# Tagged Questions

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### quasi-split algebraic group

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...
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### Is there anywhere some explicit Bruhat decompositions are written down?

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 ...
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### Complex conjugation of positive roots

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
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### Lie Groups/Lie algebras to algebraic groups

I am reading some lie groups/lie algebras on my own.. I am using Brian Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction I was checking for some other references on ...
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### The regular representation for affine group schemes

I want to understand the regular representation of an affine algebraic group. An affine algebraic group as I know it, is a functor from the category of $k$ -algebras to groups that is representable ...
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### Eigenvectors of algebraic group representation

In a paper of Kollar and Szabo there is a lemma (Lemma $1$) in which the following terminology is used: "Every representation $H\rightarrow GL(n,K)$ has an $H$-eigenvector" ($H$ is an algebraic ...
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### Dimension of the GL-orbit of d-forms in one less variable

Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
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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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### Why are $V =K^n$ and its dual isomorphic $SL(2,K)$-modules

In this paper(http://arxiv.org/pdf/1204.6131.pdf), the following statement $V =K^n$ and its dual are isomorphic $SL(2,K)$-modules, seems to be common sense. Here, $K$ is a field of ...
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### connected algebraic groups

Let $G$ be a linear algebraic group over $\mathbb C$. Let $\psi$ be a finite dimensional regular representation of $G$ into $GL(V)$. Suppose $G$ is connected. I would like to show for $v$ in $V$ the ...
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### symplectic representations: when could the center act trivially?

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
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### More on the versions of the Peter-Weyl theorem

The following three statements appear analogous: For a finite group $G$, the group algebra $\mathbb C[G]$ decomposes as $\bigoplus_{V {\rm\ irred}} V^* \otimes V$. (Peter-Weyl) For a compact group ...
It seems that, for $GL_n$, and possibly for something like complex reductive groups $G$ in general, there's an algebraic version of the Peter-Weyl theorem, which might say that the coordinate ring of ...
I understand the finite-dimensional representations of $\text{SL}(2,\mathbb C)$ as a Lie group and their correspondence with Lie algebra representations of $\mathfrak{sl}(2,\mathbb C)$. Does anyone ...