Geometry and Topology without figures

This serves as a starting point for my self-study of geometric Langlands. Currently, I am interested in geometric Satake equivalence, which establishes an equivalence between the category of representation of the Langland dual group (of a given reductive group) and the category of perverse sheaves on affine Grassmannians. Affine grassmannians are interesting objects, and appear naturally when studying stack of principal bundles. Let us now study them, and the story, of reductive groups, is often with $\mathrm{GL}_n$.  Affine Grassmannians of general linear groups via lattices Let $R$ be a commutative ring and denoteby $R[[t]]$ and $R((t))$ the formal power series and Laurent series, respectively.  Definition . A lattice $L \subset R((t))^n$ is a finite locally free $R[[t]]$-submodule of $R((t))^n$ such that the canonical morphism $L \otimes_{R[[t]]} R((t)) \longrightarrow R((t))^n$ is an isomorphism. Note that locally free means that a vector bundle at the level of schemes...

Last week, I participated in a summer school on arithmetic geometry and Langlands program. Although this is not my field of interests, I always give some respect those who work in those fields. I also did gave a motivational TA talk about couting points on elliptic curves. To be honest, I do not expect to learn much from this school but I got enough inspiration to read arithmetic stuff, which I am going to write. Since I am an algebraic geometer, I'd like to present things in a geometric way. Let $p$ be a prime number and denote $\mathbb{Q}_p$ the $p$-adic local field. Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $G_K = \mathrm{Gal}(\overline{K}/K)$ the absolute Galois group. A $\ell$- adic representation of $G_K$ is a continuous morphism (of groups) $$G_K \longrightarrow \mathrm{GL}_n(V),$$ where $V$ is a finite-dimensional $\mathbb{Q}_{\ell}$-vector space with $\ell \neq p$ a prime. We denote by $K^{unr}$ the maximal unramified extension and $I_K = \mathrm{Gal}(\overline...