Descending G-equivariant Line Bundles to GIT Quotients

Abstract

In part one, we consider descent of line bundles to GIT quotients of products of flag varieties. Let $G$ be a simple, connected, algebraic group over $mathbb{C}$, $B$ a Borel subgroup, and $T subset B$ a maximal torus. Consider the diagonal action of $G$ on the projective variety $(G/B)^3 = G/B times G/B times G/B$. For any triple $(chi_1, chi_2, chi_3)$ of regular characters there is a $G$-equivariant line bundle $mathcal{L}$ on $(G/B)^3$. Then, $mathcal{L}$ is said to descend to the GIT quotient $pi:[(G/B)^3(mathcal{L})]^{ss} rightarrow (G/B)^3(mathcal{L})//G$ if there exists a line bundle $hat{mathcal{L}}$ on $(G/B)^3(mathcal{L})//G$ such that $mathcal{L}mid_{[(G/B)^3(mathcal{L})]^{ss}} cong pi^*hat{mathcal{L}}$. Let $Q$ be the root lattice, $Lambda$ the weight lattice, and $d$ the least common multiple of the coefficients of the highest root $theta$ of $mathfrak{g}$, the Lie algebra of $G$, written in terms of simple roots. We show that $mathcal{L}$ descends if $chi_1, chi_2, chi_3 in d Lambda$ and $chi_1 + chi_2 + chi_3 in Gamma$, where $Gamma$ is the intersection over root lattices $Q_mathfrak{s}$ of all semisimple Lie subalgebras $mathfrak{s} subset mathfrak{g}$ of maximal rank. Moreover, we show that $mathcal{L}$ never descends if $chi_1 + chi_2 + chi_3 notin Q$. In part two, we discuss joint work with Shrawan Kumar. Let $mathfrak{g}$ be any simple Lie algebra over $mathbb{C}$. Recall that there exists a principal TDS embedding of $mathfrak{sl}_2$ into $mathfrak{g}$ passing through a principal nilpotent element of $mathfrak{g}$. Moreover, $wedge (mathfrak{g}^*)^mathfrak{g}$ is generated by primitive elements $omega_1, dots, omega_ell$, where $ell$ is the rank of $mathfrak{g}$. N. Hitchin conjectured that for any primitive element $omega in wedge^d (mathfrak{g}^*)^mathfrak{g}$, there exists an irreducible $mathfrak{sl}_2$-submodule $V_omega subset mathfrak{g}$ of dimension $d$ such that $omega$ is non-zero on the line $wedge^d (V_omega)$. We prove that the validity of this conjecture for simple simply-laced Lie algebras implies its validity for any simple Lie algebra. Let $G$ be a connected, simply-connected, simple, simply-laced algebraic group and let $sigma$ be a diagram automorphism of $G$ with fixed subgroup $K$. Then, we show that the restriction map $R(G) to R(K)$ is surjective, where $R$ denotes the representation ring over $mathbb{Z}$. As a corollary, we show that the restriction map in the singular cohomology $H^*(G)to H^*(K)$ is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case relies on this cohomological surjectivity.Doctor of Philosoph