1
vote
0answers
81 views

Symmetric spaces which are compact modulo the unipotent radical are compact

Is the following true? Let $X = G/H$ be a symmetric space of a reductive group over a p-adic field $F$. Let $X^0$ be an open orbit w.r.t. the action of the minimal parabolic $B$ of $G$. Let $U$ be ...
3
votes
1answer
70 views

Density of $\Gamma(N)$ in $\mathrm{Sp}_{2g}(\mathbb{Z}_{\ell})$ where $\ell \not | N$

Let $\mathrm{Sp}_{2g}$ denote the symplectic group of $2g \times 2g$ matrices for some $g \geq 1$, and let $\Gamma(N)$ be the level-$N$ principal congruence subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z})$. ...
3
votes
1answer
162 views

Weyl group action on continuous characters of the group of $\mathbf{Q}_p$-points of the torus

Let $G$ be a split reductive group over $\mathbf{Q}_p$ and assume $G$ has connected center. Let $T$ be a maximal split subtorus of $G$ and $R$ be the roots of $(G,T)$. Let $\chi : T(\mathbf{Q}_p) \to ...
1
vote
0answers
129 views

Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field. Then $G(F)$ is a p-adic group. Let $\Psi(G)$ be the lattice of algebraic characters. Let $\Lambda_G$ be the ...
4
votes
1answer
161 views

When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$. Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...
3
votes
0answers
130 views

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, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$. There is a group homomorphism : ...
4
votes
1answer
357 views

Representations of reductive groups over local fields through parahoric induction

Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as ...
0
votes
1answer
368 views

Closed subgroups of a $p$-adic algebraic group

Let $F$ be a finite extension of $\mathbb{Q}_p$ and $G$ the group of rational points of an algebraic group defined over $F$, endowed with the natural topology. Any Zariski closed subgroup $H \subset ...