All Questions
Tagged with reductive-groups rt.representation-theory
13 questions
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
9
votes
3
answers
677
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 ...
- 11.2k
27
votes
1
answer
3k
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 = \mathrm{...
- 805
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
14
votes
1
answer
1k
views
Definition of discrete spectrum and continuous and basic properties
I apologize if this is too basic for MO.
I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...
- 777
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
6
votes
2
answers
1k
views
Representations of GL(2, Q_p) and GL(2, Z_p)
The cuspidal representations of $\operatorname{GL}_n(F)$ a non archimedean field $F$ with ring of integers $o$ can be classified by inducing irreducible representation from $Z\operatorname{GL}_n(o)$.
...
- 11.2k
6
votes
2
answers
1k
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 $P(\mathbf{...
- 11.2k
6
votes
1
answer
312
views
Making sense out of intertwining operators defined by a vector valued integral
Let $G$ be the rational points of a connected, reductive group over a $p$-adic field $F$. Let $S$ be a maximal split torus of $G$ with $\Delta$ a set of simple roots corresponding to a minimal ...
- 6,180
4
votes
0
answers
288
views
Meaning of a highly ramified character for reductive groups
Let $F$ be a $p$-adic local field, and $G$ a connected reductive group over $F$. What is the meaning of a "highly ramified character" of $G(F)$? I have seen this terminology in many places in ...
- 6,180
3
votes
0
answers
282
views
Errata for Casselman's unpublished notes
In the first chapter of W. Casselman's unpublished notes on representation theory, there is at least one stated result which is not true:
A counterexample to this last result is given in the question ...
- 6,180
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
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