# Tagged Questions

**2**

votes

**2**answers

250 views

### Density of characters

Did Harish-Chandra prove that characters of irreducible representations of a $p$-adic reductive group $G$ span a dense subspace of the space of conjugation-invariant distributions on $G$? What is the ...

**1**

vote

**0**answers

118 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 ...

**6**

votes

**5**answers

464 views

### Is a unitary representation always semisimple?

I have been reading the online lecture notes by Fiona Murnaghan
http://www.math.toronto.edu/murnaghan/courses/mat1197/notes.pdf
The first lemma in p.35 says that every unitary representation of ...

**4**

votes

**1**answer

149 views

### When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$.
Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...

**3**

votes

**0**answers

90 views

### parametrization of irreducible finite dimensional representation of Weil group

Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know ...

**2**

votes

**0**answers

117 views

### Correct definition of locally algebraic parabolic induction of a locally algebraic character

Let $L$ be a finite extension of $\mathbf{Q}_p$ and $G$ the group of $L$-points of a split connected reductive group $\mathbf{G}$ over $L$, $T$ the $L$-points of a split maximal torus in $\mathbf{G}$, ...

**2**

votes

**1**answer

329 views

### On a result due to Zelevinskii

In his paper on the p-adic analogue of the Kazhdan-Lusztig hypothesis (Functional Analysis and Its Applications 15.2 (1981): 83-92), Zelevinskii proves a combinatorial proposition (outlined in Section ...

**1**

vote

**1**answer

113 views

### Modules for an idempotented algebra

Recall that an associative algebra $A$ is called idempotented provided that is the filtered union of subalgebras $eAe$ for $e \in A$ idempotent. I think sometimes people say that $A$ has approximate ...

**8**

votes

**2**answers

367 views

### Proving that some principal series representations of SL(2,F) are irreducible

I am sorry in advance if this question is not "research level".
Let $F$ be a p-adic field.
I saw, in Bumps book, a proof which I liked, showing which principal series representations of $GL(2,F)$ ...

**1**

vote

**0**answers

90 views

### Restriction and then induction of the Steinberg representation of GL(n)

Let $G_{n}=GL(n,F)$, where $F$ a locally compact non-Archimedean field, $St_{G_{n}}$ the Steinberg representation of $G_{n}$, and $B$ the standard Borel subgroup of $G_{n}$.
We denote $\pi_{n}$ the ...

**2**

votes

**2**answers

201 views

### fixed vector of a generic representation of GL(n,F)

Let $F$ be a locally compact non-archimedean field and $G_{n}$ the locally profinite group $GL(n,F)$.
Let $\Gamma_{n,k}$ be the subgroup of $G_{n}$ whose elements are the matrices of the form
$$ ...

**3**

votes

**0**answers

141 views

### p-adic Lie group vs Lie algebra cohomology with mod p coefficients

My question concerns the cohomology of a compact $p$-adic Lie group $G$ (wich is pro-$p$).
Let $M$ be a finite dimensional $\mathbb{Q}_p$-vector space with continuous linear $G$-action.
Lazard ...

**6**

votes

**0**answers

187 views

### Intersections of anisotropic tori with split Levi subgroups

Let G be a connected reductive group defined and split over a finite field k with Frobenius morphism F. Let T be an F-stable minisotropic maximal torus. Let P be an F-stable proper parabolic ...

**6**

votes

**1**answer

158 views

### When is a Moy-Prasad filtration subgroup the stabilizer of a subset of the building (up to center)?

Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an ...

**4**

votes

**1**answer

336 views

### Representations of reductive groups over local fields through parahoric induction

Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as ...

**3**

votes

**0**answers

141 views

### Reference request - Jacquet module and asymptotic of matrix coefficients

Hello,
I would like to know some nice references about the relation between asymptotics of matrix coefficients of representations of reductive groups over local fields, and the pairing between the ...

**4**

votes

**4**answers

607 views

### cuspidal types and Iwahori subgroup for $SL(2,F)$

Let $(J,\pi)$ be a cuspidal type in $SL(2,F)$, $F$ is a non-Arch. local field and let $I$ be the Iwahori subgroup of $SL(2,F)$.
Is there any possibility that $J\subset I$ or even a subgroup?

**3**

votes

**1**answer

603 views

### Discrete Series representations for $SL_{2}$ over $p$-adic field.

I am working on the chamber homology for $SL(2,F)$, and stuck at some basic stuff on D.S. reps of $SL(2,F)$.
Let $ I=\left(
\begin{array}{cc}
\mathcal{O}_{F} & \mathcal{O}_{F} \\
...