The reductive-groups tag has no wiki summary.
1
vote
1answer
48 views
Strictly contracting elements in the center of a Levi subgroup
Let $G$ be a connected reductive group over a non archimedean local field $k$.
Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z_M \subset L$ be the center of $M$ and $S_M ...
1
vote
0answers
74 views
Clarification about Tits' article in the Corvallis
I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ ...
3
votes
1answer
92 views
Supercuspidal with Iwahori fixed vector
Let $F$ be a local field. Is there a reference for the following fact:
No supercuspidal representation of $GL_2(F)$ has an Iwahori-fixed vector?
I have a proof, by I'd prefer a reference, ...
0
votes
1answer
49 views
Regular or elliptic elements in the multiplicative group of central division algebra
For an element $g$ of a connected reductive group $G$ over a field $F$,
$g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$,
$g$ is ...
3
votes
1answer
161 views
What is “special” maximal compact subgroup of algebraig group over local field?
Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...
1
vote
1answer
103 views
Compatibility of two definitions of elliptic elements in GLn
For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...
0
votes
1answer
153 views
number of simple representations
For a linearly reductive Group $G$ over a field $k$ one has that the category of finite dimensional representations of $G$ is semisimple. What can one say about the number of simple representations? ...
1
vote
0answers
109 views
Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field
Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...
7
votes
1answer
344 views
Why people usually consider reductive groups in GIT?
Where do people essentially use the reductive groups in the theory of GIT? Or how does reductive groups simplify the constructions in GIT?
I found that the property of completely reducible of ...
5
votes
2answers
245 views
Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics
The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear ...
7
votes
1answer
191 views
Open cell decomposition after applying a Weyl group element
Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step.
For ...
2
votes
0answers
74 views
Invariant vectors in supercuspidal representations of GL_2(Zp)
Let $o$ be the ring of integers in a local field $F$ with prime-ideal $p$. Let $K$ be either $GL_2(o)$ or the normalizer of the Iwahori subgroup. Let $\sigma$ be a representation of $K$ times the ...
3
votes
1answer
110 views
Semisimple group not split by an unramified extension?
Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?
6
votes
2answers
300 views
Reference request: expository text on the structure of reductive groups over non-archimedean local fields
I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner ...
3
votes
0answers
124 views
How to think about non-connected reductive groups
Suppose someone knows well the theory of connected reductive groups, over an algebraically closed field or more generally over any field, say for instance most of the content of Borel-Tits.
Is ...
4
votes
1answer
245 views
How to translate the representation theory of semisimple to reductive groups?
I am aware of the following question: Definitions of Reductive and Semisimple Groups
So let me phrase a precise question:
Is there a standard technique by which one can translate the ...
5
votes
2answers
195 views
Principal series of finite group of Lie type
I have a naive question on complex representations of finite groups of Lie type.
Let $\bf G$ be a reductive group (say connected, with connected center, for safety)
defined over a finite field ...
2
votes
0answers
119 views
Decomposition of k-split tori of p-adic reductive groups
Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$.
There is a group homomorphism :
...
2
votes
1answer
209 views
Weyl group of the restriction of scalars of split reductive group
Let $G$ be a connected algebraic group defined over a field $E$ of characteristic $0$. Suppose $G$ reductive $E$-split and let $T \subset G$ a maximal (split) torus defined over $E$.
Set $G' = ...
4
votes
1answer
170 views
Steinberg reps of reductive groups over local fields vs finite fields
Let $G$ be a reductive group over a non-archimedean field $F$ with reisdue field $f$.
Edit: The statements only make sense modulo tensoring by one-dimensional representations.
Are the unitary, ...
2
votes
2answers
183 views
a conjugacy question in quasi-split reductive groups
I have a somewhat technical question about conjugacy in
quasi-reductive groups.
Let $k$ be a field (in my main case interest, $k$ is finite), $G$ be a connected
quasi-split reductive group over ...
21
votes
1answer
1k views
Reconciling Lusztig's results with the Langlands philosophy
Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = ...
20
votes
1answer
578 views
What are the possible motivic Galois groups over $\mathbb Q$?
Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...
3
votes
1answer
214 views
Volume of PGL(2,F) \ PGL(2, A)
Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$?
This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...
1
vote
0answers
127 views
Do there exist pseudo-reductive (but not reductive) groups of small dimension?
I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is ...
10
votes
4answers
786 views
Is the normalizer of a reductive subgroup reductive?
Let $G$ be a reductive algebraic group over an algebraically closed field (of characteristic zero if it matters) and $H \subset G$ a subgroup, also reductive. Is the identity component of the ...
4
votes
1answer
254 views
Does there exist a categorical treatment of root data(systems)?
What I am looking for is an abstract description of root data with their morphisms(!) plus a comparison with the categories of reductive groups over some field, Dynkin diagrams, Lie algebras, ...
4
votes
3answers
274 views
Are all irreducible supercuspidal representation induced from compact-mod-center subgroups?
Let $G$ be a reductive group over a local non-archimedean field $F$.
Can every irreducible supercuspidal representation of $G(F)$ be realized as the induction from an open subgroup, which is compact ...
2
votes
2answers
329 views
description of an endomorphism algebra
Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).
I ...
7
votes
2answers
461 views
commuting elements in a reductive group
Does anyone know if the following holds?
Conjecture: Any two commuting elements in a reductive algebraic group G over C of rank>1 lie in a proper parabolic subgroup of G.
To make things easier, you ...
2
votes
1answer
220 views
Pseudo coefficients and orbital integrals
I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained:
"We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, ...
1
vote
1answer
195 views
Heights in reductive groups
Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular ...
3
votes
1answer
159 views
homogenous bundles
Let $G$ be a reductive algebraic group over $\mathbb{C}$ and $H$ be an algebraic subgroup of $G$. We suppose that $H$ acts on some scheme $S$, where $S$ is of finite type over $\mathbb{C}$. Then I ...
1
vote
1answer
385 views
moduli problem for flag varieties?
Hi,
Suppose $G$ is a reductive group over an algebraiclly closed field $k$
(suppose $k$ of char zero if you want at first). Let $X$ be its flag variety.
Question: What is the moduli problem that $X$ ...
1
vote
1answer
224 views
Intertwining Integral defined on a Weyl group?
Why does the intertwining integral such as the one defined in A. W. Knapp's paper "Intertwining operators for semisimple groups" depend only on an element w of a Weyl group?
...
2
votes
1answer
411 views
When is compact induction in GL(2) from an open compact group admissible?
Let $G$ be a locally profinite group and $K$ an open compact subgroup (mod the center), then Bushnell has shown that the following three statements are equivalent for a finite dimensional ...
2
votes
1answer
203 views
Abel transform is an * isomorphism for SL(2, R)
Assume we conisder $G= SL(2, R)$, $K=SO(2)$ and $N$ the strict upper triangular matrices in $G$, $A$ diagonal matrices, and the Borel supgroup $B=NA$, $W$ Weyl group.
Then we have an isomorphism of ...
3
votes
1answer
335 views
Canonical rational form for $SL(n)$
The canonical rational form helps us to parametrize the conjugacy classes in $GL(n)$ over any commutative field.
How can we parametriize the conjugacy classes in $SL_n(k)$, where $k$ is an ...
2
votes
1answer
219 views
Representation theory of G1 versus G/Z
Let $G$ be an locally compact group $G$, then every irreucible representations $\pi$ is isomorphic to $\omega_{\pi} \otimes \pi'$, where $\omega_{\pi}$ is the central character of $\pi$ and $\pi'$ an ...
1
vote
2answers
168 views
What is this measure on the affine Weyl group?
Let $G$ be $SL(n, F)$ for a non-archimedean local field $F$ with Iwahori subgroup $I$. Let $\mu$ be the Haar measure of $G$.
What are the properties of the function $w\mapsto \mu(IwI)/\mu(I)$ for ...
7
votes
2answers
547 views
Conjugation in GL(n) (p-adic setting)
In $GL(n, \mathbb{Q}_p)$, what are the orbits under conjugation of $GL(n, \mathbb{Z}_p)$?
5
votes
2answers
610 views
Parabolic induction GL(n,Zp)
Let $P$ be a parabolic subgroup of $GL(n)$ with Levi decomposition $P =MN$, where $N$ is the unipotent radical.
Let $\pi$ be an irreducible representation of $M(\mathbf{Z}_p)$ inflated to ...
3
votes
2answers
459 views
Representations of GL(2, Q_p) and GL(2, Z_p)
The cuspidal representations of $GL_n(F)$ a non archimedean field $F$ with ring of integers $o$ can be classified by inducing irreducible representation from $Z GL_n(o)$.
The general question:
...
2
votes
2answers
381 views
Parabolic subgroups and BN-pairs
We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put also, $N=N_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BN-pair of ...
2
votes
0answers
156 views
Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups
If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
4
votes
1answer
442 views
Character determines the representation?
Consider a semisimple Lie group or a $p$ adic reductive group $G$.
To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?
3
votes
2answers
243 views
Twisted Gelfand pairs (Reference and examples)
Let $G$ be a locally compact group and let $K$ be a compact group. Let $(\tau, V_\tau)$ be an irreducible representation of $K$.
We consider the space of $Endo_K(\tau)$-valued, compactly supported ...
4
votes
0answers
802 views
Cartan decomposition for upper triangular matrices
Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...
1
vote
1answer
187 views
Algorithm for the cell multiplication rule for GL(n,F)
Consider $F$ a non archimedean field and let $o$ be its ring of integer
Let $B$ be the Iwahori subgroup of $GL_n(F)$ (resp. $GL_n(o)$) and let $N$ be the normalizer of the diagonal matrices ...
2
votes
2answers
260 views
What is the normalization factor for $GL_n( \mathbb{Q}_p) // GL_n( \mathbb{\mathbb{Z}_p})$?
Consider $G = GL_n( \mathbb{Q}_p)$ and $K = GL_n( \mathbb{Z}_p)$, or more favorable replace $ \mathbb{Q}_p$ by a non archimedean field and $\mathbb{Z}_p$ by the ring of integers.
Note that the space ...
