CHARACTER STACKS ARE PORC COUNT

Abstract We compute the number of points over finite fields of the character stack associated to a compact surface group and a reductive group with connected centre. We find that the answer is a polynomial on residue classes (PORC). The key ingredients in the proof are Lusztig’s Jordan decomposition of complex characters of finite reductive groups and Deriziotis’s results on their genus numbers. As a consequence of our main theorem, we obtain an expression for the E-polynomial of the character stack.

Almost all the previous work in this area concerns the case when G = GL n , SL n , or PGL n .The only exception we know of is the unpublished thesis [Cam17].Note that from the point of view of Langlands correspondence, it is crucial to understand character stacks of all reductive groups, for Langlands central conjecture, functoriality, concerns relationship between automorphic functions (or sheaves) of different reductive groups.
The purpose of this paper is to study the arithmetic geometry of the character stack associated to a compact surface group and an arbitrary reductive group G with connected centre.This represents the first step in generalising the program of Hausel-Letellier-Villegas [HRV08, HLRV11, Let15, LRV20] from type A to arbitrary type in a uniform manner.
1.1.Main result.To state our main result, we need a definition regarding counting problems whose solutions are Polynomial On Residue Classes (PORC), cf.[Hig60].
1.1.1.Let Y be a map from finite fields to finite groupoids.For instance, Y can be a scheme or a stack of finite type over Z.We write |Y (F q )| for the groupoid cardinality of Y (F q ); i.e., Definition 1.We say Y is PORC count if there exists an integer d, called the modulus, and polynomials For instance, Spec Z[x]/(x 2 + 1) is PORC count with modulus 4 and counting polynomials ||X|| 0 = ||X|| 2 = 1, ||X|| 1 = 2, and ||X|| 3 = 0. 1.1.2.Now let Γ g be the fundamental group of a compact Riemann surface of genus g ≥ 1, G a connected (split) reductive group over Z, G ∨ the (Langlands) dual group, and X the character stack associated to (Γ g , G) as in (1).
1.1.5.According to Lusztig's Jordan decomposition [Lus84], there is a bijection between Irr(G(F q )) and the set of pairs ([s], ρ) consisting of conjugacy classes [s] of semisimple elements in the dual group G ∨ (F q ) and irreducible unipotent characters ρ of the centraliser G ∨ s (F q ).The parameterisation and degrees of unipotent representations were also determined by Lusztig [Lus84].Hence, it remains to understand centralisers of semisimple elements of G ∨ (F q ).1.1.6.The final ingredient in the proof is results of Carter and Deriziotis on centralisers of semisimple elements and genus numbers [Car78,Der85].The notion of "genus" is a reductive generalisation of Green's notion of "type".The latter is ubiquitous in point counts on character varieties in type A [HRV08, HLRV11, Let15, Mer15, LRV20].The term "genus number" refers to the number of conjugacy classes of semisimple elements whose centraliser is in the same conjugacy class.The fact [Der85] that genus numbers of reductive groups (with connected centre) are PORC is the reason that character stacks are PORC count. 1.1.7.Remarks.
(i) It is well-known that for every χ ∈ Irr(G(F q )), the quotient |G(F q )|/χ(1) is a polynomial in q, cf.[GM20, Rem.2.3.27].Theorem 2 is, however, not trivial because the sum in (3) is over a set which depends on q. (ii) Consider the representation ζ-function of G(F q ) defined by Then Frobenius' theorem (3) can be reformulated as Our approach gives an explicit expression for ζ G(Fq) (s) for any s; see 5.1.1.(iii) Note that d(GL n ) = 1 thus the GL n -character stack is polynomial count (see below).
1.2.Consequences.We now discuss some of the corollaries of our main theorem.Recall the following: Definition 3.An algebraic stack Y of finite type over F q is called polynomial count2 if there exists a polynomial ||Y || such that with G connected, is polynomial count, then the E-series of Y is a well-defined polynomial and it equals ||Y ||.In particular, one finds that the dimension, the number of irreducible components of maximal dimension, and the Euler characteristic3 of Y equal, respectively, the degree, the leading coefficient, and the value at 1 of the polynomial ||Y ||.
1.2.1.Now let X Fq := X ⊗ Z F q .As an immediate corollary of our main theorem, we obtain: Corollary 4. Suppose q ≡ 1 mod d(G ∨ ).Then X Fq is polynomial count with counting polynomial ||X|| 1 .4Thus, the E-series of X equals ||X|| 1 .1.2.2.Let rank(G) denote the reductive rank of G. Analysing the leading term of the polynomial ||X|| 1 , we find: irreducible components of maximal dimension.As observed in [LS05, Corollary 1.11], the result holds without any assumption on q because the Lang-Weil estimate implies that only the asymptotics of |X(F q )| matters.Over the complex numbers, the above numerical invariants have also been understood from other perspectives; see the Appendix for further discussions.1.2.3.The Euler characteristic of X Fq is more subtle and has not been considered in the literature.In this direction, we have: and 72 2g−2 + 8 2g−2 + 2 × 9 2g−2 , respectively.(iii) If g > 1 then the Euler characteristic of the component of the PGL n -character stack associated to 1 is equal to ϕ(n)n 2g−3 , where ϕ is the Euler totient function.Note that when g = 1, |X(F q )| equals the number of conjugacy classes of G(F q ).Thus, Part (i) can be verified by consulting tables of conjugacy classes of G(F q ).Part (ii) can also be verified by consulting character tables of G(F q ) for groups of small rank (cf.[Lue]) but we found it instructive to prove this using our approach; see §6.Part (iii) should be compared with [HRV08, Corollary 1.1.1]which states that the Euler characteristic of a component of PGL n -character stack labelled by a primitive root of unity is µ(n)n 2g−3 .1.3.Further directions.We expect the main theorem to hold for general reductive groups.The main difficulty with reductive groups with disconnected centre is that Lusztig's Jordan decomposition and genus numbers are more complicated because centralisers of semisimple elements in G ∨ may be disconnected.
We also expect the theorem to hold for fundamental groups of non-orientable surfaces.For G = GL n , this is proved in [LRV20].The main issue for general types is that the relationship between Frobenius-Schur indicators and the Lusztig-Jordan decomposition is not well-understood, cf.[TV20] for some results in this direction.
Finally, we expect the theorem to hold for fundamental groups of punctured Riemann surfaces.In this case, a careful choice of conjugacy classes at the punctures (generalising the notion of generic from [HLRV11]) will ensure that the resulting character stack and character variety are the same.This is the subject of work in progress [KNP].
1.4.Structure of the text.In §2, we review standard concepts regarding root datum and reductive groups over finite fields.In §3, we recall some results of Carter and Deriziotis on centralisers of semsimple elements and genus numbers.In §4, we review Lusztig's Jordan decomposition of irreducible characters and classification of unipotent representations.Theorem 2 and Corollary 5 are proved in §5.In §6, we provide explicit formulas for the counting polynomials of character stacks associated to simple groups of semisimple rank 2. In §7, we use Green's classification of irreducible characters of GL n (F q ) to count points on the character stack associated to GL n .Finally, in the appendix, we discuss the implications of our results for character stacks over C. 1.5.Acknowledgements.The second author would like to thank David Baraglia for introducing him to the world of character varieties and answering numerous questions and for Jack Hall for answering questions about stacks.We thank George Lusztig whose crucial comment set us on the right path.We also thank the participants of the Australian National University workshop "Character varieties, E-polynomials and Representation ζ-functions", where our results were presented.
NB is supported by an Australian Government RTP Scholarship.MK is supported by Australian Research Council Discovery Projects.The contents of this paper form a part of NB's Master's Thesis.

Reductive groups over finite fields
In this section, we recall some basic notation and facts about structure of reductive groups over finite fields, cf.[Car85, DM20, GM20].But first, we define the notion of the modulus of a root datum used in our main theorem.
Definition 7. We define the modulus of Ψ, denoted by d(Ψ), to be the lcm of the sizes of torsion parts of the abelian groups X/ Φ 1 , where Φ 1 ranges over closed subsystems of Φ and Φ 1 denotes the subgroup of X generated by Φ 1 .
2.1.1.Let G be a connected split reductive group with root datum Ψ.Then we define the modulus of G by d(G) := d(Ψ).Note that the root datum (X, X ∨ , Φ 1 , Φ ∨ 1 ) defines a connected reductive subgroup G 1 ⊆ G of maximal rank.The size of the torsion part of X/ Φ 1 equals the number of connected components of the centre Z(G 1 ).Thus, ( 4) where G 1 ranges over connected reductive subgroups of G of maximal rank.In particular, we see that d(GL n ) = 1.
2.1.2.Let G be a simple simply connected group.Then one can show that d(G) equals the lcm of coefficients of the highest root and the order of Z(G), cf.[Der85].Thus, we have: is the product of bad primes (resp.twice the product of bad primes) of G.We refer the reader to [SS70] for the definition of bad primes.
2.2.Reductive groups over finite fields.Let p be a prime, k an algebraic closure of F p , and F q the subfield of k with q elements.We use bold letters such as X for schemes, stacks, etc. over F q and script letters such as X for their base change to k.The (geometric) Frobenius F = F X : X → X is the map F 0 ⊗ id, where F 0 is the endomorphism of X defined by raising the functions on X to the q th power.
2.2.1.Let G be a connected reductive group over F q with a maximal quasi-split torus T.
Let Ψ = (X, X ∨ , Φ, Φ ∨ ) denote the root datum of (G , T ).We now explain how to encode the rational structure of G via the root datum.The Frobenius F : T → T induces a homomorphism on characters which we denote by the same letter.We then have an automorphism ϕ ∈ Aut(X) of finite order such that In other words, F acts on X as the automorphism qϕ.The rational structure of G is encoded in the automorphism F = qϕ.In particular, G is split if and only if ϕ is trivial.
The advantage of the complete root datum and root system is that q does not appear in their definition.Given a complete root datum Ψ, for every prime power q, we have a unique, up to isomorphism, connected reductive group G over F q whose complete root datum is Ψ.We call G the realisation of Ψ over F q .2.3.1.Dual group.Let G ∨ be the group over F q dual to G. By definition, this is the connected reductive group over F q whose complete root datum is given by (X ∨ , X, Φ ∨ , Φ, ϕ ∨ W ), where ϕ ∨ is the transpose of ϕ.

2.3.2.
Frobenius action on W .The action of the Frobenius on G stabilises T and N G (T ).Thus, F acts on W = N G (T )/T .We denote the resulting automorphism of W by σ.We call W := (W, σ) the complete Weyl group.Elements w 1 and w 2 in W are said to be σconjugate if there exists w ∈ W such that ww 1 σ(w) −1 = w 2 .If G is split, σ-conjugacy is just the usual conjugacy.
2.4.Finite reductive groups.Let G be a connected reductive group over F q .The finite group G(F q ) = G F is called a finite reductive group.Note that this definition excludes Suzuki and Ree groups.

Order polynomial. Let
(5) Then |G(F q )| = ||G||(q) [GM20, Remark 1.6.15].Observe that this equality may not hold if we replace q by q n .In other words, G may be not polynomial count.It is, however, polynomial count if we assume that G is split, in which case, the counting polynomial simplifies to ( 6) w∈W t l(w) .

Genus numbers are PORC
The aim of this section is to state a theorem of Deriziotis [Der85] which tells us that genus numbers for finite reductive groups are PORC.We start by recalling the definition of the genus of a semisimple element due to Carter [Car78].
3.1.Genus map.Let G be a connected reductive group over F q and G(F q ) ss the set of semisimple elements of G(F q ).For each x ∈ G(F q ) ss , let G x denote its centraliser in G.It is well-known that G x is a (possibly disconnected) maximal rank reductive subgroup of G. Thus, the root system of G • x is a closed subsystem Φ 1 ⊆ Φ.We now explain how to encode the rational structure of G • x in root theoretic terms.
3.1.1.Let W 1 ⊆ W be the Weyl group of Φ 1 and N W (W 1 ) the normaliser of W 1 in W .Let (W, σ) be the complete Weyl group of G.Note that the action of σ on W stabilises W 1 .Thus, σ acts on N W (W 1 )/W 1 and we have the notion of σ-conjugacy for this group.By a theorem of Carter [Car78, §2], the rational structure of G 1 is encoded in a σ-conjugacy class of N W (W 1 )/W 1 .
We refer to ξ as a genus and call Ξ( Φ) the set of genera of Φ.If G is split, then the complete root datum is just the same as the root datum so we denote this set by Ξ(Φ).
3.1.2.Let G [ss] (F q ) = G ss (F q )/G(F q ) denote the set of semisimple conjugacy classes of G(F q ).The above discussion implies that we have a canonical map, called the genus map, , which sends a semisimple conjugacy class to its genus.The number of points of fibres of this map are known as genus numbers.
3.2.Genus numbers.Let Ψ = (X, X ∨ , Φ, Φ ∨ , ϕW ) be a complete root datum.For each genus ξ ∈ Ξ( Φ), we define a map G [ss] ξ from finite fields to sets as follows: Given a finite field F q , let G be the realisation of Ψ over F q and set Let d(Ψ) denote the modulus as in Definition 7.
3.2.1.The freeness assumption implies that every realisation G of Ψ has simply connected derived subgroup.A theorem of Steinberg then implies that centralisers of semisimple elements of G are connected.As shown in [Der85], we have ).The reductive subgroup of maximal rank associated to ξ is the diagonal torus.Thus, G [ss] ξ (F q ) is the set of regular diagonal elements in G(F q ), up to permutation.Hence,

Note that ||G
[ss] ξ ||(q) = 0 for all q n + 1.This is just the re-statement of the fact that if q n + 1, there are no regular diagonal elements.

The polynomials ||G [ss]
ξ || i have been determined explicitly for all genera ξ of groups of exceptional type or type A, and for many genera of groups of type B, C, or D. However, as far as we understand, this problem has not been fully solved; see [Fle97] for further details.

Representations of finite reductive groups
Let G be a connected reductive group over F q .In this section, we recall deep results of Lusztig on the structure of complex representations of G(F q ).4.1.Unipotent representations.Let Irr u (G(F q )) denote the set of (irreducible) complex unipotent characters.

Theorem 11 ([Lus84,Lus93]). There exists a finite set U( W ), depending only on W , together with a function
such that the following holds: We have a bijection such that Deg(ρ)(q) is the degree of the unipotent character of G(F q ) associated to ρ.

Remarks.
(i) The pair (U( W ), Deg) has been determined explicitly by Lusztig in all types, cf. the appendix of [Lus84].(ii) If (W, σ) = (S n , id), then U(W ) equals P n the set of partitions of n.Moreover, if λ = (λ 1 , . . ., λ m ) is a partition of n with λ 1 λ 2 • • • λ m and ρ λ (q) is the corresponding unipotent representation of G(F q ), then , x (F q )] p ′ , where the subscript p ′ (resp.p) denotes the prime to p part (resp.the p part).

Counting points.
In this subsection, we prove Theorem 2. Recall that G is a connected split reductive group over Z.
(6) Recall the definition d = d(G ∨ ) from Theorem 10.Assume that q ≡ i mod d, where i ∈ {0, 1, • • • , d(G ∨ )}.Then Theorem 10 gives Note that each summand is polynomial with rational coefficients.The sum is over objects which depend only on the complete root datum Ψ i.e., they are independent of q.It follows that ||X|| ∈ Q[t].The above discussion then shows that X is PORC count with counting polynomials ||X|| i .Thus, Theorem 2 is proved.

Aside on representation
Then we have equality of complex functions 5.2.Counting polynomials in the case g = 1.In this subsection, we show that if g = 1, then ||X|| i has degree rank(G) and leading coefficient 1.This establishes Corollary 5(i).

By the above discussion
, In view of equation 7, the degree of a summand is maximal if and only if Φ 1 is empty; i.e., when ξ is the genus of a regular semisimple element.Thus, the degree of |X(F q )| equals rank(G).

Next, one can easily check that for genera (∅, [w]), the leading coefficient of ||G ∨,[ss] ξ
|| i equals 1/|W w |, where W w denotes the centraliser of w in W . Thus, the leading coefficient of ||X|| i is 1/|W w |, where the sum runs over conjugacy classes of W .By the orbit stabiliser theorem, this sum equals 1.This establishes Corollary 5(i).

5.2.3.
Here is an alternative (perhaps more intuitive) argument for this corollary.By (3), |X(F q )| is the counting polynomial for the number of conjugacy classes of G(F q ).Now the leading term of the class number equals the leading term of the polynomial counting semisimple elements.By a theorem of Steinberg, the latter equals |Z(G(F q ))|q rank([G,G]) .Thus, the leading coefficient is 1 and the degree is dim(Z(G)) + rank([G, G]) = rank(G).

5.3.
Counting polynomials in the case g > 1.In this subsection, we show that if g > 1, then the polynomial ||X|| i has degree (2g − 2) dim G + dim Z(G ∨ ) and leading coefficient |π 0 (Z(G ∨ ))|.This establishes Corollary 5(ii).5.3.1.We claim that only ξ = ([Φ ∨ ], 1) contributes to the leading term of ||X|| i .Note that a semisimple element has genus ([Φ ∨ ], 1) if and only if it is central.Thus, the claim implies that the leading term of ||X|| i (q) is the same as the leading term of |Z(G ∨ (F q ))||G ∨ (F q )| 2g−2 , which would establish the desired result.5 5.3.2.For ease of notation, set n be a genus.Thus, Φ 1 denotes a closed subsystem of the dual root system Φ ∨ .Let 5.3.4.It is clear that the above quantity is 0 if Φ 1 = Φ ∨ i.e. if ξ is central.If ξ is not central; i.e.Φ 1 is strictly smaller than Φ ∨ , then the above quantity is positive because This follows from the fact that elements of (Φ ∨ ) + −Φ + 1 span the vector space ( Φ ∨ / Φ 1 )⊗R and that n ≥ 2. This concludes the proof.
6. Examples: PGL 2 , PGL 3 , SO 5 , and G 2 In this section, we give tables containing the genera ξ ∈ Ξ(Φ ∨ ), the integer r(ξ), the size of centraliser |G ∨ ξ (F q )|, genus number |G ∨,[ss] ξ (F q )|, and the unipotent degrees of the centraliser, for simple adjoint groups G of rank 2. We assume throughout that q ≡ 1 mod d(G ∨ ).By the discussion of §5.1, the counting polynomial ||X|| 1 of the associated character stacks can be determined using these tables.
6.1.The case G = PGL 2 .In this case, G ∨ = SL 2 .In this case d(G ∨ ) = 2.So let us assume q is odd.Then the genera for PGL 2 is given in Table 1.
7. The character stack of GL n and PGL n revisited Let G = GL n and g a positive integer.Let X be the character stack of associated to (Γ g , G).In this section, we give an explicit expression for the number of points of X using the same method employed in [HRV08].The main point is that we have a good direct understanding of character degrees of G(F q ) without having to resort to Lusztig's Jordan decomposition.It would be interesting to prove directly that the polynomial obtained in this section (see ( 13)) equals the one from Definition 14. 7.1.Conjugacy classes of GL n (F q ).Let I = I(q) denote the set of irreducible polynomials over F q , except that we exclude f (t) = t.Let P denote the set of partitions.Let P n (I) denote the set of maps Λ : I → P such that Then we have a bijection between P n (I) and conjugacy classes of G(F q ).Let  Remark 15.One can show that Λ and Λ ′ have the same type if and only if the centraliser of the corresponding conjugacy classes in G(F q ) have the same genus.Thus, we have a bijection between semisimple types and genera of G.
7.1.2.The weight of a type τ is defined by Thus, the weight of an element Λ ∈ P(I) equals the weight of its type.
7.1.3.Genus number.Let A τ (q) denote the number of Λ ∈ P(I(q)) of type τ .(Equivalently, A τ (q) is the genus number of τ .)Our aim is to give an explicit formula for A τ (q).Let By convention, if T (d) = 0, then the product involving T (d) is defined to be one.We leave the above lemma as an exercise.As a corollary, we conclude that A τ (q) is a polynomial in q with rational coefficients.7.2.Irreducible characters of GL n (F q ).We have seen that irreducible characters of G(F q ) are in bijection with P n (I).Let χ Λ denote the irreducible character of G corresponding to Λ ∈ P n (I).7.2.1.Define the normalised hook polynomial associated to the partition λ ∈ P by (10) H λ (q) := q − 1 2 λ,λ (1 − q h ).
Here the product is taken over the boxes in the Young diagram of λ and h is the hook length of the box.Moreover, λ, λ := i (λ ′ i ) 2 where the sum is taken over the parts in the conjugate partition λ ′ .7.2.2.Next, define the normalised hook polynomial of Λ ∈ P n (I) by (11) H Λ (q) := (−1) n q 1 2 n 2 f ∈I H Λ(λ) (q deg f ) .
It is easy to see that H Λ is a monic polynomial in Z[q].7.2.3.Let Λ ′ be the map conjugate to Λ; i.e., Λ ′ (f ) is the partition conjugate to Λ(f ) for all f ∈ I. Then by a theorem of Green, we have (12) |G(F q )| χ Λ (1) = H Λ ′ (q).7.2.4.It is clear that the hook polynomial H Λ (q) depends only on the type of Λ; in fact, we have H Λ (q) = (−1) n q 1 2 n 2 d,λ Given a type τ , we write H τ (q) for the hook polynomial associated to τ .

T
(d) := λ∈P m d,λ .Let I d = I d (q) = |I d | denote the number of irreducible polynomials of degree d over q.By a result attributed to Gauss, we have )q d/k otherwise.Lemma 16.A τ (q) = d≥1 T (d)−1 i=0 (I d (q) − i) λ∈P m d,λ ! .