Tyler Holden | Searcharxiv

arXiv2015

Non-abelian convexity of based loop groups

If $K$ is a compact, connected, simply connected Lie group, its based loop group $ΩK$ is endowed with a Hamiltonian $S^1 \times T$ action, where $T$ is a maximal torus of $K$. Atiyah and Pressley examined the image of $ΩK$ under the moment map $μ$, while Jeffrey and Mare examined the corresponding image of the real locus $ΩK^τ$ for a compatible anti-symplectic involution $τ$. Both papers generalize well known results in finite dimensions, specifically the Atiyah-Guillemin-Sternberg theorem, and Duistermaat's convexity theorem. In the spirit of Kirwan's convexity theorem, this paper aims to further generalize the two aforementioned results by demonstrating convexity of $ΩK$ and its real locus $ΩK^τ$ in the full non-abelian regime, resulting from the Hamiltonian $S^1\times K$ action. In particular, this is done by appealing to the Bruhat decomposition of the algebraic (affine) Grassmannian, and appealing to the "highest weight polytope" results for Borel-invariant varieties of Guillemin and Sjamaar and Goldberg.

math.SG↗

arXiv2014

Generalized Equivariant Cohomology and Stratifications

For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_T^*$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $E_T^*(X)$ as an $E_T^*(\text{pt})$-module when $X$ is a direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$, $K_T^*$, and $MU_T^*$. We perform this computation on the affine Grassmannian of a complex semisimple group.

math.AT↗