On the cohomology of character stacks for non-orientable surfaces

We give a counterexample to a formula suggested by the work of Letellier and Rodriguez-Villegas (Ann l’Inst Fourier 36, 2022) for the mixed Poincaré series of character stacks for non-orientable surfaces. The counterexample is obtained by an explicit description of these character stacks for (real) elliptic curves.


Introduction
Let K be an algebraically closed field, r, k ≥ 1 be non-negative integers and C = (O 1 , . . ., O k ) a k-tuple of semisimple orbits of G = GL n (K).Consider a couple ( Σ, σ) where Σ is a Riemann surface of genus r − 1 and σ : Σ → Σ an antiholomorphic involution without fixed points.The character stack M ϵ C associated to such a couple ( Σ, σ) and C is the stacky quotient (1.0.1)where θ : G → G is the Cartan involution θ(A) = ( t A) −1 .For a more detailed definition and the relation between M ϵ C and representations of π 1 ( Σ) see Section §2.2.The stacks M ϵ C are deeply related to branes inside the moduli space of Higgs bundles: the computation of cohomology and geometry of branes is a key part in understanding mirror symmetry for the Hitchin system.References about the subject can be found for example in [1], [2], [5], [4].
Recently Letellier and Rodriguez-Villegas (see [17,Theorem 4.6]) computed the E-series E(M ϵ C , q) of these stacks over C when C is generic (for a definition of generic k-tuples of orbits see Definition 2.3.1).The E-series is a specialization of the whole (compactly supported) mixed-Poincaré series H c (M ϵ C , q, t) obtained by plugging t = −1.For a definition of the mixed-Poincaré series see §2.3.1.E-series give important information such as the number of irreducible components or non emptyness.The computation of [17,Theorem 4.6] is obtained via reduction over F q and point counting.The authors consider the F q − stack M ϵ C,Fq and compute in an explicit way a rational function In this case there is an equality E(M ϵ C,C , q) = Q(q), as shown for example in [17,Theorem 2.8].Surprisingly, the functions Q(t) appearing in this context are very similar to the ones computing E-series of character stacks for Riemann surfaces.Consider g ≥ 0, k ≥ 1 and C = (O 1 , . . .O k ) as above.The associated character stack M C for a Riemann surface Σ of genus g is the stacky quotient This can alternatively be described as the quotient stack where {x 1 , . . ., x k } is a set of k points of Σ and each y i is a small loop around x i .E-series for these stacks and generic orbits were computed in [10,Theorem 1.2.3].As observed in [17,Remark 1.5], for r = 2h we have an equality E(M ϵ C , q) = E(M C , q) where M C is associated to a Riemann surface of genus h.Even for r odd, the formulas for the E-series of M ϵ C are very similar to those of E(M C , q) (see §2. 3

.3 for more details).
There is a longstanding conjecture about the whole mixed Poincaré series of character stacks for Riemann surfaces [10,Conjecture 1.2.1]: in [17,Theorem 4.8] the authors verified that a completely analogous formula holds for M ϵ C for r = 1 and k = 1 (see §2.3.3 for more details).It would therefore have been natural to expect a similar formula to hold for all r.The main result of this paper (see §3.2) is an explicit description of some of these spaces and their cohomology in the case r = 2 giving a counterexample to the expected formula.The main theorem is: C is a µ 2 -gerbe over C * .In particular, its mixed Poincaré series is To prove Theorem 1.0.1 we need some results of independent interest concerning the geometry of the spaces M ϵ C for k = 1 and the orbit {e πdi n } (see §3.1).To summarize these results, let M ϵ n,d be the GIT quotient associated to M ϵ C and M n,d be the GIT quotient of the character stack associated to the Riemann surface Σ (of genus r − 1) for k = 1 and the orbit {e 2πdi n }.There is an involution on M n,d , which we denote again by σ, which sends a representation ρ ∈ M n,d to σ(ρ) = θ(ρ)(σ * ) (for more details and a definition of σ * see §2.1, §2.2).In §3.1 we show that: The Theorem 1.0.2(and the others in §3.1) are probably known to the experts but we could not locate a reference in the literature.We review them here for the sake of completeness.In the paragraph §3.Acknowledgements.I would like to thank Emmanuel Letellier for bringing this subject to my attention.It is a pleasure to thank also Florent Schaffhauser for many useful discussions about the topics dealt in this paper.

Fundamental groups of punctured non-orientable surfaces
Let Σ be a Riemann surface of genus g and σ : Σ → Σ be an antiholomorphic involution σ without fixed points.The quotient Σ := Σ/σ is endowed with the structure of a non-orientable surface: topologically Σ is the connected sum of r := g + 1 real projective planes.We denote by p the quotient map p : Σ −→ Σ.
Finally, we denote by σ * the morphism on π 1 ( Σ − p −1 (S)) given by Notice that the morphism σ * is not an involution in general.More precisely, σ 2 * is the conjugation by the element σ(λ σ )λ σ ∈ π 1 ( Σ − p −1 (S)) which in general is not the identity.We recall that there are explicit presentations of the above fundamental groups: and Example 2.1.1.Let us look at the case of r = 2 and k = 1.We consider the elliptic curve Ẽ associated to the lattice ⟨1, i⟩ ⊆ C i.e Ẽ ∼ = C/ ⟨1, i⟩ and let π be the projection π : C → E. Let σ : Ẽ → Ẽ be the involution without fixed points defined by σ(z) = z + 1 2 and p : Ẽ → E := Ẽ/σ be the associated quotient.

Character stacks for non-orientable surfaces
We fix an algebraically closed field K (which for us will be either C or F q ).We denote by G the general linear group GL n (K) and by θ the Cartan involution g → ( t g) −1 .The corresponding semidirect product will be denoted by We consider the variety where p : G → ⟨θ⟩ is the natural projection and h : G → G + the natural inclusion.Given the explicit presentation of π 1 (Σ − S) we can rewrite Hom ϵ C (Π, G + ) as The variety Hom ϵ C (Π, G + ) is endowed with a G-action defined by: The character stacks we will consider are the quotient stacks As Hom ϵ C (Π, G + ) is affine and G is reductive, we can also consider the GIT quotient The stacks M ϵ C admit an alternative description in terms of the so-called real σ-invariant representations (which can be found in [ It is therefore natural to ask conversely which representations ρ of Π can be lifted to a morphism ρ : Π → G + which makes the diagram (2.2.2) commute.To answer to the question, it is necessary to precisely describe monodromies around the punctures, as explained in [17,Remark 4.2].
Let p −1 (S) = {y 1,1 , . . ., y k,1 , y 1,2 , . . .y k,2 } where σ(y i,1 ) = y i,2 for i = 1, . . ., k.We can rewrite the standard presentation 2.1.2 of Π as where each x i,j is a path around y i,j .Let C be the and Hom C ( Π, G) be the affine variety For a representation ρ ∈ Hom C ( Π, G) we say that ρ is σ-invariant if ρ ∼ = θ ρ(σ * ).This is equivalent to asking for the existence of an element h σ ∈ G which verifies We say that ρ is quaternionic if there exists h σ as in eq.(2.2.3) such that ρ(σ If the conditions of Equations ( 2 The variety ŨC is endowed with a G-action defined by The arguments above imply the following Proposition Proposition 2.2.2.There is an isomorphism of quotient stacks Remark 2.2.3.If ρ is an irreducible representation and h is such that there is an equality We can define a morphism f : M ϵ C → M σ C which maps a couple (ρ, h) as in Equation (2.2.5) to the representation ρ.In a slightly more involved way, it would be possible to lift the map f to a morphism of quotient stacks F : M ϵ C → M σ C .These morphisms are in general not even surjective: we will describe the image of f in certain cases in Proposition 3.1.2.

Cohomology computation
In this paragraph we will briefly review the results obtained by Letellier and Rodriguez-Villegas in [17] about the character stacks M ϵ C .Let us first recall the definition of the E-series and the mixed Poincaré series of an algebraic stack and the combinatorics needed for the formulas for the E-series E(M ϵ C , q).

Mixed Poincaré series
Let X be an algebraic stack of finite type over an algebraically closed field k.For K = C, we will consider the compactly-supported cohomology groups H * c (X ) := H * c (X , C) with coefficients in C. For K = F q , we will denote by H * c (X ) the compactly supported étale cohomology with coefficients in Q ℓ .When K = C, each vector space H k c (X ) is endowed with the weight filtration W k • (see [7,Chapter 8] for a definition and [17, Section 2.2] for the analogous one for stacks X over F q ).We define the mixed-Poincaré series H c (X , q, t) as The specialization H c (X , 1, t) of H c (X , q, t) at q = 1 is equal to the Poincaré series of the stack X .When ) is finite for each m, we define the E-series: For a quotient stack X = [X/G] where G is a connected linear algebraic group and X an affine variety, the E-series E(X , q) is well defined and E(X , q) = E(X, q)E(BG, q) where BG is the classifying stack of G: for a proof see [17, Theorem 2.5].

Combinatorics
We fix integers m, k ≥ 0 and we denote by P the set of partitions.Let x 1 = {x 1,1 , x 1,2 . . .}, . . . . . ., x k = {x k,1 , . . .} be k sets of infinitely many variables and let us denote by Λ k = Λ(x 1 , . . ., x k ) the ring of functions separately symmetric in each set of variables.On Λ k there is a natural bilinear form obtained by extending by linearity where ⟨, ⟩ is the bilinear form making the Schur functions s µ an orthonormal basis.For a multipartition µ = (µ 1 , . . ., µ k ) ∈ P k we denote by We consider the hook functions and the associated series where H λ (x i , q, t) are the (modified) Macdonald symmetric polynomials (for a definition see [9, I.11]).We define the functions H µ,m (z, w) by the following formula: where Log is the plethystic logarithm (for a definition see for example [10, Section 2.3.3]).

E-polynomials and conjectures
Let us now explain one of the main results of [17] about the stacks Definition 2.3.1.The k-tuple C is said to be generic if the following property holds.Given a subspace V of K n which is stabilized by some In [17,Theorem 4.6] the authors showed that for a generic C, the following equality holds: where µ = (µ 1 , . . ., µ k ) is the multipartition given by the multiplicities of the eigenvalues of O 1 , . . ., O k respectively and This result is surprinsigly similar to the analogous one obtained in [10, Theorem 1.2.3] about character stacks for Riemann surfaces.Fix a Riemann surface Σ of genus g and a set of k-points S = {y 1 , . . ., y k } ⊆ Σ.The associated character stack M C is the quotient stack In [10, Theorem 1.2.3] the authors showed that the following equality holds: where agrees thus with the one of M C for a Riemann surface Σ of genus h.
In the same paper [10, Conjecture 1.2.1], the authors were also able to give a conjectural formula for the whole mixed Poincaré series of the stacks M C , naturally deforming Equation (2.3.7).The conjectural identity for the mixed Poincaté series of M C is The conjectural identity (2.3.8) is generally believed to be true.The Poincaré series of the stacks M C were computed by Mellit in [19,Theorem 7.12] and his result agrees with the specialization of Formula (2.3.8) at q = 1.In [17,Theorem 4.6] it is proved that a formula analogous to Formula (2.3.8)holds in the non-orientable setting for r = k = 1 i.e that the following equality holds (2.3.9) It would therefore have been natural to expect that such a formula holds for all r, k i.e that for a generic C. The main result of this paper is a counterexample to Formula (2.3.10),obtained by an explicit description of these spaces in the case r = 2.
3 Main results

Character stacks for k = 1 and generic orbit
In this section we assume that K = C.We fix r ≥ considered for example in [13].As d and n are coprime, the representations ρ ∈ M n,d are irreducible (as shown in [13,Lemma 2.2.6]).In this case, given an element ρ ∈ Hom ϵ C (Π, G + ) corresponding to a couple (ρ, h σ ) with ρ ∈ Hom C ( Π, G) we have Stab G (ρ) = ±1 (see [23,III.5.1.3]).The stack M ϵ n,d is thus a µ 2 -gerbe over the affine variety M ϵ n,d .Remark 3.1.1.The canonical morphism q : M ϵ n,d → M ϵ n,d , being a µ 2 -gerbe, is proper.The proper base change for Artin stacks implies that for every x ∈ M ϵ n,d and for every i ∈ Z we have As the rational higher cohomology of B(µ 2 ) vanishes, R i q * C = 0 if i ̸ = 0 and q * C = C.The Leray spectral sequence for cohomology with compact support implies that ).The cohomology of the quotient stack is isomorphic to that of the GIT quotient: in particular, the (compactly-supported) cohomology of M ϵ n,d is 0 in negative degrees.
The main result of this paragraph is the following proposition: where M σ,+ n,d , M σ,− n,d are given by real/quaternionic representations respectively and there is an isomorphism 4 is an isomorphism.
Before proving Proposition 3.1.2,we notice that the quaternionic and the real representations form disjoint subsets by Remark 2.2.3.To see that there are no quaternionic representations for r odd, we will use the equivalence between quaternionic representations and quaternionic Higgs bundles.As this correspondence is crucial for the study of the varieties M ϵ n,d , let us briefly review it here.For more details, see for example [5], [21], [1], [2], [4]

Real and quaternionic Higgs bundles
A Higgs bundle over Σ is a couple (E, Φ) where E is a vector bundle over Σ and Φ a morphism The moduli space of (stable) Higgs bundle over Σ of rank n and degree d is denoted by M Dol,n,d (for a definition of stability see for example [1, Section 4.1] or [5, Definition 2.3]).It is a fundamental result (see for example [24]) that there is a homeomoprhism (called non abelian Hodge correspondence) We consider the involution on M Dol,n,d , which we denote again by σ, given by and we say that a Higgs bundle (E, Φ) is σ-invariant if there exists an isomorphism α : (E, Φ) → σ((E, Φ)).Real Higgs bundles are couples ((E, Φ), α) such that In a similar way, quaternionic Higgs bundles are defined by asking for the equality In [5, Proposition 5.6], [4,Theorem 4.8] it is shown that the homemorphism (3.1.1)restricts to a homeomorphism M σ n,d ∼ = M σ Dol,n,d .In loc.cit it is shown moreover that this bijection sends real/quaternionic representations into real/quaternionic Higgs bundles respectively.We will denote the subsets of M Dol,n,d given by real/quaternionic Higgs bundles by M σ,+ Dol,n,d /M σ,− Dol,n,d respectively.
Notice that, as σ : M Dol,n,d → M Dol,n,d is antiholomorphic, the fixed points locus M σ Dol,n,d is not a complex algebraic variety anymore but is identified with the set of R-points of M Dol,n,d with respect to the real structure induced by σ.
For odd r, if a quaternionic couple (ρ, h) existed (i.e M σ,− n,d ̸ = ∅) there would exist a stable quaternionic Higgs bundle (E, Φ) on Σ.Its determinant det(E) would be a quaternionic line bundle of degree d over Σ: the quaternionic condition is preserved under taking the determinant as n is odd.The existence of a quaternionic line bundle for odd r is ruled out by the topological criterion of [20,Theorem 2.4].
To prove Proposition 3.1.2,we will need the following preliminary Lemma.
The projection map ψ : Z → Y is a principal G m -bundle for the étale topology.
Proof.The variety Z is endowed with the G m action t • (ρ 1 , ρ 2 , h) = (ρ 1 , ρ 2 , th).This action is free and transitive on the fibers of ψ, as all the representations inside Hom n,d ( Π, G) are irreducible.Moreover ψ(t • z) = ψ(z) for all z ∈ Z.We are thus reduced to show that ψ is locally trivial for the étale topology.
As the map q : Hom n,d ( Π, G) → M n,d is a principal PGL n -bundle for the étale topology, there exists an étale open covering It is enough to show that the pullback map ψ : Z U i → Y U i is locally trivial in the étale topology for each i ∈ I. Fix then i ∈ I and put U i = U .Notice that the variety Y U admits the following isomorphism: In a similar way, the variety Z U is isomorphic to so that ψ corresponds to the morphism ψ(u, g, h, s) = (u, g, h).We can view Y U as a subset of U × PGL n × PGL n × PGL n as Via these identifications, the map ψ corresponds to the restriction of the morphism U ×PGL n × PGL n × GL n → U ×PGL n × PGL n × PGL n given by the identity on the first three factors and the quotient map GL n → PGL n on the last one.This is a principal G m -bundle because GL n → PGL n is so.
We now prove Proposition 3.1.2.We keep the notations of Lemma 3.1.3.
) and similarly for quaternionic representations Hom n,d ( Π, G) σ,− = q−1 (M σ,− n,d ).The variety Hom n,d ( Π, G) σ is isomorphic to the closed subvariety Y σ of Y given by: ) and similarly Y σ,− .From Remark 2.2.3 there is a well-defined morphism p is thus a principal G m -bundle.The G-action on Hom n,d (Π, G + ) ϵ defined by the Formula (2.2.6) induces an action of the center Z G = G m which differs from the one coming from the principal G m -bundle structure by a square factor.The morphism Hom We deduce the following chain of isomorphisms: To end the proof of Proposition 3.1.2,it actually remains to show that M σ,+ n,d , M σ,− n,d are isomorphic if r is even.For r even there exists a quaternionic representation τ ∈ M σ,− 1,0 of rank 1 over Σ (see [21,Theorem 2.4]).Taking the tensor product by τ gives then an isomorphism −⊗τ : M σ,+ n,d → M σ,− n,d : the same proof was carried out for real and quaternionic vector bundles in [21, Theorem 1.1].

Character stacks for (real) elliptic curves
We focus now on the case r = 2.We consider the elliptic curve Ẽ and the antiholomorphic involution σ introduced in Example 2.

2 ,
we describe the variety M σ n,d for r = 2 and find an isomorphism M σ,+ n,d ∼ = C * which allows to prove Theorem 1.0.1.

1
and a Riemann surface Σ of genus g := r − 1 with an antiholomorphic involution σ : Σ → Σ.Consider a point z 1 ∈ Σ = Σ/σ and the subset S := {z 1 } ⊆ Σ (i.e k = 1).Let d, n ∈ N such that d is even and (d, n) = 1.Let C be the generic semisimple orbit of GL n (C) given by C = {e πi d n I n }.We denote the associated character stacks in this case by M ϵ n,d := M ϵ C and M n,d := M C respectively and similarly the associated GIT quotients by M ϵ n,d and M n,d respectively.As C is a central orbit, the character stack M n,d is the twisted character stack