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Equivariant cellular models in Lie theory

  • Garnier, Arthur
Publication Date
Dec 10, 2021
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This work aims to construct explicit cellular structures on spaces arising in Lie theory, that are equivariant with respect to the action of a Weyl group $W$. In general, the main purpose of studying such structures on a $W$-space $X$ is to exhibit a well-defined complex in the bounded homotopy category $\mathcal{K}^b(\Z[W])$ of $\Z[W]$-modules, which is a model for the derived functor of global sections $R\Gamma(X,\underline{\Z})$ in the derived category $\mathcal{D}^b(\Z[W])$.The two classes of spaces we focus on are flag manifolds and maximal tori of compact Lie groups. More precisely, given a simple compact Lie group $K$ and a maximal torus $T<K$, we give a general explicit simplicial structure on $T$, equivariant with respect to the action of the Weyl group $W:=N_K(T)/T$ and we describe the associated $W$-dg-ring, depending on the character lattice of $T$. For non-crystallographic finite Coxeter groups, we construct compact hyperbolic manifolds which may be seen as analogues of tori, using hyperbolic extensions rather than affine extensions. In the case of dihedral groups, these are arithmetic Riemann surfaces. Concerning flag manifolds, we study three different $\Sym_3$-equivariant cellular decompositions of the real flag manifold $\mathcal{F}_3(\R)$ of $\R^3$, which is the first non-trivial example. The first one starts with the Goresky-Kottwitz-MacPherson graph of $\Sym_3$ and an algebraic embedding $\mathcal{F}_3(\R)\hookrightarrow\R\Pro^7$, the second one uses the fact that the universal cover of $\mathcal{F}_3(\R)$ is the $3$-sphere and yields a particularly nice and simple cellular homology chain complex. The third one is perhaps the most promising one, as it relies on a Dirichlet-Voronoi fundamental domain, defined using only a normal homogeneous Riemannian metric on $\mathcal{F}_3(\R)$. Therefore, this method is expected to be generalizable to other flag manifolds. We give some preliminary results in this direction.

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