Deligne's character-valued polynomiality conjecture for parabolic-bundle counts

Let $b_1>\cdots>b_f$, let $\mathbf{m}=(m_1,\dots,m_f)\in\mathbb{Z}_{\geq 0}^f$, and set

$$\mathcal{E}^{{\bf b};\mathbf{m}}=\bigoplus_{i=1}^f\mathcal{O}(b_i)^{m_i}.$$

Let $\boldsymbol{\mu}$ be an $l$-tuple of partitions of $n$, let $S_{\boldsymbol{\mu}}$ be its stabilizer in $\mathfrak{S}_l$, and let $\mathcal{P}_{\boldsymbol{\mu}}$ be the product of partition sets determined by the multiplicities of its distinct coordinates. For $\boldsymbol{\lambda}\in\mathcal{P}_{\boldsymbol{\mu}}$, let $D$ be a reduced divisor compatible with $\boldsymbol{\lambda}$, and let $\mathcal{A}^{{\mathcal{E}^{{\bf b};\mathbf{m}}}}_{\boldsymbol{\mu},\boldsymbol{\lambda},D}(q)$ denote the number of isomorphism classes of geometrically indecomposable parabolic structures of the prescribed types.

Deligne's conjecture. There exists a polynomial $A^{{\bf b};\mathbf{m}}_{\boldsymbol{\mu}}(t)\in\mathcal{C}h(S_{\boldsymbol{\mu}})[t]$, with coefficients in the positive cone $\mathcal{C}h(S_{\boldsymbol{\mu}})^+$, such that for every finite field $k=\mathbb{F}_q$, every $\boldsymbol{\lambda}\in\mathcal{P}_{\boldsymbol{\mu}}$, and every compatible reduced divisor $D$, one has

$$\mathcal{A}^{{\mathcal{E}^{{\bf b};\mathbf{m}}}}_{\boldsymbol{\mu},\boldsymbol{\lambda},D}(q)=A^{{\bf b};\mathbf{m}}_{\boldsymbol{\mu},w}(q),$$

for any $w\in S_{\boldsymbol{\mu}}$ of cycle type $\boldsymbol{\lambda}$.

The conjecture strengthens ordinary polynomiality by packaging the counts as evaluations of a character-valued polynomial. It is attributed in the source to Deligne and is proved there when $n=2$.