# Tagged Questions

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94 views

### quasi-split algebraic group [migrated]

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...

**2**

votes

**1**answer

162 views

### Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
...

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**0**answers

93 views

### Arthur's refinement of parameters for unitary automorphic representations

In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...

**5**

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**3**answers

485 views

### A question on non-archimedian Fourier transform

Let $M(n)$ be the vector space of $n\times n$ matrices over a local non-archimedian field $K$.
Let $\mathcal S$ denote the space of locally constant compactly supported functions
on $M(n)$. Similarly, ...

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**0**answers

58 views

### nonvanishing of global theta lifting from U(1) to U(1?)

I understanding nonvanishing of theta lifting, either global or local, is a difficult and open problem. But I wanna know if there is an answer for the following simplest case.
Let $E/F$ be a ...

**6**

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156 views

### mod $p$ Jacquet-Langlands correspondence

Let $F$ be a local field of characteristic $0$. Let $D$ be division algebra over $F$ of dimension $n^2$. The construction of irreducible complex representations of $D^*$ is known by Howe, Zink, and ...

**6**

votes

**2**answers

208 views

### Whittaker models for $GL_n$ and Fourier coefficients

Let $G$ be a compact abelian group. Then we know, because of the Peter-Weyl theorem, that $L^2(G)$ decomposes as a Hilbert space direct sum of 1 dimensional representations of $G$.
Let $\mathbb{A}$ ...

**11**

votes

**1**answer

265 views

### Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...

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244 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

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vote

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368 views

### Chinese Remainder Theorem backwards

I have the following situation, that is much alike the Chinese Remainder Theorem. Let $\phi_d(\alpha)$ be the $d^{th}$ cyclotomic polynomial in the variable $\alpha$ (I'm not specifying the ...

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votes

**2**answers

220 views

### Which Weil group over a $p$-adic field?

For simplicity, call the Weil group of a local nonarchimedean field $F_v$ to be the following extension:
$$1\longrightarrow F^\times_v\longrightarrow W_{F_v}\longrightarrow\text{Gal}(F_v/\mathbb ...

**14**

votes

**1**answer

337 views

### Local-global principle for split extensions of Galois representations

I guess the following is well-known (and probably follows from Chebotarev's density theorem, but I'm not very comfortable with it):
Define some notation:
$K$ a global field,
$G$ the absolute Galois ...

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votes

**2**answers

627 views

### An application of Maschke's theorem

I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...

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**3**answers

524 views

### What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...

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**0**answers

320 views

### What's the status of Arthur's announced classification for GSp(4)?

In "Automorphic representations of GSp(4)" (2004) (see http://www.math.toronto.edu/arthur/), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no ...

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**2**answers

1k views

### Current Status on Langlands Program

The Langlands Program was launched almost fifty years ago, and progress has been made gradually, much of it hard earned. Langlands himself wrote a survey on the functoriality conjecture in 1997, Where ...

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**1**answer

377 views

### Local Langlands for $GL(2,\mathbf{C})$ and reducible principal series

My naive picture of the local Langlands correspondence for $GL(2,\mathbf{C})$ is this. The Weil group of $\mathbf{C}$ is canonically $\mathbf{C}^\times$. On the Galois side then we're looking at ...

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**0**answers

185 views

### Why Whittaker functions are useful?

Whittaker functions appears in Langlands program. Recently, it is shown that some Whittaker functions can be obtained by integrating a function related to decoration over a geometric crystal in ...

**4**

votes

**2**answers

227 views

### Gelfand pair and double coset decomposition

Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus ...

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**0**answers

90 views

### Joint representation of the semi-direct product of the metaplectic group and Heisenberg group

Given a symplectic space $W$ over a local field $F$ and a additive character $\psi$ of $F$, we can construct the Weil representation $\omega_\psi$, which can be viewed as a representation of the ...

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79 views

### Bounding global matrix coefficient for PGL_2

I'm trying to find a reference that gives a bound for the adelic matrix coefficient for $\text{PGL}_2$ using the bound towards Ramanujan conjecture. More specifically:
Let $G=\text{PGL}_2$. Let $F$ ...

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votes

**1**answer

148 views

### Indefinite orthogonal groups over p-adics

Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...

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**1**answer

236 views

### Can Galois conjugates of lattices in SL(2,R) be discrete?

Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...

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**2**answers

479 views

### Etymology of cuspidal representations

In the literature on representation theory of $GL_2(\Bbb F_p)$ and $GL_2(\Bbb Q_p)$, the irreducible representations with trivial Jacquet module are often called "cuspidal" or "supercuspidal". Why are ...

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**3**answers

417 views

### Definition of Hecke operators

I am confused about the definition of Hecke operators. It will be great if someone provides some references.
Shimura's 'Arithmetic Theory of Automorphic forms' says: Let $\Gamma$ be acting in the ...

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votes

**1**answer

212 views

### Is Mazur's deformation ring R integral?

Consider the absolutely irreducible Galois representation
$\overline{\rho} \colon G_{\Bbb Q} \to {\mathrm{GL}}_2({\Bbb F}_p)$. We apply the Mazur's deformation theory on the lift ${\rho} \colon ...

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**1**answer

234 views

### Iwasawa theory for Mazur's deformation ring R

The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...

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vote

**2**answers

197 views

### On size of Hecke algebras.

Let $G$ be a subgroup in $SL_2(\mathbb{Z})$ and $S_k(G)$ be the space of cusp (automorphic?) forms invariant by any element of $G$ of weight $k$.
Question 1: Generally for two arithmetic subgroups ...

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337 views

### Representation-theoretic operations on modular forms

Let $A$ and $B$ be Hecke eigenforms of some weight $k$ and level $N$. We know that there are irreducible representations $\rho_a$, $\rho_b$ of the absolute Galois group of $\mathbb{Q}$ whose trace of ...

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187 views

### 0-dimensional Gorenstein local ring.

Assume the following condition for the ring T = F_p[[X,S]]/I:
Condition 1. T is NOT a zero ring.
Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements.
Then, is T a ...

**3**

votes

**1**answer

212 views

### Conductor of a representation of a $p$-adic group

Let $G$ be a connected split reductive group over $\mathbb{Z}$. Let $F$ be a local non-Archimedean field. Let $\rho$ be an irreducible smooth representation of $G(F)$. How does one define the ...

**3**

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**1**answer

206 views

### On a unitary automorphic representation

I sometimes come across this notion called "unitary automorphic representation". But I have never seen the precise definition. When they say $(\pi, V)$ is a unitary automorphic representation of a ...

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**5**answers

481 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

210 views

### Is $(G,K)$ a strong Gelfand pair?

Let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$. When $G={\rm GL}_n$, it is a classical result that $(G(F),G(\mathcal{O}))$ is a Gelfand pair. Is it actually a strong Gelfand pair? I ...

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230 views

### Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...

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231 views

### Which de Rham representations are trianguline?

Let $K/\mathbf{Q}_p$ be a finite extension, and let $V$ be an $n$-dimensional $\overline{\mathbf{Q}_p}$-vector space with a continuous action of $G_K$. Suppose $V$ is de Rham, so potentially ...

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**0**answers

83 views

### Large prime divisors in entries of matrix powers

Are any examples known of an integer matrix $A$, such that the largest prime divisor of some specified entry of $A^n$ grows exponentially in $n$? How about where it just grows strictly faster than any ...

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177 views

### square-tiled surfaces and the Euler phi function

In Billiards in Rectangles with Barriers, Eskin-Masur-Schmoll count the number of primitive square-tiled surfaces with two cone points with angle $4\pi$ for one cone point with angle $6\pi$: (See ...

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**1**answer

137 views

### centralizer of a n-cyclic permutation matrix over F_2 in GL(n,2)

This is a continuation of this question, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit ...

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117 views

### Bessel function for $GL_3(\mathfrak{R})$?

In the $GL_2(\mathfrak{R})$ case, assume that $\pi$ is an irreducible unitary representation and $W_{\pi}(g)$ is the Whittaker functional associates with $\pi$. Then there is a Bessel function ...

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257 views

### What can we say about the Local Langlands Correspondence for GL_n without using Bernstein-Zelevinski?

I have two specific questions regarding the LLC for $GL_n$, and in particular, what we can say about the conjecture if we don't have the ideas of Bernstein and Zelevinski, which reduce the problem to ...

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373 views

### A possible application of representation theory to Galois classes of L-functions

I define the notion of a Galois class of L-function as follows:
$A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true:
1) $A$ is a subset of the ...

**2**

votes

**1**answer

260 views

### L functions of Langlands Quotients of essentially-square-integrable representations

I'm reading Kudla's Article on the Local Langlands Conjecture for $p$-adic general linear groups, and specifically I'm trying to understand how the ideas of Bernstein-Zelevinski yield show that you ...

**13**

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**1**answer

676 views

### What is the Twisted Trace Formula?

I am studying the trace formula using "An Introduction to the Trace Formula" by James Arthur. I would like to understand the twisted trace formula, but unfortunately I never came across a good ...

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**0**answers

103 views

### New vectors for representations of GSp4 with nontrivial central character

Roberts and Schmidt have developed a theory of new vectors for generic irreducible smooth representations of $\operatorname{PGSp}_4(F)$ for $F$ a nonarchimedean local field, using the "paramodular ...

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**0**answers

277 views

### What are limits of discrete series and which are cohomological?

Perhaps this is a bit too 'standard' for MO, but I'm struggling to dig it out of the literature (and maybe other people have had a similar experience), so it feels like a useful thing to ask and I ...

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votes

**1**answer

168 views

### Does every equivalence class of Hecke characters contain a distinguished element?

Let $k$ be a number field and let $I_k$ denote the idele group of $k$. Let
$$|\cdot|: (x_v) \mapsto \prod_{v \in \Omega_k} |x_v|,$$
denote the adelic norm map.
If $I_k^1$ denotes the kernel of this ...

**5**

votes

**1**answer

176 views

### Subgroups of algebraic groups

Is anyone aware of a result (or a counterexample) along the following lines: let $G$ be an algebraic group over $\mathbf Z$. Let $H$ be a finite group such that $H$ occurs as a subgroup of ...

**2**

votes

**1**answer

398 views

### Would a proof of Ramanujan Conjecture together with other known results about automorphic L-functions imply the Grand Riemann Hypothesis?

The question is in the title: if I'm not mistaken, what misses to prove that all automorphic L-functions belong to the Selberg class is a proof of the Ramanujan Conjecture. But the Selberg class is ...

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204 views

### Transferring addition and multiplication over finite fields to $\mathbb{Z}$

It seems to me that the most basic wisdom on why many number-theoretic conjectures are hard is because the interplay between addition and multiplication is subtle and delicate (much of the lay chatter ...