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Results tagged with algebraic-groups
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user 43107
Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.
7
votes
Preimage of a maximal compact open subgroup in the simply connected cover
The answer is negative even for ${\rm{PGL}}_2$: the stabilizer of an edge in the building is a counterexample (with Iwahori preimage in ${\rm{SL}}_2(F)$). It likewise fails for ${\rm{PGL}}_n$ for eve …
- 1,623
7
votes
Accepted
rationality question while dealing with an isogeny
The answer is negative (even under all of the given hypotheses); this expresses a standard difficulty in the arithmetic aspects of connected semisimple groups over local function fields in contrast wi …
- 1,623
1
vote
Weyl group action on continuous characters of the group of $\mathbf{Q}_p$-points of the torus
The answer is false due to torsion $\chi$, such as for PGL$_2$ (center trivial, hence connected!) with $\chi({\rm{diag}}(x,1)) = (-1)^{{\rm{ord}}_p(x)}$. [EDIT: This is wrong, as the OP notes below.] …
- 1,623
2
votes
About structure of parabolic subgroups of finite classical algebraic groups
It is a general fact that if $P$ is a parabolic $k$-subgroup of an arbitrary connected reductive group $G$ over an arbitrary field $k$ then $U := \mathscr{R}_u(P)$ has a canonical $P$-equivariant filt …
- 1,623
13
votes
Accepted
Orbits of group scheme action
Presumably you meant to assume the schemes are finite type over $k$. To work naturally with orbit questions for such schemes one just has to bring in appropriate use of flatness to adapt intuition an …
- 1,623