# Tagged Questions

For questions about groups which have additional structure as a algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. Consider using with the (group-theory) tag.

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### Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
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### p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
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### Why are parabolic subgroups called “parabolic” subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...
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### Representation theory of the general linear group over a finite prime field

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire ...
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### Is there a classification of finite abelian group schemes?

There is a well-known classification of finite abelian groups into products of cyclic groups. What about finite abelian group schemes, where we may put in the qualifiers "affine", "etale", or "...
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### Motivation?: Lie algebra and algebraic group Cohomology

This is just an a-priori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
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### Weil Pairing on Linear Algebraic Groups

I've been studying the Weil pairing on elliptic curves recently and discovered that it has a generalisation to an abelian variety $A$ with its dual $A^{\vee}$, which then becomes a pairing on an ...
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### Hecke characters and Unitary groups

Let $M/F$ be a quadratic extension of number fields, with Galois group $G=\{1,\tau\}$. Consider the following unitary group $$U_1(R)=\{z\in (R\otimes_FM)^\times :zz^\tau=1\},$$ where $R$ is an $F$-...
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### Galois cohomology of Unitary groups

From Hilbert 90 (or, more precisely, a generalisation thereof), we know that the first (Galois) cohomology group of $GL_n$ is trivial, no matter the field of definition. However, for unitary groups, ...
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### Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
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### quotient group scheme

assume I have a group $G$ over a field of char 0 and $H$ a closed subgroup. When is it true that the group $N(H)/H$ is representable? If $G$ has nice properties, like to be reductive or unipotent is ...
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### Spherical building of an exceptional group of Lie type

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
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### When is a torus the closure of the cyclic subgroup generated by one of its elements?

$K$ is an algebraically closed field. And $T$ is an algebraic group conatined in $GL(n,K)$. Assume $K$ is not the algebraic closure of a finite field. If $T$ is a torus, show that $T$ is the ...
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### Questions about Affine algebraic group scheme over an infinite field K

For an easily comprehension of my questions I write some definitions: An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...
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### homomorphism between smooth algebraic groups of the same dimension

For Lie groups, we have a theorem: Suppose $G$ and $G'$ are Lie groups of the same dimension, $G'$ is connected, and $f : G \to G'$ is a homomorphism of Lie groups with discrete kernel. Then, $f$ ...
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### Identifying the cotangent bundle of the flag variety

Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book "...
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### Is a connected unipotent subgroup always contained in a Borel subgroup?

As the question says, is a connected unipotent subgroup $U$ of a linear algebraic group scheme $G$ always contained in a Borel subgroup of $G$? I have an argument for why the answer is yes, and I ...
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Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given. Also, I calculated the following regarding the Bruhat decomposition of $\... 0answers 91 views ### Making the definition of dual root unambiguous In 5.4 of his book Lectures on Invariant Theory, Igor Dolgachev introduces the dual of a root by requiring that$\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)(\alpha\circ\check\...
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I am trying to understand the Borel-Weil theorem, but I am very confused because of the different conventions used in different sources. I am especially confused about two things: (1) the definition ...
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### What is $\rho^{\vee}(-1)$?

What is $\rho^{\vee}(-1)$? By definition, $\rho^{\vee}$ is the sum of all positive coroots. I have some difficulty in computing $\rho^{\vee}(-1)$. For example, in the case of $SL_3$, all positive ...
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### Embedding $\mathbb{G}_a$ into $GL_2$

Let $k$ be an algebraically closed field of characteristic $p$. I'd like to find interesting examples of closed embeddings $\mathbb{G}_a(k)\hookrightarrow GL_2(k)$, where $\mathbb{G}_a(k)$ is $(k,+)$....
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### When is the adjoint representation self-dual?

Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question ...
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### Any local algebraic group is birationally equivalent to an algebraic group

In this paper, page $6$ the authors state the following: By Weilâ€™s theorem $[17]$, any local algebraic group is birationally equivalent to an algebraic group. Where $[17]$ A.Weil. On ...
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