4
votes
0answers
53 views
Rational points with small denominator in $U(n)$
Fix integers $n,d>0$. (I'm probably thinking about $n\leq 6$ and $d\leq 2000$.) Let $X$ be the set of matrices $A\in U(n)$ such that the entries of $dA$ lie in $\mathbb{Z}[i]$.
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2
votes
1answer
138 views
Group scheme over a DVR whose special fibre is the image of points under reduction mod p
Let $R$ be a complete discrete valuation ring with maximal ideal
$\mathfrak{p}$ and algebraically closed residue field $k$. Denote
the field of fractions of $R$ by $F$. Let $G$ be …
1
vote
1answer
133 views
Smooth map to the stack of G-bundles
Let $G$ a semisimple group and $B$ a Borel subgroup.
We denote by $Bun_{G}$ the stack of G-bundles.
Is it true that a certain open subset $Bun_{B,r}$ maps smoothly to $Bun_{G}$?
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1
vote
0answers
222 views
For what fields is $GL_n(k)$ a rational variety?
I know that every linear algebraic group is rational over algebraically closed fields. To what extent is that true for other fields? For example: is $GL_n(\mathbb{Q}_p)$ a rational …
2
votes
2answers
243 views
Equivariant Cohomology of a Complex Projective Variety
Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would lik …
2
votes
1answer
198 views
Quotient of algebraic groups in the étale topology
Let $G$ be an affine algebraic group over $\mathbb{C}$. According to SGA3, any closed normal subgroup $N$ is representable by an affine algebraic group, as is the quotient $G/N$.
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3
votes
1answer
137 views
differential of the characteristic polynomial
Let $\chi:GL_{n}(\mathbb{C})\rightarrow \mathbb{C}^{n}$ the map given by the coefficients of the characteristic polynomial.
Let $A$ a regular semisimple matrix, do we have a formul …
0
votes
1answer
70 views
Reference on elements of finite order in principal congruence subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $ …
1
vote
1answer
73 views
On the $F$-rational points of the derived group of a connected reductive algebraic group
Let $F$ be a local non-archimedean field and let $G$ be a connected reductive algebraic group defined over $F$. Let $G_{der}$ denote the algebraic derived group of $G$; this is co …
4
votes
1answer
105 views
Levi decomposition in disconnected linear algebraic group (characteristic 0)?
For algebraic groups or Lie groups, the subject of Levi decompositions tends to be surrounded by some mystery in the literature (and in an older question raised here). While I po …
2
votes
2answers
254 views
Why are these parabolic subgroups opposed?
I am reading notes of Michel Brion on spherical varieties.
Consider a reductive group $G$, a Borel $B$ in $G$, a finite dimensional $G$-module $M$ and a closed orbit $Y$ of $G$ in …
7
votes
1answer
141 views
Counting conjugacy classes in simple groups of Lie type
Finite groups of Lie type include those obtained as rational points of a connected simple algebrraic group over a finite field $k = \mathbb{F}_q$ of characteristic $p$: these are s …
0
votes
0answers
98 views
A simple question on unitary group.
Hi!
Let $E/F$ be a quadratic extension of number fields and $A_E$ and $A_F$ are adele rings of $E$ and $F$, respectively.
Then, I am wondering whether all 1-dimension hermition s …
1
vote
0answers
30 views
open immersion, affine grassmanian and negative loop group
Let $G$ a semisimple group over $k=\bar{k}$.
Let the $k$-indgroup, $LG^{-}\subset G(k[t^{-1}])$ be the kernel of the reduction. We know by Faltings that the multiplication map:
$ …
2
votes
0answers
77 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 gr …

