Newton-Okounkov polyhedra for character varieties and configuration spaces (Q2790720)

The author constructs families of Newton-Okounkov bodies for the free group character varieties \( \mathcal{X}(F_g, G) \) and configuration spaces \( P_{\vec{\lambda}}(G) \) of any connected, complex reductive group \( G \) with maximal torus \( T \). Here, \( F_g \) is a free group and \( \vec{\lambda} = \lambda_1, \ldots, \lambda_n \) is a tuple of dominant weights in \( G \). A key step is the construction of Newton-Okounkov bodies for the affine ``master'' configuration spaces \( P_n(G) \) (introduced in Example 4.6), from which \( P_{\vec{\lambda}}(G) \) can be obtained as a right \( T^n \) GIT quotient. As an application, he shows that \( \mathcal{X}(F_g, G) \) and \( P_n(G) \) are Cohen-Macaulay and that \( P_{\vec{\lambda}}(G) \) is arithmetically Cohen-Macaulay.NEWLINENEWLINEA Newton-Okounkov body is a convex set \( C_v \) which is given as the convex hull of the image \( v(A) \subset \mathbb{R}^M \) of a maximal rank valuation \( v : A \to \mathbb{Z}^M \subset \mathbb{R}^M \), where \( A \) is a commutative domain of finite Krull dimension. If \( v(A) \) is finitely generated, there is a flat degeneration of \( \mathrm{Spec}(A) \) to the toric variety attached to \( C_v \).NEWLINENEWLINEThe method for obtaining the valuations is by building ``strong'' increasing filtrations for the coordinate rings. The condition of being strong provides that the data of such a filtration is equivalent to a valuation as above (Proposition 3.2). The author's construction is divided into two steps. First, he makes use of the ordering on dominant weights to build filtrations which are not yet fine enough, but which reduces the problem to the case of \( P_3(G) \) (Propositions 4.9 and 4.11). In section 5, these filtrations are enhanced using \textit{G. Lusztig}'s construction of the dual canonical basis [J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008)] and a filtration on the string parameters of the latter.NEWLINENEWLINEFurther, the existence of a valuation with the desired properties is proven for \( P_3(G) \) (Theorem 5.9).NEWLINENEWLINEAfter giving the proof for the main theorems (section 6), the author discusses examples in the final section.