_{N} is a transversal slice for the action of U_{d} on Y_{N}.

Proof. First we remark that, by construction, the space Y_{N} is compatible with the decomposition R_{d} = ⨁_{s,t} R_{d}^{s,t}. Thus, a complement E_{N} as above exists by Lemma 2.4 (iii).

(i) is a consequence of Lemma 2.4 (ii), (iii) and the decomposition 2.4 of ϕ. Now we prove (ii). Fix a complement E_{N} and let X be in E_{N}. We claim that

X is the unique element of E_{N} in the orbit U_{d}.X. Suppose that Y = U.X ∈ E_{N}, with U ∈ U_{d}. The component of U (induced by the decomposition V_{d} = ⨁_{s} V_{d}^{s}) belonging to ⨁_{i} Hom((V_{d}^{s})_{i}, (V_{d}^{t})_{i}) is denoted by U_{s,t} for s ≥ t. Note that U_{s,s} is the identity for all s = 1, …, ν.

We prove by induction on s − t > 0 that U_{s,t} = 0, for s > t. Fix a pair (s, t) for 1 ≤ t < s ≤ ν. It is easily seen by a weight argument that the induction hypothesis implies that the component Y_{s,t} of Y has the following form : Y_{s,t} = U_{s,t}N_{t}−N_{s}U_{s,t}. So, we have U_{s,t}N_{t} − N_{s}U_{s,t} = Y_{s,t} ∈ E_{N} by the hypothesis on E_{N}. This implies that Y_{s,t} ∈ E_{N} ∩ ϕ(𝔲_{d}) = {0}. By Lemma 2.4, this gives U_{s,t} = 0. The claim is proved.

In particular, this implies that the action of the group U_{d} on Y_{N} is free. Thus, |E_{N}| = \frac{|Y_{N} |}{|U_{d}|} = |Y_{N}/U_{d}|. This equality, together with the claim just proved, gives (ii). □

The following corollary can be seen as a generalization of Riedtmann’s formula, [9].

Corollary 2.6. Let E_{N} be as in the previous proposition. Then,

|F_{N_{ν},...,N_{1}}^{M}| = \frac{|Aut(M)| · |E_{N} ∩ O_{M}|}{∏_{s} |Aut(N_{s})|},

where F_{N_{ν},...,N_{1}}^{M} denotes the set of filtrations 0 = M^{ν} ⊂ … ⊂ M^{1} ⊂ M^{0} = M of the representation M with successive subquotients M^{s−1}/M^{s} isomorphic to N_{s} for 1 ≤ s ≤ ν.

Proof. By construction, we have F_{N_{ν},...,N_{1}}^{M} = p_{1}^{−1}(M)∩X\overline{N}. Moreover, as U_{d} ⊂ G_{d}, the previous proposition implies that E_{N} ∩ O_{M} is a transversal slice for the action of U_{d} on Y_{N} ∩ O_{M}. The corollary then follows from Proposition 2.3. □

It is known [11] that there exists a polynomial F_{N_{ν},...,N_{1}}^{M}(t) ∈ ℤ[t], called the generalized Hall polynomial, whose value at any prime power q equals the number of 𝔽_{q}-rational points of the variety F_{N_{ν},...,N_{1}}^{M}.

3. Hall algebras and coefficients of the bar automorphism.

3.1. We define the Euler form < , > on ℕI by < d, e >:= ∑_{i∈I} d_{i}e_{i} −∑_{α:i→j} d_{i}e_{j}. We suppose in this section that k is a field with q = v^{2} elements for some v ∈ ℂ. The dimension type d and the representations M, N are no longer fixed. For all finite sets X on which G_{d} acts, we denote by ℂ_{G_{d}}[X] the set of G_{d}-invariant functions from X to ℂ. Define

H_{v}(Q) = ⨁_{d∈ℕI} ℂ_{G_{d}}[R_{d}].

The space H_{v}(Q) is endowed with a structure of ℕI-graded ℂ-algebra by the convolution product :

(f.g)(X) = v^{<d,e>} ∑_{U⊂X} f(X/U)g(U), f ∈ ℂ_{G_{d}}[R_{d}], g ∈ ℂ_{G_{e}}[R_{e}], X ∈ R_{d+e},

where U runs over all subrepresentations of X of dimension type e. It is known [11] that this product defines the structure of an associative algebra on H_{v}(Q), which is called the (twisted) Hall algebra of the quiver Q.

For any representation M of Q with isoclass \overline{M}, let e_{M} = e\overline{M} be v^{dimEndN−dimN} times the characteristic function of the orbit O_{M}. It is clear that \{e\overline{M}\}_{\overline{M}∈ \overline{mod} kQ} is a basis of H_{v}(Q).

Let S_{i} be the simple representation corresponding to the vertex i in I. It is known [12] that there exists an isomorphism η from the Hall algebra H_{v}(Q) to the positive part U_{q}(𝔤)^{+} of the quantum enveloping algebra associated to Q, such that η maps e_{S_{i}} to the canonical generator e_{i} of U_{q}(𝔤)^{+}. Note that the basis e_{M} = e\overline{M} is sent to the so-called Poincaré-Birkhoff-Witt basis of U_{q}(𝔤)^{+} which corresponds to a reduced decomposition of the longest Weyl group element naturally associated to Q, see [5, 4.12].

3.2. We consider the inner product on H_{v}(Q), called Green form, defined by

(e_{M}, e_{N}) = v^{2dimEndN}a_{M}^{−1}δ_{N,M},

where δ is the Kronecker symbol and a_{M} := a_{M}(v^{2}) = |Aut(M)|. Hence, we obtain the dual PBW type basis by setting e_{M}^{∗} = v^{−2dimEndN}a_{M}e_{M}.

The following lemma is an easy consequence of the definition of the convolution product in the Hall algebra and of the properties of the decomposition N = ⨁_{s} N_{s}.

Lemma 3.1. Suppose that N = ⨁_{s} N_{s} is the decomposition of N into powers of indecomposables as above. Then,

(i) e_{N} = e_{N_{1}} … e_{N_{ν}}, (ii) e_{N}^{∗} = e_{N_{1}}^{∗} … e_{N_{ν}}^{∗}, (iii) e_{N_{ν}} … e_{N_{1}} = ∑_{M} v^{S−dimEndM}F_{N_{ν},...,N_{1}}^{M}(v^{2})e_{M}, where S = ∑_{s} dimEndN_{s} + ∑_{s>t} < \underline{dim}N_{s}, \underline{dim}N_{t} >.

3.3. Using the basis elements e_{M} introduced above, the multiplication in the Hall algebra reads as follows:

e_{M} · e_{N} = ∑_{\overline{X}} v^{dimEndM+dimEndN+<\underline{dim}M,\underline{dim}N>−dimEndX}F_{M,N}^{X}(v^{2}) · e_{X}.

Since the F_{M,N}^{X} are polynomials, we can thus take the above formula as the definition of structure constants for a ℚ(v)-algebra, the generic (twisted) Hall algebra [12], which will still be denoted by H_{v}(Q).

We define a ℚ-linear involution on ℚ(v) by \overline{v} = v^{−1}. We define on H_{v}(Q) :

– a \overline{·}-linear involution by \overline{e_{i}} = e_{i}, called the bar involution,

– the ℚ(v)-linear antiinvolution σ by σ(e_{i}) = e_{i}.

Denote by ω_{M,N} the e_{N}-coefficient of \overline{e_{M}} in the PBW-basis. It is clear that ω_{M,N} is zero if M and N do not have the same dimension type. Following Lusztig [5], we use the normalization Ω_{M,N} = v^{dimO_{N} −dimO_{M}} ω_{M,N} = v^{dimEndM−dimEndN}ω_{M,N}. We say that M degenerates to N if O_{N} belongs to the closure of O_{M}. The following is proved in [5].

Lemma 3.2. For any two representations M, N in modkQ, we have:

(i) if Ω_{M,N} ≠ 0, then M degenerates to N, (ii) Ω_{M,M} = 1, (iii) Ω_{M,N} ∈ ℤ[v^{−2}].

We now want to give a geometric interpretation of the polynomial \overline{Ω_{M,N}}. This will be provided by Theorem 3.4. The following proposition precises a result of [3, Proposition 3.1], where the formula was asserted up to a power of v.

Proposition 3.3. Let M, N be two representations in modkQ, and let N = ⨁_{s} N_{s} be the decomposition into powers of indecomposables as above. Then, the polynomial \overline{Ω_{M,N}} ∈ ℤ[v^{2}] is given by

\overline{Ω_{M,N}} = F_{N_{ν},...,N_{1}}^{M}(v^{2})\frac{∏_{s} a_{N_{s}}}{a_{M}} .

Proof. We first calculate ω_{M,N} = (\overline{e_{M}}, e_{N}^{∗}) by using the adjoint of the bar automorphism for the Green form. From [7, 1.2.10.], we have :

\overline{ω_{M,N}} = (−v)^{−dimM}v^{−<\underline{dim}N,\underline{dim}N>}(e_{M}, σ(\overline{e_{N}^{∗}})).

From Lemma 3.1 (ii), this gives :

\overline{ω_{M,N}} = (−v)^{−dimM}v^{−<\underline{dim}N,\underline{dim}N>}(e_{M}, σ(\overline{e_{N_{ν}}^{∗}}) … σ(\overline{e_{N_{1}}^{∗}})).

By [2], the elements e_{N_{s}}^{∗} belong to the dual canonical basis. Hence, by [8, Lemma 4.3.],

σ(e_{N_{s}}^{∗}) = (−v)^{∑ dimN_{s}}v^{<\underline{dim}N_{s},\underline{dim}N_{s}>}e_{N_{s}}^{∗}

for all s = 1, …, ν. We deduce that

\overline{ω_{M,N}} = v^{− ∑_{s≠t}<\underline{dim}N_{s},\underline{dim}N_{t}>}(e_{M}, e_{N_{ν}}^{∗} … e_{N_{1}}^{∗}).

It remains to calculate the e_{M}^{∗}-component of the product e_{N_{ν}}^{∗} … e_{N_{1}}^{∗} in the dual PBW-basis. This is obtained from Lemma 3.1 :

(e_{M}, e_{N_{ν}}^{∗} … e_{N_{1}}^{∗}) = v^{T} F_{N_{ν},...,N_{1}}^{M}a_{N_{1}} … a_{N_{ν}} a_{M}^{−1},

where T = − ∑_{s} dimEndN_{s} + dimEndM + ∑_{s>t} < \underline{dim}N_{s}, \underline{dim}N_{t} > . Now, from the interpretation of < , > as the homological Euler form in modkQ, namely < \underline{dim}M, \underline{dim}N >= dimHom(M, N) − dimExt^{1}(M, N), we easily obtain

dimEnd(N) = ∑_{s} dimEnd(N_{s}) + ∑_{s≤t} < \underline{dim}N_{s}, \underline{dim}N_{t} >,

and the claimed formula follows.

□

3.4. Our efforts are rewarded in that we can deduce a geometric interpretation of the coefficient \overline{Ω_{M,N}}.

Theorem 3.4. Let k be the finite field 𝔽_{v^{2}}. Fix a dimension type d in ℕI, and fix representations N, M in R_{d}. Let E_{N} be a graded complementary of the tangent space of O_{N} at N as in Proposition 2.5. Then, the value of the polynomial \overline{Ω_{M,N}} at v^{2} equals the cardinality of the set E_{N} ∩ O_{M}.

Proof. This is Proposition 3.3 combined with Corollary 2.6. □

Note that the theorem implies the following curious identity :

Corollary 3.5. Let d be in ℕI and N in R_{d}. Then,

∑_{P} \overline{Ω_{P,N}} = |E_{N}| = q^{dimExt^{1}(N,N)},

where P runs over the set of isoclasses of representations of dimension type d.

We finish the section with the following remark. The (generalized) Hall polynomials are known to have leading coefficient equal to one. Hence, by the Lang-Weil theorem, all the varieties E_{N} ∩O_{M} have a unique irreducible component of maximal dimension.

4. The preprojective variety and coefficients of the bar automorphism.

4.1. For any arrow α : i → j in the quiver Q, we define i(α) = i and h(α) = j. Let Q^{op} be the opposite quiver, having the same vertices as Q, and an arrow α^{∗} : j → i for each arrow α : i → j in Q.

Fix a dimension type d = ∑_{i} d_{i}i in ℕI. As above, G_{d} acts on R_{d}(Q) = R_{d} and on R_{d}(Q^{op}). Note that the map sending a linear map to its adjoint induces an isomorphim R_{d}(Q) → R_{d}(Q^{op}), M ↦ M^{∗}.

Let Π_{d} be the preprojective variety (see [6]):

Π_{d} := {((M_{α})_{α}, (N_{α^{∗}})_{α}) ∈ R_{d}(Q) × R_{d}(Q^{op}), for all i ∈ I :

∑_{α∈Q_{1},h(α)=i} N_{α^{∗}}M_{α} ∑_{α∈Q_{1},i(α)=i} M_{α}N_{α^{∗}}} ⊂ R_{d}(Q) × R_{d}(Q^{op}).

We denote by p (resp. p^{op}) the canonical projection Π_{d} → R_{d}(Q) (resp. Π_{d} → R_{d}(Q^{op})). We also define a twisted projection p̃: Π_{d} → R_{d}(Q) by mapping (M, N) to M + N^{∗} using the identification R_{d}(Q) ≃ R_{d}(Q^{op}) above.

4.2. Observe that, for all N in R_{d}, the identification R_{d}(Q) → R_{d}(Q^{op}) maps the orbit G_{d}.N to G_{d}.N^{∗} and the tangent space T_{N}(O_{N}) to T_{N^{∗}}(O_{N^{∗}}).

We consider the non degenerate pairing on R_{d}(Q) × R_{d}(Q^{op}) given by

< (M_{α}), (N_{α^{∗}}) >= ∑_{α∈Q_{1}} T r(M_{α}N_{α^{∗}}).

We have the following

Lemma 4.1. With respect to the pairing above, we have T_{N}(O_{N})^{⊥} = p^{op}(p^{−1}(N)), for all N in R_{d}(Q).

Proof. By Lemma 2.4 (ii), an element (f_{α^{∗}}) of R_{d}(Q^{op}) belongs to T_{N}(O_{N})^{⊥} if and only if for all X in 𝔤_{d}, we have

∑_{α∈Q_{1}} T r((X_{h(α)}N_{α} − N_{α}X_{i(α)})f_{α^{∗}}) = 0.

By well known properties of the trace form, this is equivalent to

∑_{α∈Q_{1}} T r(X_{h(α)}N_{α}f_{α^{∗}}) − T r(X_{i(α)}f_{α^{∗}}N_{α}) = 0,

thus

∑_{i∈I} T r(X_{i}( ∑_{i(α)=i} N_{α}f_{α^{∗}} − ∑_{h(α)=i} f_{α^{∗}}N_{α})) = 0.

Since the trace form is non-degenerate, this gives

∑_{i(α)=i} N_{α}f_{α^{∗}} − ∑_{h(α)=i} f_{α^{∗}}N_{α} = 0,

which proves (f_{α^{∗}}) ∈ p^{op}(p^{−1}(N)) as required. □

For any morphism π : X → Y of k-varieties, and any function f (resp. g) on X (resp. Y), we define as usual :

π^{∗}(g) : X → k, x ↦ g(π(x)),

π_{∗}(f) : Y → k, y ↦ ∑_{π(x)=y} f(x).

For a subset A of R_{d} or R_{d}(Q^{op}), we denote by 1_{A} the corresponding characteristic function. The previous lemma gives the following interpretation of the polynomials \overline{Ω_{M,N}} in terms of the geometry of the preprojective variety:

Proposition 4.2. For M in R_{d}(Q) and N^{∗} in R_{d}(Q^{op}), we have

(p^{op})_{∗}(p̃)^{∗}(1_{O_{M}})(N^{∗}) = \overline{Ω_{M,N}}.

Proof. Obviously, (p^{op})_{∗}(p̃)^{∗}(1_{O_{M}}) belongs to ℂ_{G_{d}}[R_{d}(Q^{op})].

Fix N in R_{d}. By the definitions, (p^{op})_{∗}(p̃)^{∗}(1_{O_{M}})(N^{∗}) can be rewritten as (p^{op})_{∗}(f_{M})(N^{∗}), where the function f on Π_{d} is defined by f_{M}(A, B^{∗}) = 1 if A + B ∈ O_{M}, and 0 otherwise. Hence, by the previous lemma,

(p^{op})_{∗}(p̃)^{∗}(1_{O_{M}})(N^{∗}) = ∑_{A∈T_{N^{∗}} (O_{N^{∗}})^{⊥}} f_{M}(A, N^{∗}) = |O_{M} ∩ (N + T_{N^{∗}}(O_{N^{∗}})^{⊥})|.

By Theorem 3.4, this gives (p^{op})_{∗}(p̃)^{∗}(1_{O_{M}})(N^{∗}) = \overline{Ω_{M,N}}. □

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

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

Mathematisches Institut, Universität Münster, 48149 Münster, Germany

E-mail address: reinekem@math.uni-muenster.de