Skip to main content
MathOverflow

Return to Question

deleted 57 characters in body; added 58 characters in body
Source Link
Arkandias
Arkandias
  • 991
  • 7
  • 15

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi^+$$\Phi$.

There is a group homomorphism : $$S(k) \to \mathbb{Z}^{\Delta}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Phi^+}$$$$S(k) \to \mathbb{Z}^{\Delta}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Delta}$$ with finite cokernel and kernel equal to $S_0 (Z(k) \cap S)$$S_0 (Z(k) \cap S(k))$ where $S_0 \subset S(k)$ is the maximal compact subgroup and $Z$ the center of $G$.

This is used in many papers (e.g. Tit's "Reductive groups over local fields") but I could not find a reference for the proof.

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi^+$.

There is a group homomorphism : $$S(k) \to \mathbb{Z}^{\Delta}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Phi^+}$$ with finite cokernel and kernel equal to $S_0 (Z(k) \cap S)$ where $S_0 \subset S(k)$ is the maximal compact subgroup and $Z$ the center of $G$.

This is used in many papers (e.g. Tit's "Reductive groups over local fields") but I could not find a reference for the proof.

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$.

There is a group homomorphism : $$S(k) \to \mathbb{Z}^{\Delta}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Delta}$$ with finite cokernel and kernel equal to $S_0 (Z(k) \cap S(k))$ where $S_0 \subset S(k)$ is the maximal compact subgroup and $Z$ the center of $G$.

This is used in many papers (e.g. Tit's "Reductive groups over local fields") but I could not find a reference for the proof.

added 1 characters in body
Source Link
Arkandias
Arkandias
  • 991
  • 7
  • 15

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus and, $\Phi(G,S)$ the relative root system. Let and $\Phi^+$ be$\Delta$ a subsystembasis of positive roots$\Phi^+$.

There is a group homomorphism : $$S \to \mathbb{Z}^{\Phi^+}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Phi^+}$$$$S(k) \to \mathbb{Z}^{\Delta}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Phi^+}$$ with finite cokernel and kernel equal to $S_0 (Z \cap S)$$S_0 (Z(k) \cap S)$ where $S_0$$S_0 \subset S(k)$ is the maximal compact subgroup and $Z$ the center of $G$.

This is used in many papers (e.g. Tit's "Reductive groups over local fields") but I could not find a reference for the proof.

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus and $\Phi(G,S)$ the relative root system. Let $\Phi^+$ be a subsystem of positive roots.

There is a group homomorphism : $$S \to \mathbb{Z}^{\Phi^+}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Phi^+}$$ with finite cokernel and kernel equal to $S_0 (Z \cap S)$ where $S_0$ is the maximal compact subgroup and $Z$ the center of $G$.

This is used in many papers (e.g. Tit's "Reductive groups over local fields") but I could not find a reference for the proof.

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi^+$.

There is a group homomorphism : $$S(k) \to \mathbb{Z}^{\Delta}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Phi^+}$$ with finite cokernel and kernel equal to $S_0 (Z(k) \cap S)$ where $S_0 \subset S(k)$ is the maximal compact subgroup and $Z$ the center of $G$.

This is used in many papers (e.g. Tit's "Reductive groups over local fields") but I could not find a reference for the proof.

Source Link
Arkandias
Arkandias
  • 991
  • 7
  • 15

Decomposition of k-split tori of p-adic reductive groups

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus and $\Phi(G,S)$ the relative root system. Let $\Phi^+$ be a subsystem of positive roots.

There is a group homomorphism : $$S \to \mathbb{Z}^{\Phi^+}, s \mapsto (\mathrm{ord_p} \alpha(s))_{\alpha \in \Phi^+}$$ with finite cokernel and kernel equal to $S_0 (Z \cap S)$ where $S_0$ is the maximal compact subgroup and $Z$ the center of $G$.

This is used in many papers (e.g. Tit's "Reductive groups over local fields") but I could not find a reference for the proof.