Counting absolutely cuspidals for quivers

Abstract

For an arbitrary quiver $Q=(I,\Omega )$ and dimension vector $\mathbf {d} \in \mathbb {N}^I$ we define the dimension of absolutely cuspidal functions on the moduli stacks of representations of dimension $\mathbf {d}$ of a quiver Q over a finite field $\mathbb {F}_q$, and prove that it is a polynomial in q, which we conjecture to be positive and integral. We obtain a closed formula for these dimensions of spaces of cuspidals for totally negative quivers.

Appendix A. A trivial variant of Kac’s conjecture

Proposition A.1

Let $Q=(I,\Omega )$ be any quiver, and let $Q^{re}=(I^{re},\Omega ^{re})$ be the full subquiver of Q whose vertices are real. Then for any dimension vector $\mathbf {d}\in \mathbb {N}^I$ we have

$$\begin{aligned} A_{Q,\mathbf {d}}(0)=\dim (\mathfrak {g}_{Q^{re}}[\mathbf {d}]). \end{aligned}$$

Proof

It is enough to show that $A_{Q,\mathbf {d}}(0)=0$ whenever $\mathbf {d}$ is not supported on $I^{re}$ (and then we are reduced to the usual Kac conjecture proved by Hausel). Let $H_{Q,\mathbf {d}}(t) \in \mathbb {N}[t]$ be the polynomial counting all isomorphism classes of representations of dimension $\mathbf {d}$. The relation to $A_{Q,\mathbf {n}}(t)$ reads

$$\begin{aligned} \sum _{\mathbf {d}} H_{Q,\mathbf {d}}(t)z^{\mathbf {d}}=\text{ Exp }_{t,z}\left( \sum _{\mathbf {d}} A_{Q,\mathbf {d}}(t)z^{\mathbf {d}}\right) . \end{aligned}$$

(A.1)

Evaluating (A.1) at $t=0$, we see that $A_{Q,\mathbf {d}}(0)=0$ for all $\mathbf {d}\not \in \mathbb {N}^{I^{re}}$ if and only if $H_{Q,\mathbf {d}}(0)=0$ for all $\mathbf {d}\not \in \mathbb {N}^{I^{re}}$. Writing $H_{Q,\mathbf {d}}(t)=c_{Q,\mathbf {d}} + t K_{Q,\mathbf {d}}(t)$ with $c_{Q,\mathbf {d}} \in \mathbb {N}$ and $K_{Q,\mathbf {d}}(t) \in \mathbb {N}[t]$ we immediately see that $H_{Q,\mathbf {d}}(0)=0$ of and only if $q\;|\; H_{Q,\mathbf {d}}(q)$ for all prime powers q. Let us now fix a finite field $\mathbf {k}$ and $\mathbf {d}\not \in \mathbb {N}^{I^{re}}$. Let $i \in I \backslash I^{re}$ be an imaginary vertex for which $\mathbf {d}_i \ne 0$, and let $h \in \Omega$ be an edge loop at i. Define an action of the additive group $\mathbb {G}_a(\mathbf {k})=(\mathbf {k},+)$ on the set $Ind_{Q,\mathbf {d},\mathbf {k}}$ of isomorphism classes of indecomposable representations of dimension $\mathbf {d}$ by setting

$$\begin{aligned} \lambda \cdot (x_\gamma )_{\gamma \in \Omega } =(x_{\gamma } + \lambda \delta _{\gamma ,h} Id_{\mathbf {k}^{\mathbf {d}_i}})_{\gamma \in \Omega }. \end{aligned}$$

Considering the eigenvalues of $x_h$, we see that this action is free. This proves that $q\;|\; H_{Q,\mathbf {d}}(q)$ as wanted. $\square$

Remark

A more interesting variant of Kac’s conjecture in the context of quivers with edge loops is proved in [3]: the constant term of the nilpotent Kac polynomial $A^{1}_{Q,\mathbf {d}}(t)$ is equal to the multiplicity of the root $\mathbf {d}$ in the generalized Borcherds algebra $\mathfrak {g}_Q$ associated to Q in [2].