All Questions
52
questions
2
votes
1
answer
239
views
Normalizer of Levi subgroup
Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$.
Associated with this ...
- 53
5
votes
1
answer
301
views
Parabolic subgroups of reductive group as stabilizers of flags
$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
- 497
3
votes
1
answer
124
views
Connected components of a spherical subgroup from spherical data?
This question is in a similar spirit to this one by Mikhail Borovoi.
Let $G$ be a reductive group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety.
Losev proved that the spherical $...
- 2,386
3
votes
1
answer
316
views
Does a quasi-split reductive group scheme admit a maximal torus?
Let $G \to S$ a reductive group scheme over arbitrary base. Following the conventions from Conrad's Reductive Group Schemes notes, we define a Borel subgroup to be an $S$-subgroup scheme $B \subseteq ...
- 565
0
votes
1
answer
140
views
Calculating relative root systems
Let $\mathbf{G}$ be a connected semisimple algebraic group defined over a field $k$. Let $T$ be a maximal torus of $\mathbf{G}$ defined over $k$, and let $S \subset T$ be a maximal $k$-split torus. ...
- 43
3
votes
2
answers
213
views
Reductive groups over arbitrary fields with disconnected relative root systems
Let $\mathbf{G}$ be a connected reductive group over a field $k$, not necessarily algebraically closed. Let $\Phi$ be the relative root system for $\mathbf{G}$ with respect to $k$, and assume that $\...
- 131
1
vote
0
answers
94
views
Connected stabilisers for actions of reductive groups
Let $G$ be a connected split reductive group over a field $k$ acting on a variety $X$ over $k$. For each $x\in X$, let $G_x$ be the stabiliser. In general, $G_x$ may be disconnected.
Now suppose $G$ ...
- 2,711
3
votes
1
answer
233
views
Is Deligne's braiding functorial?
$\newcommand{\ssc}{{\rm sc}}
\newcommand{\ad}{{\rm ad}}
\newcommand{\Fbar}{{\overline F}}
$
Let $F$ be a field and $\Fbar$ be a fixed algebraic closure of $F$.
Let $G$ be a (connected) reductive group ...
- 13.8k
3
votes
1
answer
242
views
Embeddings of reductive groups over algebraically closed fields
Let $K/k$ be an extension of fields, not necessarily algebraic; let $G$ and $H$ be split, reductive groups over $K$; and let $f : H \to G$ be an embedding of groups.
Do there exist split, reductive ...
- 12k
2
votes
0
answers
98
views
Number of points of parabolic Springer fibres for general reductive groups
My question is the same as this post but for an arbitrary reductive $G$ instead of just $\mathrm{GL}_n$.
Let $G$ be a connected split reductive group over a finite field $k$.
Let $P$ be a parabolic ...
- 2,711
3
votes
0
answers
202
views
A quantity computed from weights of representations -- Have you seen it?
The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...
- 3,039
4
votes
0
answers
138
views
Centraliser of a maximal $k$-split torus of a reductive $k$-group
Let $G$ be a connected reductive group defined over a field $k$, and let $S$ be a maximal $k$-split $k$-torus of $G$. Then the centraliser $\mathscr Z_{G}(S)$ is defined over $k$. In fact, it is a ...
- 61
3
votes
1
answer
185
views
Regular embeddings of a reductive groups with induced center
Let $G$ be a reductive group over the finite field $\mathbb{F}_q$. Then a regular embedding of $G$ is an $\mathbb{F}_q$-rational embedding $\iota \colon G \rightarrow G'$ into a second reductive group ...
- 765
5
votes
1
answer
400
views
Lifting $\mathfrak{sl}_2$-triples
Let
$k$ be an algebraically closed field,
$G$ a (smooth, connected) reductive algebraic group over $k$,
$H$ a (smooth, connected) reductive group of semisimple rank 1, and
$T$ a maximal torus in $H$.
...
- 12k
4
votes
1
answer
295
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 ...
- 1,585