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
467 questions
2 votes
0 answers
114 views
The precise structure of the centralizer subgroup of an $\mathfrak{sl}_2$ in a complex simple Lie algebra
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 ...
- 731
5 votes
0 answers
160 views
Almost unipotent characters
Let us consider a split adjoint simple group $G$ over $\overline{\mathbb{F}_q}$. Then we have a Frobenius map $F$, and we can consider a finite group of Lie type, $G^F$. (We can assume that the ...
- 325
4 votes
1 answer
195 views
Congruent subgroups of $\mathrm{PGL}_2$ vs $\mathrm{SL}_2$
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(\...
- 129
0 votes
0 answers
116 views
Haar measures on reductive groups via top degree differential forms
Let $G$ be a connected reductive group over a local non archimedean field $k$. An invariant differential form is an element of dual of the $\wedge^{top} Lie(G)$. Let $\omega_G$ be an invariant ...
- 253
14 votes
1 answer
407 views
What changes for reductive groups when the smoothness assumption is dropped for the unipotent radical?
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 ...
- 284
2 votes
1 answer
173 views
Center and Levi subgroups
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 ...
- 1,025
4 votes
0 answers
198 views
$B(\mathcal{O})$-orbits in the affine Grassmannian
Let $G$ be a complex reductive group with Borel subgroup $B$, $\mathcal{O}=\mathbb{C}[\![t]\!]$, $\mathcal{K}=\mathbb{C}(\!(t)\!)$ and $\operatorname{Gr}_G=G(\mathcal{K})/G(\mathcal{O})$ the affine ...
- 4,087
5 votes
2 answers
528 views
Closed $G$-orbits in an affine variety
Let $k$ be an algebraically closed field (of characteristic $2$ in the case I am interested in). Let $G$ be a connected reductive group over $k$ and $X$ be an affine variety over $k$, on which $G$ ...
- 1,968
10 votes
1 answer
430 views
Plücker relations for generalized flag varieties
Let $G$ be a complex simply-connected semisimple group. Then I know that the flag variety of $G$ can be described as the subscheme of $\prod_{\lambda} \mathbb{P}(V_\lambda)$ (with the product over all ...
- 4,087
4 votes
1 answer
327 views
Invariants of a maximal unipotent group in GL(n) acting by conjugation on n by n matrices
Let $G = \operatorname{GL}(n, \mathbb{C})$ and let $U \subset G$ be the usual maximal unipotent subgroup (upper triangular matrices with ones on the diagonal). Let $U$ act by conjugation on the space $...
- 41
4 votes
1 answer
216 views
Compact quotients of $p$-adic reductive groups
Fix a prime $p$ and let $G$ be a connected reductive group over a $p$-adic local field $K$. Let $H \subset G$ be a connected reductive subgroup such that $G(K)/H(K)$ is compact in the $p$-adic ...
- 579
1 vote
0 answers
125 views
Examples of Harish-Chandra Schwartz functions coming from tempered representations
Let $F$ be a local field and (for simplicity) $G$ a semisimple group split over $F$. Then for in both the archimedean and non-archimedean cases, one has the space $\mathcal{C}(G)$ of Harish-Chandra ...
- 513
1 vote
1 answer
181 views
Restriction of homogeneous line bundles on G/B to P/B
Let $G$ be a semisimple simply connected algebraic group over $\mathbb C$ and let $B\subseteq G$ be a Borel subgroup. Furthermore let $T\subseteq B$ be a fixed maximal torus, $\lambda \in X^*(T)$ an ...
- 177
4 votes
0 answers
103 views
Tits's proof that order-$3$, anisotropic elements normalise a maximal split torus in characteristic $3$
In "Unipotent elements and parabolic subgroups. II", on p. 266, Tits says: Under these conditions [that $G$ is quasi-simple and $K$-split], we conjecture that any anisotropic element of ...
- 14k
5 votes
1 answer
383 views
Is it possible to construct or characterize the integral parahoric group schemes before defining parahoric subgroups abstractly?
I am a novice in Bruhat-Tits theory, so I apologize if I misrepresent something. As far as I can determine the theory is ordered as follows (say, for a quasi-split reductive group over a ...
- 736
2 votes
0 answers
82 views
Existence of quasi-invariant smooth function with eigencharacter for reductive Lie group
What is known about existence of quasi-invariant smooth function with some eigencharacter on Lie algebra of a reductive Lie group? Consider reductive Lie group $G$ and its Lie algebra $\mathfrak{g}$. ...
- 131
3 votes
0 answers
97 views
Reductivity of integral model associated to a concave function in Bruhat--Tits theory
Let $G$ be a reductive group over a henselian discretely valued field $k$ whose ring of integers is denoted by $\mathfrak{o}$. Let $\mathscr{B}$ be the (extended) Bruhat--Tits building. We consider a ...
- 383
3 votes
1 answer
156 views
Two definitions of Moy–Prasad filtration of induced torus
Let $L/K$ be a field extension of two local fields. Let us consider an induced torus $T= \text{Res}_{L/K}(\mathbb{G}_m)$. For $n\in\mathbb{Z}_{+}$, we have two definition of the Moy--Prasad filtration ...
- 383
12 votes
1 answer
1k views
What is the answer to the Kneser–Tits problem over a finite field?
Let $k$ be a field and $G$ a reductive $k$-group. The Whitehead group of $G$ is defined as the quotient $$W(k,G) := G(k)/G(k)^+$$ where $G(k)^+$ is the subgroup of $G(k)$ generated by the $k$-rational ...
- 977
6 votes
0 answers
101 views
Computing extensions of irreducible (g,K)-modules (with same infinitesimal characters)
Let $G$ be a real connected reductive group with Lie algebra $\mathfrak{g}$, and $K$ be a maximal compact subgroup with Lie algebra $\mathfrak{l}$. All irreducible admissible $(g,K)$-modules could be ...
- 7,442
5 votes
1 answer
324 views
Invariants of tuples of matrices under $\mathrm{GL}(p)\otimes \mathrm{GL}(q) \subseteq \mathrm{GL}(n)$?
$\DeclareMathOperator\GL{GL}$Consider $\GL(n)$ over some field of characteristic zero (I'm thinking of either the rationals, reals or complexes) and the subgroup $\GL(p)\otimes \GL(q)$ which embeds ...
- 235
5 votes
1 answer
161 views
An algebraic group $G$ over $L^+$ such that $G_{\mathbb{R}}$ is compact for almost all embeddings
Does there exist a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is ...
- 127
5 votes
1 answer
364 views
Galois action on Borovoi's algebraic fundamental group
In Borovoi's paper Abelian Galois cohomology of reductive groups, the algebraic fundamental group of a connected reductive group $G$ over a field $K$ of characteristic zero is defined as $$\pi_1(G, T):...
- 53
6 votes
1 answer
327 views
Centralizers in semisimple Lie group
For a semisimple complex Lie algebra $\mathfrak{g}$ and a regular element $X\in \mathfrak g$ the centralizer of $X$ in $\mathfrak g$ is a Cartan subalgebra (see Knapp, 'Lie Groups beyond an ...
- 63
4 votes
0 answers
132 views
Levis, parabolics and Bruhat-Tits over Henselian local rings
Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$. The paper "Reductive ...
- 7,442
3 votes
0 answers
329 views
What do we do when $G$ doesn't have a Shimura variety?
Let $G$ be a reductive group. If one can associate to $G$ a Shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for the ...
user546695
1 vote
0 answers
44 views
Behavior of the number of components of disconnected reductive groups when intersecting a Levi subgroup
Let $G$ be a connected reductive group over $\mathbb{C}$. Let $P=MN$ be a parabolic subgroup of $G$ with its Levi decomposition ($N$ the unipotent radical, $M$ a Levi). Let $H\subset M$ be a finite ...
- 731
5 votes
1 answer
286 views
Representations with finitely many nilpotent orbits
Let $G$ be a reductive group over $\mathbb{C}$ and let $V$ be a finite dimensional representation of $G$. We can define the ``nilpotent cone'' of $V$ as $$\mathcal{N}(V):=\{ v\in V\;: \; 0\in\overline{...
- 2,611
7 votes
1 answer
347 views
What are the intermediate semisimple groups of type A?
Background: The first examples one sees of reductive groups over a field $k$ are $\text{GL}_n$, $\text{SL}_n$, and $\text{PGL}_n$. We all know the definitions of $\text{GL}_n$ and $\text{SL}_n$, and ...
- 977
1 vote
0 answers
79 views
Linear algebraic group, absolute root system, computing roots
Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
- 11
2 votes
0 answers
130 views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 3,346
3 votes
0 answers
80 views
Root systems of maximally noncomact Cartan subalgebras
Let $G$ be a real reductive Lie group, and $K$ a maximal compact subgroup in $G$. Write $\mathfrak{g}$ for the Lie algebra of $G$, and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the Cartan ...
- 961
2 votes
0 answers
207 views
Centre of centralisers in connected reductive groups
Let $G$ be a connected reductive group over an algebraically closed field. Let $T$ be a maximal torus and $x\in T$. Let $G_x$ denote the centraliser of $x$ in $G$. Question: What is an explicit ...
- 3,003
7 votes
1 answer
188 views
Intersection of integral points with a unipotent and its opposite
This is a follow-up to Does the bruhat decomposition induces decomposition on integral points (on an open cell)? Given a split connected reductive group $G$ over a $p$-adic local field $F$ with ring ...
- 505
3 votes
0 answers
182 views
Does the Bruhat decomposition induces decomposition on integral points (on an open cell)?
Edit: both questions are resolved in comments. Let $F$ be a local field and $O$ its integral points. Let $G$ be a split reductive group over $O$. The Bruhat decomposition states that there is a ...
- 460
2 votes
0 answers
69 views
Nonvanishing of Jacquet modules of principal series subquotients
Let $G$ be a connected reductive group over a $p$-adic field $F$. For simplicity, we assume $G$ to be split. Fix a Borel $B=TU$ with its Levi decomposition ($U$ unipotent radical, $T$ maximal torus). ...
- 731
1 vote
0 answers
112 views
Asymptotic behavior of Shalika germs near non-regular elements
Let $G$ be a connected reductive group over a $p$-adic field $F$. Let $T\subset G$ be a maximal torus. Fix a special maximal compact subgroup $K$ of $G(F)$ and for any closed subgroup $H\subset G(F)$ ...
- 731
3 votes
1 answer
229 views
Factoring out an element of a root subgroup to make a conjugation integral
Fix a nonarchimedean local field $L/\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, uniformizer $\varpi$, and residue field $k$. If I take a matrix $$\begin{pmatrix} a & \varpi b \\ c & d \...
- 505
2 votes
0 answers
96 views
Number of rational points of a connected reductive group in a compact subset
Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...
- 71
4 votes
1 answer
237 views
Centralizer of conjugacy classes
Let $\mathrm{G}$ be a complex reductive group and let $\mathrm{O}_g$ be the adjoint orbit of $g\in \mathrm{G}$. I wonder is the centralizer $\mathrm{C}_{\mathrm{G}}(\mathrm{O}_g)$ still a reductive ...
- 563
4 votes
1 answer
167 views
Do parabolic inductions share a composition factor if and only if the inducing data are associate?
Let $F$ be a local field of characteristic zero and $G$ a connected reductive group over $F$. Let us call an inducing datum a triple $(P,M,\sigma)$, where $P$ is a parabolic subgroup of $G$, $M$ is a ...
- 387
3 votes
1 answer
542 views
Normalizer of Levi subgroup
Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$. Associated with this ...
- 73
7 votes
1 answer
392 views
Nilpotent orbits of a parabolic subgroup
Suppose $G$ is a reductive group over an algebraically closed field of characteristic $0$ with parabolic $P$, Levi quotient $M$, and unipotent radical $U$. We denote the nilpotent elements of $\mathrm{...
- 963
6 votes
4 answers
569 views
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group). By the Tannakian formalism, $G(k)$ can be ...
- 4,087
1 vote
0 answers
125 views
Injection of $G(k)/Z(k)$ into $(G/Z)(k)$
In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...
- 131
3 votes
0 answers
133 views
Representations of a reductive Lie group vie central character and K-types
Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...
- 721
3 votes
0 answers
125 views
$\mathbb{Z}_p$-points of a $\mathbb{Z}_p$-model of a reductive linear algebraic $\overline{\mathbb{Q}}_p$-group
Let $G$ be a (connected) reductive linear algebraic group over $\overline{\mathbb{Q}}_p$. By definition, this means that $G$ is a closed subgroup of some $\mathrm{GL}_n$. We can always find a ...
- 31
4 votes
3 answers
346 views
Does every nilpotent lie in the span of simple root vectors?
Let $G$ be a reductive group (for simplicity, we can work over $\mathbb{C}$). Let $N \in \operatorname{Lie} G$ be nilpotent. Does there exist a Borel pair $(B,T)$ of $G$ such that $N$ lies in the span ...
- 963
7 votes
1 answer
701 views
Parabolic subgroups of reductive group as stabilizers of flags
$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
- 627
4 votes
1 answer
243 views
Are isomorphic maximal tori stably conjugate?
Let $F$ be a field and $G$ a reductive $F$-group. For various applications it is important to understand the "classes" of maximal ($F$-)tori of $G$. Here "class" can mean the ...
- 977