# Questions tagged [reductive-groups]

A reductive group is an algebraic group $G$ over an algebraically closed field such that the unipotent radical of $G$ is trivial

314 questions
Filter by
Sorted by
Tagged with
90 views

### Action of split torus on positive root spaces

Let $G$ be a connected reductive group over a field $k$ (not necessarily algebraically closed). Let $S$ be a maximal split torus in $G$ with relative root system $\Phi = \Phi_k(S,G)$. Let $\Phi^+$ ...
73 views

### How to read the paper of Arthur on trace formula on general reductive groups (Reference Request)

My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily. I would like ...
194 views

### Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
28 views

72 views

### When representations of reductive Lie group in a Banach space and in its Garding space have the same length?

Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of ...
154 views

### The norm of the principal series intertwining operator for $\operatorname{GL}_2$

Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}_2(\mathbb Q_p)$? Background: For a character $\chi = (\chi_1,\chi_2)$ of the ...
131 views

### Maximal torus of linear algebraic group over a ring

Let $G$ be a linear algebraic group over a $k$-algebra $A$, where $k$ is an algebraically closed field. Consider the structure morphism $G\rightarrow U={\rm Spec}(A)$. Assume that for every $k$-point ...
46 views

### Maximal compact subgroups of p-adic orthogonal groups

Let F be a non-archimedean local field, and G be split orthogonal groups of odd degrees $\geq$ 3. In this setting, my question is; Is there explicit descriptions of maximal compact subgroups of G? I ...
62 views

### Calculating the residue of Eisenstein series from the residue of the intertwining operator

I've been reading the article Forms of $\operatorname{GL}(2)$ from the analytic point of view by Gelbart and Jacquet in Corvallis and am confused on a particular claim (equation 5.17 on page 232). The ...
47 views

172 views

### Schubert cells in G/P for reductive G

All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
200 views

### Finiteness of the volume of $G(F) \backslash G(\mathbb A)$

Let $G$ be a semisimple algebraic group over a number field $F$ with trivial center. Let $\mathfrak S \subset G(\mathbb A)$ be a Siegel domain (defined in terms of a given maximal split torus and ...
125 views

### Harmonic analysis on reductive groups over $\mathbb{R}$

A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...
117 views

### Centralizers of semisimple subgroups

$\DeclareMathOperator\GL{GL}$If $G$ is a simple Lie group, and $\rho: G \to \GL(V)$ is a representation, then by Schur's lemma, the group of automorphisms of $\rho$ is a reductive subgroup of $\GL(V)$....
346 views

### Branching laws for smooth representations

Let $E / F$ be a quadratic extension of nonarchimedean local fields (characteristic 0 if it matters), and $\pi$ an irreducible infinite-dimensional smooth representation of $GL_2(E)$. Let $B$ be the ...
47 views

202 views

### Real forms of complex reductive groups

I have a collection of related (to me) questions, which stem from the fact that I feel like I have a bunch of pieces, but not a full clear picture. I'm curious about forms of reductive groups in ...
130 views

### Globalising tori and weak approximation

Let $G$ be a semi-simple and simply connected reductive group over $\mathbb{Q}$ and let $T \subset G_{\mathbb{Q}_p}$ be a maximal torus. A classical result of Harder tells us that we can find a ...
80 views

54 views

### Reference request: boundedness for semistable principal bundles on a family of curves

We work over an algebraically closed field $k$. Let $G$ be a reductive group and $X$ be a smooth projective curve over $k$. It is proven in [1, Theorem 1.2] that the moduli of semi-stable principal $G$...
59 views

Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(... 0answers 75 views ### Bruhat-Tits theory: how does the normalizer act on an apartment? Let$G$be the points of a split, simply connected, semisimple algebraic group over a$p$-adic field$k$. Let$T$be a maximal torus of$G$,$T_c$its unique maximal compact subgroup, and$N$the ... 0answers 101 views ### Jordan normal form in a reductive group Let$G$be a connected complex reductive group. Given an element$g \in G$, there is a canonical decomposition$g = s u$such that$s$is semisimple,$u$is unipotent and$s$and$u$commute. ... 0answers 189 views ### Definition of Iwahori subgroup independently of the Bruhat-Tits building Let$G$be the points of a connected, semisimple algebraic group over a$p$-adic field$k$. To make life easy, let's assume the underlying group scheme is simply connected. The Bruhat-Tits building$...
I am reading Knapp's book "Lie groups beyond an introduction" (2nd edition). I am struggling to understand the following point. Recall that $G$ is a reductive Lie group. If the Lie algebra \$\...