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 $\mathbb{Z}[i]$. …

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 …

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 $Bun_{G}$? …

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 …

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 …

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 $G/N$. …

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 …

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 $ …

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 …

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 …

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 …

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 …

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 …

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: $ …

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 …