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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.

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Let $k,n \in \mathbb{N}$. Consider the conjugation action of $\mathrm{Sp}_{2n}$ on $\mathrm{Sp}_{2n}^k$ and the corresponding invariant algebra $\mathbb{C}[\mathrm{Sp}_{2n}^k]^{\mathrm{Sp}_{2n}}$. Is ...
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This is a reference/literature request. Given a field $K$ and an endomorphism $x \colon V \to V$ of a finite-dimensional $K$-vector space it is well-known that the Jordan-Chevalley decomposition of ...
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Recently I am studying the paper Leonid Vaserstein, Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups, Annals of Mathematics 171 issue 2 (2010) pp. 979–1009, ...
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$\newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} $I have asked this question on MSE, but got no answers or comments, so I decided to try my luck on MO. Consider a non-connected reductive ...
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Consider a quadratic separable extension $E/F$. Let $\sigma$ be the non-trivial element in the Galois group. Then $c$ acts naturally in $R \otimes_F E$ by $id_{R} \otimes_{F} \sigma$. For a matrix $J \...
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I'm currently working through Tate's paper on $p$-divisible groups and have come to an impasse in his discussion of tangent spaces of $p$-divisible groups. I'd like to show that for an abelian variety ...
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Let $A$ be any commutative unital ring and let $V:=A\{e_1,..,e_n\}$ be the free rank $n$ $A$-module on the basis $B:=\{e_i\}$. Let $1 \leq d_1 < \cdots < d_k < n$ be integers and let $V_i:=A\{...
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I'm looking for the following result in SGA quoted in Grothendieck's "Groupes de Barsotti-Tate et Cristaux de Dieudonne" via a translation of Peterson: Let $S$ be a scheme over $\mathbb{F}_p$...
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Mostow's theorem classically states that, over a field of characteristic zero, any short exact sequence of algebraic groups $$ 1 \to U \to G \to L \to 1, $$ for which $L$ is reductive and $U$ is ...
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Does there exist a group-object $Mp^{an}$ in the category of rigid/Berkovich/adic spaces over a non-archimedean field $k$, with a homomorphism $Mp^{an}\to Sp^{an}$ to the analytification of the ...
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For a connected reductive group $G$ over $\mathbb{C}$, we have the affine Grassmannian $Gr_G:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ and we have the Cartan decomposition $G(\mathbb{C}((t))) = \coprod_{...
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I am currently reading Brion's notes. In it, he shows that $\textrm{Pic}(G/B)\to \textrm{Pic} (X_w)$ is a surjection as follows: He considers the Bott-Samelson desingularization $\pi: Z_w \to X_w$, ...
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I am currently reading Klyachko's paper "Toric varieties and flag varieties", where he first proves that the permutahedral variety $Y$ is a generic torus orbit closure $Y_p\subset G/B$ of a ...
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Let $\mathfrak g$ be a simple Lie algebra over an algebraically closed field $k$ of good characteristic $p>0$ (for example, $\mathfrak{sl}_n(k)$ with $p \nmid n$). Consider a maximal Lie subalgebra ...
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Let $\pi: X' \to X$ be an étale $\Gamma$-cover of curves (over an algebraically closed field) and $G$ a group scheme (e.g. reductive). Of course, $\operatorname{Bun}_G(X)$ can be identified with the ...
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I'm trying to understand the last inclusion in the proof of Lemma 1.6.1 in Haines, Kottwitz, and Prasad's Iwahori-Hecke Algebras, which I will restate for convenience. We have $G$ as a split connected ...
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I think this question might be folklore to experts (to what extent it can be answered, and to what extent an expected answer is inaccessible), but since I am just a beginner in this direction, please ...
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In real or complex algebraic matrix groups, one can define the Lie algebra using the ordinary matrix exponential as the set of matrices $M$ such that $\exp(tM) \in G$ for all sufficiently small $t \in ...
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Let $K$ be a non-archimedean local field of characteristic $p>0$ and $\mathcal{O}_{K}$ its valuation ring with maximal ideal $\mathfrak{m}$. Let $\mathbf{G}$ be a connected, simply connected $K$-...
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Exercise 4.3 of Mine's notes on Shimura Varieties (https://www.jmilne.org/math/xnotes/svi.pdf) says: Show that the image in $\mathrm{PGL}_2(\mathbb{Q})$ of a congruence subgroup in $\mathrm{SL}_2(\...
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Let $\mathfrak{g}$ be a Kac-Moody algebra a Borel pair $(\mathfrak{b},\mathfrak{t})$, let $R^{+}$ be the set of positive roots, $\alpha_1,\dots,\alpha_n$ the simple roots. For $\alpha=\sum n_{i}\...
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Consider the abelian three dimensional Lie algebra $$\mathfrak{g}= \begin{pmatrix} 0&a & b&c \\ 0 &0 &0 &a \\ 0 &0 & 0 & b\\ 0 &0 &0 &0 \end{pmatrix} $...
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This is one of those things everyone repeats, but I can’t seem to find a proof: Let $n\in \mathbb{N}$ and $1\leqslant k\leqslant n$. Consider the standard action of $\mathrm{GL}_n(\mathbb{R})$ on $\...
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I encountered this problem in Unsolved problems in group theory but I couldn’t find its original source or even what it’s called, nor whether it has been solved The problem: Suppose that $G$ is an ...
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Let $G$ be an adjoint simple algebraic group over $\mathbb{C}$. Denote by $\mathrm{Dyn}(G)$ the automorphism group of the Dynkin diagram of $G$. Then $\mathrm{Dyn}(G)$ is a finite group, either ...
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Let $\mathfrak{g}$ be a simple complex Lie algebra of type $D_4$, and let $\sigma$ be an automorphism of $\mathfrak{g}$. Suppose the fixed-point subalgebra $\mathfrak{g}^\sigma$ is simple and of type ...
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I'm interested in proving vanishing of $\mathrm{Ext}^*(\mathbf{G}_\mathrm{m}, \mathbf{G}_\mathrm{a})$ in the category of abelian sheaves over $\mathbf{Q}$. This seems like a very classical question, ...
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Let $K$ be a global field and let $V_K$ denote the set of places of $K$. Let $G$ be a connected reductive group over $K$, and let $\xi\in H^1(K,G)$ be a Galois cohomology class. Let $S(\xi)\subset V_K$...
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The definition of a reductive group over a field $k$ is that it is smooth (and let us say connected, although not all authors require this, this is the most common definition), and has no non-trivial ...
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In Braden-Macpherson's paper https://arxiv.org/abs/math/0008200, they take $U$ to be an affine variety with a $T$-action containing a $T$-fixed point $x$. We have that $C_x$ is some closed subvariety ...
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Let $\mathbf{U}$ be a quantized enveloping algebra defined by a root datum as in Lusztig's book. Even in finite type, this is slightly more general than something like $\mathbf{U}(\mathfrak{g})$, ...
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If you take a Bialynicki-Birula decomposition, it is not necessarily a stratification. However, I have heard (for instance at the beginning of page 2 of this paper), https://arxiv.org/pdf/math/0008200,...
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Let $G$ be a reductive group over $\mathbb{C}$ such that $Z_G \neq 1$. Can we always find an element $\lambda \in Z_G$ such that We have that $\lambda \in [G,G]$ For any Levi subgroup $L \subsetneq ...
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I'm certain the following type of thing exists in the literature on buildings, since it seems not so hard to work out, but I'm not sure what the correct words are: the Tits building of $\mathrm{GL}_n(...
xir's user avatar
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In [Milne, Example 12.10], the author states: "Later (12.22, 15.39) we shall see that an extension of diagonalizable groups is diagonalizable if if it is commutative, which is always the case if ...
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This is a slightly nitpicky question and somewhat related to a previous question of mine: If $T'$ is a Tannakian subcategory of a Tannakian category $T$, then is $T'$ necessarily closed under ...
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We have an embedding of the complex simple Lie algebra $G_2$ into $\mathfrak{so}_7$. Is this embedding unique up to $\mathrm{SO}_7$-conjugacy? Note it is easy to see the embedding of $G_2$ into $\...
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Let $G$ be a smooth unipotent algebraic group over the integers, say of dimension $n$, and let $p$ be a prime, with $k$ the finite field of order $p$. Let $G_1$ denote the (first) Frobenius kernel of ...
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Can anyone help me to find references for the statement: Let $H$ be a connected semisimple real algebraic group with Lie algebra $\mathfrak{h}$. Let $\Gamma$ be a discrete group, and suppose $\rho : \...
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Let $\mathcal{C}$ be a rigid abelian $k$-linear tensor catgory, where $k$ is an algebraically closed field. I assume that there is a commutativity constraint, but I do not assume that it is symmetric. ...
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Let $\mathfrak g$ be a complex simple Lie algebra with a chosen Cartan subalgebra $\mathfrak h$ and a choice of positive roots. Let $G$ be a complex Lie group having $\mathfrak g$ as its Lie algebra. ...
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In the paper of Batyrev and Blume, they mention that by projecting along the normal of each Weyl hyperplane, we get a map from the fan of the permutahedral variety $X$ to the fan of $\mathbb{P}^1$ and ...
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Over a field of characteristic $0$, consider a finite group scheme $G$ and its quotient by a subgroup scheme $H$. Does there exist a schematic section of the morphism from $G$ to its quotient $G\over ...
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There is an open locus $U$ of $GL_n/B$ such that if you take a point $p$ in $U$, then its torus orbit closure $\overline{T\cdot p}$ is isomorphic to the permutahedral variety. Equivalently, by Atiyah'...
cacha's user avatar
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$\newcommand{\ad}{{\rm ad}} $Let $G$ be a simply connected semisimple group over a field $K$. Let $Z_G$ denote the center of $G$, and let $G^\ad=G/Z_G$ be the corresponding adjoint group. Consider the ...
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Let $\mathbf{G}^F={\rm Sp}_{10}(q)$, and assume that |$\mathbf{T}^F|=(q^4-1)(q+1)$. In the manual of CHEVIE, it is said that the function DeligneLusztigCharacter(W,w) returns the Deligne-Lusztig ...
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Given a subgroup $\Gamma\subset PSL(2,\mathbb{Z})$, let $H(\Gamma)$ be the set of $PSL(2,\mathbb{R})$-conjugates of $\Gamma$ which are contained in $PSL(2,\mathbb{Z})$, and let $h(\Gamma)$ be the ...
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Let G be a reductive group over $\bar{\mathbb{F}}_q$ with nonsplit $\mathbb{F}_q$-structure. Let $F$ be the twist Frobenius. Let $B$ be a $F$-stable Borel subgroup and $P$ be a $F$-stable standard ...
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If $G$ is a reductive algebraic group (I am interested in $G={\rm GL}_n, {\rm SL}_n,{\rm PGL}_n$), $X$ is a smooth complex algebraic variety (I am interested in a smooth quasi-projective variety $X$) ...
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Let $G$ be a connected complex reductive group, and let $K \subseteq G$ be a connected reductive subgroup. It is well known that the invariant algebra $\mathbb{C}[\mathfrak{g}]^G$ is a polynomial ring,...
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