4 problems

Let $G$ be a reductive group over $F$, let $\rho$ define the reductive monoid $X_\rho$, and write $\mathcal{S}_{\mathrm{ES}}(X_\rho(F)):=C^\infty_c(X_\rho(F))$. View these function…

Let $G$ be a reductive group over $F$, let $\rho$ define the reductive monoid $X_\rho$, and let $C^\infty_c(X_\rho(F))$ be the space of compactly supported locally constant functio…

Let $G$ be a reductive group over $F$, let $\rho$ be the representation defining the reductive monoid $X_\rho$, and let $C^(X_\rho(F))$ be the space of locally constant functions o…

Assume that $\rho$ factors through $\widehat{G}(\mathbb{C})\rtimes\operatorname{Gal}(E/F)$ for a finite Galois extension $E/F$, and let $M_\rho$ be Ngô's reductive monoid attached…