All Questions
Tagged with reductive-groups lie-groups
62 questions
6
votes
0
answers
86
views
Generic representations of $\mathrm{GL}_n(\mathbb{R})$
Let $F$ be a local field of characteristic $0$, $G=\mathrm{GL}_n(F)$.
When $F$ is $p$-adic, Bernstein and Zelevinsky classified the irreducible generic representations. The statement is:
Let $\delta_{...
- 709
6
votes
1
answer
158
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
1
vote
0
answers
18
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 ...
- 709
4
votes
0
answers
87
views
Doubling constructions beyond classical groups: general principles?
The doubling method for constructing integral representations of L-functions has been successfully applied to classical groups, as demonstrated in this paper. However, extending this method to a wider ...
- 111
2
votes
0
answers
97
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 $C_G(g)$ ...
- 2,907
3
votes
0
answers
50
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 ...
- 951
2
votes
0
answers
157
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 ...
- 2,751
4
votes
1
answer
170
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 ...
- 395
7
votes
1
answer
335
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{...
- 953
1
vote
0
answers
95
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
107
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-...
- 460
4
votes
3
answers
274
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 ...
- 953
3
votes
0
answers
65
views
One parameter subgroups of reductive algebraic groups
If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
- 63
2
votes
0
answers
110
views
On the character of a representation of $\mathrm{GL}(n,\mathbb{R})$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ad{Ad}$Let $G=\GL(n,\mathbb{R})$. Given a continuous admissible irreducible representation of $G$ in a Frechet (or a Banach) space. Then its character ...
- 21.8k
7
votes
2
answers
314
views
Holomorphic discrete series vs. discrete series
(I apologize in advance if this question is too naive for experts.) Let $G$ be a real semisimple Lie group. I know that holomorphic discrete series representations are only a part of all the discrete ...
- 709
0
votes
1
answer
175
views
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 $\...
- 833
2
votes
0
answers
71
views
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/...
- 21
8
votes
0
answers
267
views
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 ...
- 1,856
2
votes
0
answers
160
views
Principal bundles with no trivializable extensions
Let $Q \to M$ be a principal $G$-bundle. Given a homomorphism $\phi: G \to H$, we can ‘extend the structure group’ of $Q$ to $H$, by defining an associated principal $H$-bundle: $Q_{H} := (Q \times H)/...
- 491
6
votes
1
answer
678
views
Cartan decomposition of loop group
Let $G$ be a complex reductive group. Let $LG$ and $L^+ G$ denote the formal loop spaces given by maps from the punctured formal disk and the formal disk, respectively, to $G$. The quotient $LG/L^+ G$ ...
- 553
3
votes
3
answers
581
views
Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
- 143
3
votes
0
answers
40
views
Generating $K$-types of a $(\mathfrak g,K)$-module for $K$ disconnected
Let $G$ be a real reductive Lie group, let $K$ be a maximal compact subgroup of $G$, and let $V$ be a $(\mathfrak g,K)$-module. For $\sigma\in\widehat{K}$ we denote the $\sigma$-isotypic component of $...
- 741
4
votes
2
answers
120
views
When representations of reductive Lie group in a Banach space and in its Garding space have the same length?
Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of ...
- 21.8k
3
votes
1
answer
284
views
Schubert cells in G/P for reductive G
All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
- 1,677
3
votes
0
answers
190
views
Harmonic analysis on reductive groups over $\mathbb{R}$
A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...
- 131
4
votes
1
answer
700
views
Centralizers of semisimple subgroups
$\DeclareMathOperator\GL{GL}$If $G$ is a simple Lie group, and $\rho: G \to \GL(V)$ is a representation, then by Schur's lemma, the group of automorphisms of $\rho$ is a reductive subgroup of $\GL(V)$....
- 491
2
votes
0
answers
190
views
Conjugacy classes in centralizers
Let $G$ be a complex reductive group, let $g$ be an element, and let $C$ be the connected component of its centralizer. I'm curious about what is known about the intersection of conjugacy classes in $...
- 491
15
votes
2
answers
2k
views
Why are coroots needed for the classification of reductive groups?
As we know reductive groups up to isomorphism corresponds to root data up to isomorphism. My question is why in the definition of root data do we need the coroots?
Let's break it down to two questions:...
- 2,071
7
votes
1
answer
371
views
Gelfand pair, weakly symmetric pair, and spherical pair
I am a bit confused with the relations among Gelfand pairs, weakly symmetric pairs, and spherical pairs defined in the book "Harmonic analysis on commutative spaces" written by professor ...
- 951
2
votes
0
answers
175
views
Jordan normal form in a reductive group
Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
- 491
4
votes
0
answers
211
views
Books on integration on semisimple Lie groups
Can anyone suggest me some good books where I can find integration theory on semisimple Lie groups (using KAK, KAN and other type of decompositions)?
I have read Knapp's book "Lie groups beyond ...
- 1,879
3
votes
0
answers
252
views
Reductive Lie groups and existence of maximal compact subgroup
I am reading Knapp's book "Lie groups beyond an introduction" (2nd edition). I am struggling to understand the following point. Recall that $G$ is a reductive Lie group. If the Lie algebra $\...
- 1,879
3
votes
1
answer
578
views
Conjugacy of elements in a parabolic subgroup
Let $G$ be a complex connected reductive group, and let $P \subseteq G$ be a parabolic subgroup. My question is the following: if $g$ and $h$ are elements of $P$ which are conjugate as elements of $G$,...
- 491
6
votes
1
answer
541
views
Is the connected centralizer of a semisimple element in a connected reductive group also a centralizer?
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field and let $g\in G$ be semisimple. Write $C=\mathrm{C}_G(g)$ and $C^\circ=\mathrm{C}_G(g)^\circ$ for the ...
- 993
10
votes
1
answer
375
views
Invariants for $SO_n \backslash \mathfrak{gl}_n / SO_n$
Is there a nice theorem about the algebra of invariants $\mathbb{C}[\mathfrak{gl}_n]^{SO_n \times SO_n}$, where the action is by left and right multiplication? I'm hoping for something along the lines ...
- 4,991
4
votes
0
answers
105
views
Siegel Levi in $\operatorname{GSpin}(2n+1)$ and image into $\operatorname{SO}(2n+1)$
Let $T$ be a maximal torus of split $\operatorname{SO}_{2n+1}$ with basis $e_1, ... , e_n$. Let $$\Delta = \{e_1 - e_2, ... , e_{n-1} - e_n, e_n\}$$ be a set of simple roots of $T$ corresponding to a ...
- 6,180
1
vote
0
answers
100
views
Embedding of discrete series
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals that of its maximal compact subgroup. Let $G'$ be a reductive subgroup of $G$ with equal rank. If $\tau$ is a discrete ...
- 951
9
votes
2
answers
419
views
$G(k)/H(k)$ as a submanifold of $G/H(k)$
Let $k$ be a local field (if necessary, assume characteristic zero). In general, if $X$ is a smooth variety of finite type over $k$ of dimension $n$, then the set of $k$-rational points $X(k)$ is an ...
- 6,180
4
votes
1
answer
324
views
Possible groups appearing in a Shimura datum
Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for ...
- 61
4
votes
1
answer
1k
views
Complexification of compact Lie groups and complex algebraic linear reductive groups
I'm studying complexifications of compact Lie groups on "Representation of compact Lie groups- Dieck Brocker".
I found on internet that there is a bijection between complexifications of compact Lie ...
- 73
2
votes
1
answer
271
views
Discrete decomposability of unitary representation
[INTRODUCTION]
Let $G$ be a non-compact simple Lie group, and $G'$ a reductive subgroup of $G$. Suppose that $\pi$ is a non-trivial (hence, infinite dimensional) irreducible unitary representation of ...
- 951
5
votes
0
answers
210
views
rational cohomology of classifying spaces of complex reductive Lie groups
I am looking for a reference or an ad-hoc proof of the following fact, which seems to be known to experts: Let $\mathbf{G}$ be a complex algebraic group with maximal (algebraic) torus $\mathbf{T}$ and ...
- 2,928
3
votes
1
answer
396
views
Choosing canonical representatives of Weyl group elements, some questions
Let $G$ be a connected, reductive group which is quasisplit over a field $k$ of characteristic zero. Let $B$ be a Borel subgroup defined over $k$, containing a maximal torus $T$ defined over $k$. ...
- 6,180
8
votes
3
answers
699
views
Centralizers of subtori in reductive groups, derived subgroups
Let $G$ be a split, almost-simple connected reductive group over a field $F$ with split maximal torus $T$. I am trying to understand precisely the groups $[G_{\alpha}, G_{\alpha}]$, where $\alpha$ is ...
- 123
6
votes
0
answers
1k
views
Definition of Admissible Representation
Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...
- 6,180
1
vote
0
answers
955
views
Iwasawa decomposition and compact subgroups
Let $G$ be the $k$-points of a connected, reductive group $\mathbf G$ over a local field $k$. I have heard several statements about compact subgroups and Iwasawa decomposition, mostly in the context ...
- 6,180
8
votes
1
answer
167
views
Symmetries of the flag variety
Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\mathcal B=G/B$ be the associated Flag variety.
Is it true that the obvious map
$$
\mathfrak g\to \Gamma (T\...
- 43.2k
4
votes
1
answer
349
views
Existence of lattices in reductive Lie groups
What is known about existence of lattices in reductive Lie groups? The best results I know about existence of lattices in connected Lie groups are either about semisimple groups or nilpotent groups ...
- 43
2
votes
1
answer
280
views
Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$....
- 1,111
2
votes
1
answer
661
views
Unipotent conjugacy classes
Consider a connected reductive group G over the complex numbers. Is there a `simple' formula for the number of conjugacy classes of unipotent elements in G?
- 1,190