All Questions
43 questions
6
votes
4
answers
414
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 ...
- 3,446
13
votes
0
answers
509
views
Is there a simple proof that representations of GL(n,k) are determined by their restriction to diagonal matrices?
Let $k$ be a field of characteristic zero. The general linear group $\mathrm{GL}(n,k)$ has a subgroup $\mathrm{D}(n,k)$ consisting of invertible diagonal matrices. These are linear algebraic groups ...
- 22.3k
2
votes
1
answer
268
views
Stabilizer of a Levi subgroup in the Weyl group and its quotient
(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.)
For simplicity, let $G$ be a connected reductive ...
- 709
4
votes
1
answer
185
views
Canonicality of group of integers for reductive groups over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $G$ be a semisimple (but I think there is no obstruction to assume it to be reductive) algebraic group over a non-Archimedean local field $K$ and $\mathcal{O}_K$ be ...
- 6,018
4
votes
1
answer
299
views
Can any pair of associate parabolics be related by opposite parabolics?
Let $G$ be a reductive group, say over an algebraically closed field of characteristic zero.
We have the following definitions for a pair of parabolic subgroups $P_1$ and $P_2$ with Levi quotients $...
4
votes
0
answers
77
views
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 ...
- 12.9k
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
101
views
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 ...
- 2,751
3
votes
0
answers
203
views
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 "...
- 3,079
3
votes
1
answer
156
views
Automorphisms of étale-by-torus groups
Automorphisms of connected, reductive groups are well understood: the outer automorphism group is an essentially combinatorial object associated to the root datum. I am trying to understand ...
- 12.9k
3
votes
2
answers
221
views
Number of involutions in finite reductive groups
Let $G$ be a connected split reductive group over $\mathbb{Z}$. Let $n$ be a positive integer. Let $i_n(q)$ be the number of elements of $G(\mathbb{F}_q)$ satisfying $x^n=1$.
Question: Is there a &...
- 2,751
3
votes
1
answer
187
views
Regular embeddings of a reductive groups with induced center
Let $G$ be a reductive group over the finite field $\mathbb{F}_q$. Then a regular embedding of $G$ is an $\mathbb{F}_q$-rational embedding $\iota \colon G \rightarrow G'$ into a second reductive group ...
- 755
3
votes
1
answer
245
views
Can non-geometrically reduced reduced subschemes happen for reductive groups?
The title is meant to be punchy, but also a tongue-in-cheek acknowledgement of the prevalence of ‘reduce’-derived words in this area. (Unfortunately, I overlooked the fact that the question in the ...
- 12.9k
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
5
votes
0
answers
122
views
Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$
Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...
- 71
1
vote
0
answers
59
views
Distributivity property for smooth parabolic induction
Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $ B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. ...
- 473
3
votes
0
answers
248
views
Representation of Levi subgroup $L\subset P \subset G$
Let $G$ be a split connected reductive group over a finite field extension of $\mathbb{Q}_p$ with split maximal torus $T$ of rank $d$ and simple roots $\Delta$. Furthermore associated to $I\subset \...
- 473
9
votes
1
answer
545
views
Showing subgroups with equal Lie algebras are equal
Let $k$ be a field. It might as well be algebraically closed, but I do not want to assume that it has characteristic $0$. I will write "group" for "affine group scheme over $k$", ...
- 12.9k
7
votes
1
answer
239
views
Definition of functions in the induced space from parabolic induction
Let $P$ be a parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field. Let $M$ be a Levi subgroup of $P$, and let $N$ be the unipotent radical of $P$. If $(\pi,V)$ is a smooth, ...
- 6,180
5
votes
1
answer
222
views
Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?
Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
- 6,180
1
vote
0
answers
63
views
Question on the proof that the Jacquet module preserves admissibility
Let $P = MN$ be a parabolic subgroup of a reductive group $G$ over a $p$-adic field. For $(\pi,V)$ an admissible representation of $G$, the Jacquet module $(\pi_N,V_N)$ is defined by the action of $\...
- 6,180
1
vote
0
answers
141
views
Do we have $K \cap P = (K \cap M)(K \cap N)$?
Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...
- 6,180
1
vote
0
answers
230
views
Group schemes and Hyperspecial maximal compact subgroups
Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...
- 223
5
votes
1
answer
280
views
Integral structures via lattices
I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ...
- 223
5
votes
1
answer
372
views
Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper
In Kottwitz's 1985 Compositio paper,
Isocrystals with additional structure, first page, paragraph 4:
Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...
- 53
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
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
- 6,180
9
votes
1
answer
584
views
Endoscopic group that is not a subgroup
The question is a very little more than what's in the title. It is easy (for some values of ‘easy’) to produce examples of endoscopic groups that are not subgroups. When I asked a colleague, he ...
- 12.9k
17
votes
2
answers
3k
views
What's the point of a Whittaker model?
Let $G$ be a quasi-split connected reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup which is defined over $F$, with $B = TU$, $T$ defined over $F$. The choice of $T$ and $B$ ...
- 6,180
7
votes
1
answer
256
views
On existence of a certain irreducible character of $SL(5, q)$
Let $q=p^f$ be a prime power such that $q \equiv 1 \pmod 5$. According to the list of irreducible (complex) character degrees of $SL(5, q)$ in Frank Luebeck's homepage (here), $SL(5, q)$ has 20 ...
- 143
2
votes
0
answers
142
views
Unipotent characters of (disconnected) centralizers of semisimple elements: Why these two definitions are equivalent?
Assume that $\mathcal{G}$ is a simple simply-connected algebraic group over $k$, where $k$ is algebraic closure of a finite field of characteristic $p>0$, and $F$ is a Frobenius endomorphism. Let $(...
- 143
2
votes
0
answers
142
views
Iwahori subalgebra as maximal solvable
I think the following is true, but haven't came up with a proof myself. Thanks in advance!
Let $G$ be a semisimple (to avoid more words) algebraic group over $\mathbb{C}$. Write $F=\mathbb{C}((t))$ ...
- 1,856
3
votes
0
answers
145
views
Correspondence between dual center and linear characters of finite reductive group
Let $(G,F)$ be a connected reductive group defined over $\mathbb{F}_q$ via the Frobenius $F$ and let $(G^*,F^*)$ be a group in duality with $(G,F)$ with respect to rational maximal tori $T \subseteq G$...
- 309
8
votes
2
answers
1k
views
Examples to keep in mind while reading the book 'The Admissible Dual...' by Bushnell and Kutzko and the importance of Interwining of representations
I am a beginner in the field of representation theory. I was reading the book 'The Admissible Dual of $GL(N)$ Via Compact Open Subgroups' by Bushnell and Kutzko.
Let me first describe the book a ...
- 685
1
vote
0
answers
166
views
Conjugacy scheme, fppf versus GIT
I would be glad to have some guidance in the following.
Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
- 5,562
51
votes
2
answers
4k
views
Which philosophy for reductive groups?
I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be ...
- 5,424
3
votes
2
answers
471
views
Representations of complex semi-simple algebraic group "defined over $\mathbf{Z}$"?
If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over $\mathrm{Spec}(\...
- 163
2
votes
2
answers
663
views
Regular embeddings of reductive groups
A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where $G'$...
- 309
11
votes
2
answers
2k
views
Representation theory of the general linear group over a finite prime field
I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.
The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...
- 4,902
5
votes
1
answer
1k
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 ...
- 945
2
votes
0
answers
415
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 ...
- 1,457
3
votes
1
answer
611
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 unitary/...
- 11.2k
2
votes
2
answers
839
views
Possible Borel subgroups of GL_n?
I am trying to understand the interaction between Borel subgroups of $GL_n$ and its roots. Is it correct to say that for any choice of roots among each pair of reciprocal roots
there is a Borel ...
- 2,842