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I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can a …

$\DeclareMathOperator\Sym{Sym}$Let $G$ be a compact lie group. Chern–Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (\Sym^{\bullet} \mathfrak{g^*})^G$$ which in our case …

Let $S_n$ denote the symmetric group on $n$-letters and let $\mathrm{Rep}(S_n)$ denote its representation ring. For every $n$ restriction along the inclusion $S_{n-1} \to S_n$ induces a ring homomorph …

I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}_m$ action. Here's what I have so far: Let $X=Spec\,A$ be an affine scheme (after this case is setteled I imagine it …

Let $G$ be a finite group and let $Rep(G)$ be its representation ring (as a group it is the free $\mathbb{Z}$-module on the irreducible complex reps). The collection of permutation representations $\m …

Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question wi …

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $Rep_ …

Motivating examples: Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$ The L …

Let $H < G$ be a subgroup of a finite group $G$. Let $X:=G/H$ and $\mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $X$ (w.r.t. left multiplication) associated to a finite dimensional representatio …

Let $k$ be a field, $\mathsf{Vect}$ the category of finite dimensional vector spaces, and $\mathsf{C} = Fun(\mathsf{Vect},\mathsf{Vect})$ the abelian category of pointed endofunctors (sending $0$ to $ …

This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway. If I understand correctly, for any reductive algebraic group $G$ th …

I'll be using homological grading throughout this question. Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods: $$H^{\bullet}( …

All sets and groups in the question are finite. In order to understand equivariant sheaves better I'm trying to prove some basic facts from Mackey theory using equivariant sheaves. The main obstacle …

Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators. Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category …

Let $\mathfrak{gl}_n(x)= \mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}(x)$ be the algebra of matrices taking values in rational functions. Definition: Two matrices $A, B \in \mathfrak{gl}_n(x)$ ar …