# Questions tagged [reductive-groups]

A reductive group is an algebraic group $G$ over an algebraically closed field such that the unipotent radical of $G$ is trivial

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### Faithful representations of integral models

I am reposting a question that I had asked on stackexachage three weeks ago. Let $G/\mathbb{Q}$ be a connected reductive group, and $\mathcal{G}/\mathbb{Z}$ be an integral model (i.e. flat affine ...
1 vote
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### Relative position of flags for the general linear group

This question is related my question in An example of a Deligne–Lusztig variety for a general linear group, and the obtained answer. Situation I am working with the general linear group. Specifically, ...
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### Relative position of flags and the Robinson-Schensted correspondence

This question is related my question in An example of a Deligne–Lusztig variety for a general linear group, and the obtained answer. I am currently reading Steinberg, Robert, An occurrence of the ...
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### Springer sheaf and Deligne-Lusztig induction

Let $G=Gl_n$ be the general linear group over the algebraic closure of a finite field $\overline{\mathbb{F}}_q$ and let $F:G \to G$ be the standard Frobenius. On $G$ there is the Springer (perverse) ...
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### Is Deligne's braiding functorial?

$\newcommand{\ssc}{{\rm sc}} \newcommand{\ad}{{\rm ad}} \newcommand{\Fbar}{{\overline F}}$ Let $F$ be a field and $\Fbar$ be a fixed algebraic closure of $F$. Let $G$ be a (connected) reductive group ...
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### Principal series representations for complex groups

Let $G$ be a complex semisimple group. In Bernstein-Gelfand, "Tensor products of finite and infinite dimensional representations of semisimple Lie algebras" (http://www.numdam.org/article/...
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### How do characters of representations in cohomology depend on the (positive-characteristic) field?

The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
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### Embeddings of reductive groups over algebraically closed fields

Let $K/k$ be an extension of fields, not necessarily algebraic; let $G$ and $H$ be split, reductive groups over $K$; and let $f : H \to G$ be an embedding of groups. Do there exist split, reductive ...
1 vote
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### Eigenvalues of orthogonal group element

Let $q$ be a quadratic form over a nonarchimedean local field $F$, and let $\operatorname{O}(q)$ be the corresponding orthogonal group. Let $g\in\operatorname{O}(q)$ be semisimple. Can we know ...
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### Intersection of certain subsets in a split connected reductive group $G$ related to affine open cover of $G/B$

Let $k$ be a field of characteristic zero and $G$ a split connected reductive group over $k$. Moreover, let $T$ be a split maximal torus of $G$ and $B\supset T$ a Borel subgroup. Additionally, we ...
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### Is there a "spherical building" for a reductive group over a Henselian local ring?

Let $A$ be a Henselian local ring and let $G$ be a split reductive $A$-group. I'm interested in some notion of a "building of parabolic subgroups" for the group scheme $G$. In my specific ...
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### Number of points of parabolic Springer fibres for general reductive groups

My question is the same as this post but for an arbitrary reductive $G$ instead of just $\mathrm{GL}_n$. Let $G$ be a connected split reductive group over a finite field $k$. Let $P$ be a parabolic ...
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### When does the null-cone consist entirely of eigenvectors?

Let $V$ be a finite-dimensional representation of a complex reductive Lie algebra $\mathfrak g$. For our purposes, we may define the null-cone like this: $v\in V$ belongs to the null-cone if and only ...
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### Image of tori in locally symmetric spaces and homology

Suppose we have a reductive group $G$ over $\mathbb{Q}$, a compact subgroup of the adelic points $K_f\subset G(\mathbb{A}_{\mathbb{Q}})$, and the associated locally symmetric space Y_K := G(\mathbb{...
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### A quantity computed from weights of representations -- Have you seen it?

The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...
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### Does the F-unitary group isomorphism arises from a conformal isometry?

Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$. Question 1 Does ...
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### Reference request: Criterion for a subgroup of $\mathrm{GL}_{n}(\mathbb{C})$ being reductive in terms of the trace

Let $G$ be a complex algebraic group embedded into $\operatorname{GL}_{n}(\mathbb{C})$. A criterion for $G$ to be reductive is the following. Let $\mathfrak{g}$ be the Lie algebra of $G$, and let $B$ ...
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### Dimension of the $G$-orbit $\mathcal O_{I,J}(w)$ given by Bruhat decomposition in $G/P_I \times G/P_J$

Let $G$ be a reductive group over an algebraically closed field. Fix a maximal torus $T$ and a Borel subgroup $B$ containing $T$. Let $(W,S)$ be the Coxeter system associated to $(B,T)$, where $S$ ...
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### History of points of view on Eisenstein series

What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them? There are, as far as I know, two major perspectives on what Eisenstein series are. The first is in ...
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### Is every compact simply-connected reductive p-adic group perfect?

Let $k$ be a nonarchimedean local field and $G$ a reductive $k$-group, which we assume to be semisimple and simply-connected. Recall that an abstract group $H$ is perfect if it is generated by ...
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### Centraliser of a maximal $k$-split torus of a reductive $k$-group
Let $G$ be a connected reductive group defined over a field $k$, and let $S$ be a maximal $k$-split $k$-torus of $G$. Then the centraliser $\mathscr Z_{G}(S)$ is defined over $k$. In fact, it is a ...