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I’m trying to better understand the continuous spectrum of $ G = \operatorname{GL}_2(\mathbb A_{\mathbb Q})$ , which is the direct integral of induced representations $ \mathbf H(s) = \operatorname{Ind}_{Z(\mathbb A)N(\mathbb A)T(\mathbb R)^0T(\mathbb Q)}^{G(\mathbb A)} \omega_s$ . The reference I’m using is Forms of GL(2) from the analytic point of view by Gelbart and Jacquet inRead more

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 representation $ \pi$ of $ H$ with character $ \chi$ . Let $ Ind_H^G(\chi)$ be the characterRead more

Let $ X, Y$ , let’s say, homotopy commutative $ H$ -spaces, $ f,g$ maps from $ X$ to $ Y$ . (Actually we only need $ Y$ to be homotopy commutative $ H$ _space, but the statement is easier if we also suppose $ X$ to be one). It is not difficult to showRead more

Let $ X$ be a proper $ k$ -scheme and $ k \subset k’$ a field extension. Consider the fibre product \ base change $ X’ = X \otimes _k k’$ . Let $ \mathcal{F} \in Coh(X)$ and $ p: X’ \to X$ the canonical projection (I think that it is a affine morphism (whyRead more

Suppose we have a graph $ G$ with $ n$ vertices. If we color the edges of $ G$ by two colors, then we can conclude by Konig’s Theorem that there exists a monochromatic subtree $ T$ with at least $ n/{\alpha}$ vertices, $ \alpha$ is the size of maximum independent set in the graph.Read more