Geometric Invariant Theory of Peterson Varieties

Abstract

We study the GIT quotients of the Peterson variety $\mathrm{Pet}_n\subset \mathrm{GL}(n,\mathbb C)/B$ under a one-parameter subgroup $\lambda:\mathbb G_m \to T$ with respect to the linearization $\mathcal L(\chi)$ given by a regular dominant character $\chi$ in the root lattice. Using the Richardson stratification, we describe the semistable and stable loci explicitly in terms of subsets of simple roots. This determines the GIT chamber decomposition and the corresponding wall-crossing morphisms. In the deep chamber, the quotient is shown to be isomorphic to the weighted projective space $\mathbb P(1,2,\ldots,n-1)$. We obtain a complete chamber-theoretic characterization of normality and describe how normality varies with the choice of linearization. We also prove that the quotient is smooth if and only if $n\le3$, independently of the regular dominant linearization. These results describe how the singular geometry of the Peterson variety is reflected in the variation of its GIT quotients.