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 ...
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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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An example of a Deligne–Lusztig variety for a general linear group
Take $G=\operatorname{GL}_3$, defined over the algebraic closure of a finite field $\mathbb{F}_q$ and let $X$ be the set of Borel subgroups of $G$.
The Frobenius morphism $F:G\to G$ induces a map $F:...
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Definition of locally symmetric space of reductive groups
This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless.
In my attempt to study Shimura varieties, I came across ...
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Weight of the defining function of a Bruhat cell in a simply connected semisimple group
Let $G$ be a connected semisimple simply connected group over an algebraically connected field. Let $w_0$ be the longest element in the Weyl group $W$ and $s_i$ be a simple reflection in $W$. Choose a ...
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Residue of a local $\gamma$-factor and its relation with adjoint $\gamma$-factor
I met the following relation (if my understanding is correct) of local $\gamma$-factors when I was reading Hiraga-Ichino-Ikeda's paper "Formal Degrees and Adjoint $\gamma$-Factors": Let $F$ ...
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Connected stabilisers for actions of reductive groups
Let $G$ be a connected split reductive group over a field $k$ acting on a variety $X$ over $k$. For each $x\in X$, let $G_x$ be the stabiliser. In general, $G_x$ may be disconnected.
Now suppose $G$ ...
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$L$-parameters and parabolic induction
I apologize in advance if the answer to this question is well-known to experts.
So let $F$ be a $p$-adic field, and $G$ a reductive group over $F$. Let $P$ be an $F$-parabolic subgroup of $G$ and $M$ ...
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Find an analogue of Weyl chamber structure
Let $G$ be a split reductive group and let $T$ be a split maximal torus whose rank is $l$. Is it possible to find a base $\gamma_1,..., \gamma_l$ of the weight lattice $X(T)$ such that the cone $C$ in ...
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Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
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Does $\mathrm{Ext}^i_G(\pi,\pi')$ vanish if $\pi$ and $\pi'$ are smooth irreducible representations of $G$ with different central characters?
Let $G$ be a $p$-adic group, ie. the group of $F$-points of some connected reductive group over $F$, where $F$ is a $p$-adic field. We consider complex smooth representations of $G$. Any irreducible ...
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Centralizer of a reductive subgroup
Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...
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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 ...
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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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The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$
I'm reading James Arthur's notes on the trace formula and am confused on a point on pages 65 and 66. For $G$ a reductive group over $\mathbb Q$ we are going over the decomposition of the space $L^2(G(...
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What are the good maximal compact subgroups in $p$-adic unitary groups?
Let $E/\mathbb Q_{p}$ be a quadratic extension and let $V$ be an $n$-dimensional $E$-hermitian space. Denote the hermitian form by $(\cdot,\cdot):V\times V \rightarrow E$. Let $G := \mathrm{U}(V)$ be ...
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Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
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Asymptotic behavior of matrix coefficients
I'm reading Casselman's notes "Introduction to the theory of admissible representations of p-adic reductive groups". In chapter 4 "The asymptotic behavior of matrix coefficients", ...
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Constructing a Kac-Moody group as a quotient of the free product of its root subgroups
The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven'...
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Comparing cohomology of quotient by algebraic group and Borel subgroup
Let $X$ a variety over an algebraically closed field $k$ (which we can assume to be actually $k=\mathbb{C}$) and $G$ a connected reductive algebraic group acting freely on $X$ (we can actually assume $...
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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 ...
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Is there a Chevalley map for spherical varieties?
If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $...
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Cohomology of compact open subgroups of semisimple groups over local fields
Let $E$ be a local field, $\mathcal{O}$ its ring of integers, $k$ its residue field, and $G$ a split semisimple group over $\mathcal{O}$. Let $K$ be an open subgroup of $G(\mathcal{O})$; more ...
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Conjugates of relative root groups by an element of the Weyl group
Let $G$ be a reductive group (over an algebraically closed field), $T$ a maximal torus, and $\Phi$ the root system of $(G,T)$. Then for each root $\alpha \in \Phi$ there is a unique connected $T$-...
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Is the product of unipotent radicals of opposite Borels a closed immersion?
Let $G$ be a reductive group over a scheme $S$ and let $B \subset G$ and $B' \subset G$ be opposite Borel subgroups with their unipotent radicals $U \subset B \subset G$ and $U' \subset B' \subset G$. ...
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Can closure of an orbit under a reductive action contain infinitely many orbits?
I posted this on math.se a week ago, currently it has 23 views and no other feedback.
Here on MO there are several questions about orbit closures but I could not find anything about what I need.
To be ...
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$\mathbb G_{\mathrm{m}}$-gerbes are to (derived) Azumaya algebras as $G$-gerbes are to …?
Let $X$ be a quasicompact quasiseparated scheme over a field $k$. The connection between Azumaya algebras over $X$ and $\mathbb G_{\mathrm{m}}$-gerbes over $X$ is well-known: there exists an injection ...
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Semisimple covers of varieties
Let $X$ be an algebraic variety.
The finite étale covers of $X$ are measured by the étale fundamental group $\pi_1^{\rm et}(X)$.
On the other hand, the Cox ring ${\rm Cox}(X)$ of $X$ (finitely ...
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Reference for character sheaves over $\mathrm{GL}_n(q)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$I know a little bit about complex representation theory of finite reductive groups as $\GL_n(q),\SO_n(q)$ etc via Deligne-Lusztig induction and ...
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A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$
I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
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Nef cone of a GIT quotient
I want to know how to calculate nef cone of a GIT quotient. In particular let $X$ be a projective variety and $L$ be an ample line bundle on $X$ and $G$ be a reductive algebraic group and chosen a $G$ ...
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Divisibility by 2 of invariants forms on reductive Lie algebras and anomaly cancellation for gauge theories
Let $G$ be a connected reductive group over $\mathbb C$ and let $\rho:G\to \operatorname{Sp}(2n,\mathbb C)$ be a homomorphism. You can think about $\rho$ as a linear symplectic representation of $G$ ...
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Gindikin-Karpelevich formula in the quasi-split case
In Euler Products (1971), Langlands established the Gindikin-Karpelevich formula for split reductive groups over the local fields $\mathbb{Q}_p$. On the other hand, the archimedean Gindikin-...
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Can we compare $K$-spherical representations of $p$-adic groups for varying special maximal subgroups $K$?
In this question, I'm borrowing the notations from Minguez' paper on unramified representations of unitary groups. Let $F$ be a $p$-adic field and let $G$ be a connected reductive group over $F$. Let $...
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