Motivic (Representation) Stability of Representation Varieties and Character Stacks

Let $S$ be a variety over an algebraically closed field $k$ of characteristic 0. In this paper, we will work with $G$-varieties, namely, varieties $X$ over $S$ equipped with an action of a linear algebraic group $G$ over $k$ so that 1) the map $X\to S$ is $G$-equivariant (with the trivial action of $G$ on $S$) and 2) $X$ can be covered by $G$-equivariant open affines. Morphisms of $G$-varieties are morphisms of varieties over $S$ that are $G$-equivariant.

In the case where $G$ is the trivial group, we obtain the Grothendieck ring of varieties over $S$, that we will denote by $\textup{K}_{0}(\textup{\bf{}Var}_{S})$. For a $G$-variety $X$ over $k$, the class $[X]\in\textup{K}_{0}(\textup{\bf{}Var}^{G}_{k})$ is called the motivic (or virtual) class of $X$. We denote the virtual class of $\mathbb{A}^{1}$ (with the trivial $G$-action) by $q$.

Let $G$ be a finite group and $H$ a subgroup of $G$. On the level of representations, we have induction and restriction functors. Corresponding to these functors, we have maps on the corresponding Grothendieck rings.

It is easy to see that these functors descend to the Grothendieck ring of varieties providing maps of Abelian groups

A little bit more is true. The restriction map $\operatorname{Res}_{H}^{G}:\textup{K}_{0}(\textup{\bf{}Var}_{S}^{G})\to\textup{K}_{0}(\textup{\bf{}Var}_{S}^{H})$ is a ring homomorphism and it can be defined for any morphism of linear algebraic groups $H\to G$. However, for general linear algebraic groups, more care is needed in the case of the induction functor. In fact, we may face three kinds of problems: 1) the GIT quotient appearing in the induction functor may not exist, 2) the GIT quotient may not be a variety, and 3) the GIT quotient may not be motivic. However, in the case when $H$ is a closed subgroup of a linear algebraic group $G$, all these issues are resolved. Indeed, in this case, the quotient $(G\times X)/H$ exists [19], and since the action of $H$ is free on $G\times X$, the GIT quotient is motivic (see, for instance, Theorem 4.2.11 of [9]). As a result of the discussion above, we have the following.

The goal of this section is to provide a framework in studying representation stability via virtual classes. Since, the proofs of the representation stability results [20] for representation varieties, character varieties and character stacks do not depend on the actual cohomology theory, it is expected that representation stability holds on the motivic level.

The difficulty in approaching this problem using the Grothendieck ring of varieties is twofold. First, it is not clear how to decompose a variety with an $S_{n}$-action into subvarieties corresponding to representations of $S_{n}$. Second, in order to do any kind of computations, one needs that this decomposition satisfies certain properties, for instance, Künneth-formuala, etc.

These difficulties cannot be solved. To avoid the first problem, we have to make a choice on how to decompose a variety according to the representations of $S_{n}$. The choice is basically an appropriate set of subgroups in $S_{n}$. To deal with the second problem, we will show that, even though not every $S_{n}$-varieties can be used in Künneth-type formulas, there are still sufficient $S_{n}$-varieties that can be used for computational purposes.

We begin by addressing the first problem. The representations of $S_{n}$ are parametrized by Young tableaux: for a partition $\lambda$ of $n$, we denote the corresponding representation by $V_{\lambda}$. For a general partition $\lambda=(\lambda_{1},\lambda_{2},...)$ (with $\lambda_{1}\geq\lambda_{2}\geq...$) and for an integer $n\geq|\lambda|+\lambda_{1}:=2\lambda_{1}+\lambda_{2}+...$, we denote the partition of $n$ given by $(n-|\lambda|,\lambda_{1},\lambda_{2},...)$ by $\lambda[n]$. In representation stability, one is interested in the stability of the representations of $V_{\lambda[n]}$.

Corresponding to a partition $\lambda=(\lambda_{1},\lambda_{2},...)$ of $n$, we have a subgroup

Thus, for the sequence of representations $V_{\lambda[n]}$, we obtain a sequence of subgroups $S_{\lambda[n]}\leq S_{n}$. After setting up the notation, we arrive at the definition of motivic representation stability.

In many of our cases, $G$ will be the trivial group. The following observation is key to understanding motivic representation stability.

In the rest of the section, we motivate why we believe that Definition 2.10 is a right definition for motivic representation stability in the context of the Grothendieck ring of varieties.

Let $G$ be a finite group and consider, $R_{\mathbb{Q}}(G)$, the representation ring of $G$ over $\mathbb{Q}$ that is the Grothendieck group of finite dimensional $\mathbb{Q}$-representations of $G$ where the ring structure is induced by the tensor product of representations. In this section, we wish to decompose a $G$-variety over a variety $S$ with respect to the rational representations of $G$, in other words, we wish to construct a ring map

so that for a variety $X$ over $S$ the part that corresponds to the trivial representation is $[X/G]$ (indeed, the tensor product on the right-hand side has a natural structure of a ring, because both $\textup{K}_{0}(\textup{\bf{}Var}_{S})$ and $R_{\mathbb{Q}}(G)$ are rings). We denote the desired motivic decomposition by $[X]_{G}\in\textup{K}_{0}(\textup{\bf{}Var}_{S})\otimes_{\mathbb{Z}}R_{\mathbb{Q}}(G)$ for a $G$-variety $X$.

This goal will not be achieved in general. The main difficulties arise from a) the lack of a motivic Künneth-formula, see Example 3.6.17. of [26], and b) over-constraints coming from subgroups of $G$, see Example 3.6.9 of [26].

The representation ring, $R_{\mathbb{Q}}(G)$, is equipped with the standard inner product $\langle,\rangle$ that is given on two irreducible representations, $V_{1}$, $V_{2}$, by $\langle V_{1},V_{2}\rangle=0$ if the irreducible representations are not isomorphic and $\langle V_{1},V_{2}\rangle=1$, if the representations are isomorphic. This inner product can be extended to an inner product

by $\langle[X_{1}]\otimes V_{1},[X_{2}]\otimes V_{2}\rangle=[X_{1}][X_{2}]\otimes\langle V_{1},V_{2}\rangle.$

Our key observation is that good sets of conjugacy classes of subgroups provide motivic decompositions.

In many cases, natural choices exist for good sets of conjugacy classes of subgroups.

The example above motivates Definition 2.10, in fact, the following holds.

We expect that as the genus gets larger and larger the product of commutators $\prod_{i=1}^{g}[A_{i},B_{i}]$ spreads evenly across the commutator subgroup $[G,G]$. In fact, this happens in the case of finite groups $G$. In this case, using a classical result of Frobenius [6], $\operatorname{Rep}_{G}(M_{g})$ is a finite set of cardinality

Thus, in the case of finite groups, the limit 1 exists, namely

The above computation suggests that such limit should exist motivically in the case of connected linear algebraic groups over $\mathbb{C}$.

In this section, we show that the limit 1 exists in the context of the motivic measure given by the E-polynomial (see Remark 2.9) for reductive groups $G$ with connected center.

The $G$-representation varieties of surfaces, $\operatorname{Rep}_{G}(M_{g})$, are PORC count (polynomial on residue classes) [2] meaning that $\#\operatorname{Rep}_{G(\mathbb{F}_{q})}(M_{g})$ is a polynomial, $q_{G,g}(t)\in\mathbb{Q}[t]$, in terms of $q$ supposing that $q\equiv 1$ module $d(G^{\vee})$, the modulus of the Langlands dual group. Using a result of Katz, this implies that the E-polynomial of the $\operatorname{Rep}_{G}(M_{g})$ is given as $q_{G,g}(uv)$.

Furthemore, the summand of the right-hand side of Equation 2 corresponding to 1-dimensional representations

is also PORC count, showing the sum of the rest of the summands together is also PORC count. Using Remark 5.2 of [2], the rest of the summands of the form $\left(\frac{G}{\chi(1)}\right)$ are of smaller dimensions than $G$, so their contribution is negligible in the limiting E-polynomial (since we raise these pieces to the $2g$-th power).

To summarize the above, we have the following.

We conjecture, based on the finite field heuristics and Corollary 3.1, that the limit 1 exists.

We verify the conjecture in two main cases: 1. in the case of $G=\textup{SL}_{2}(k)$ and in the case of upper triangular matrix groups $G$ of rank up to 5. In fact, the virtual classes of $G$-representation varieties of surface groups are notoriously difficult to compute, namely, only the cases above have been solved successfully [9, 13, 25]. In all such cases, the computations were performed using Topological Quantum Field Theories.

The assumption that the linear algebraic group is connected is crucial in Conjecture 3.2. In fact, in the case of non-connected groups, Lang’s theorem [17] fails, and the finite field heuristics fail. To give an example, we consider $G=\mathbb{G}_{m}\rtimes\mathbb{Z}/2\mathbb{Z}$, where the action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{G}_{m}$ is given by $x\mapsto x^{-1}$. In [11], the authors described the virtual class of the corresponding representation variety as

providing

in $\textup{K}(\textup{\bf{}Stck}/k)$.

The problem is even more severe if one considers the corresponding character stack. In fact, in [11], the class of the character stack is computed as

Using this computation, it is clear that the limit in Conjecture 3.2 does not exist.

In this section, we study the motivic representation stability of the representation varieties $G^{n}:=\textup{Hom}_{Gp}(F^{n},G)$ corresponding to the free groups $F_{n}$ on $n$ letters, and of the corresponding character stacks $[G^{n}/G]$ where the group $G$ acts by conjugation on $G^{n}$. The symmetric groups, $S_{n}$, act by permuting the coordinates of $G^{n}$, and the conjugation action by the group $G$ commutes with the action of $S_{n}$.

First, we show that the representation varieties are motivically representation stable which follows from the following easy lemma.