# Tagged Questions

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264 views

### What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?

What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?
When $F$ is a local field, the representations of $GL_n(F)$ are classified by Bernstein and Zelevinsky in ...

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**0**answers

99 views

### Algebraic construction of the modular representation of $\mathrm{SL}_2(\hat{\mathbf Z})$

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if ...

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**0**answers

268 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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votes

**1**answer

152 views

### What is the logarithmic derivative of an (intertwining) operator?

The constant term of the Eisenstein series (for an adele group $GL_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for ...

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votes

**1**answer

166 views

### reference help about a result on representation theory

I read the following theorem in a paper without a proof, which I don't understand well. Let $F$ be a global function field, and $v$ be a place of $F$, use $G_r$ to denote $GL_r$.
Theorem: For any ...

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**0**answers

158 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 ...

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votes

**2**answers

200 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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votes

**1**answer

80 views

### Siegel domains and cuspidal functions

Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$.
Do we have a analog of Siegel subset for the quotient ...

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**0**answers

67 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

**3**answers

381 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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194 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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votes

**1**answer

183 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

463 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 ...

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votes

**1**answer

207 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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**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 ...

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111 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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57 views

### support of Bessel distributions and a question about notations in Shalika's multiplicity one paper

Let's consider $G_n=GL(n,F)$ where $F$ is a p-adic field.$\pi$ a generic irreducible representation of $G_n$, $\hat{\pi}$ its contragredient with Whittaker model $\mathcal{W}(\psi^{-1})$. If $f\in ...

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**0**answers

493 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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**1**answer

252 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 ...

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**1**answer

274 views

### Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.

The definition I know of for a cuspidal automorphic representation of, say, $G=\mathrm{GL}_2$ over a number field $F$ (relative to a choice of compact open subgroup $K_f$ of $G(\mathbf{A}_F^\infty)$ ...

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**1**answer

165 views

### On the Cartan decomposition of unitary group

Hello. I have some question on Cartan decomposition of unitary group, especially $U(2)$.
I am interested in local situation, that is p-adic or archimedian.
Let $F$ be a local field and $E$ be its ...

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**0**answers

124 views

### Conceptual reason behind Shimura lifts

Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G ...

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**1**answer

325 views

### Is Eisenstein series not identically zero

How does one prove that an Eisenstein series (adelically formulated as in the book of Moeglin-Waldspurger) is not identically zero? Namely how does one prove that the sum
$\sum_{\gamma\in ...

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**1**answer

210 views

### Arthur-Clozel Prop 3.1 for Function Fields?

The subject says it all. I would like to know if Proposition 3.1 in
Arthur-Clozel's book on the trace formula holds for local fields of positive
characteristic.
Thanks!
EDIT: Here is Prop 3.1 of ...

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**0**answers

176 views

### The operator \boxtimes and \boxplus in automorphic representations

Given two automorphic representations $\pi_1, \pi_2$ of $GL_2(\mathbb A_Q)$ and $GL_3(\mathbb A_Q)$ respectively. Let $\pi_i =\otimes_v \pi_{i, v}$.
Now, for each $v$, let $\pi_{1, v}\boxtimes ...

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185 views

### On the theta lift and its L-function

I am wondering how the relation is between of the automorphic L-function and its lift's.
More precisely,
Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over E of ...

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votes

**1**answer

225 views

### On the restriction of cusp. irr. representation and period.

Let's consider only global case.
Let $G_n$ be classical algebraic group over global # field (eg, $GL(n),SO(n), U(n)$...) and let $\pi_n$ be its irr. cusp. reps of $G_n$.
Then we can define the ...

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**1**answer

153 views

### Global square integrability ensures local sq. integrability?

This might be a stupid question for expert in this area.
I am considering automorphic representation of algebraic group.
In studying it, local tempered, local square integrable representations ...

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**1**answer

215 views

### Extending cuspidal representation to more bigger group.

I am thinking of extending an irreducible cuspidal representation to more bigger group. My question is almost same with the earlier one posted by Neal Harris except the only one.
Let me first ...

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**1**answer

323 views

### On the Weil representation of unitary groups.

I suppose I am the first one who asked about Weil representation here.
In studying Weil representation, I fell into a slough and so determined to ask you for shedding a light. I think your responses ...

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votes

**0**answers

201 views

### On the L-function of unique subrepresentation of induced representation.

In studying the L-functions of induced representation, it is not easily come up with me the papers or books dealing the L-function of irreducible subrepresentation of induced representation, while the ...

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votes

**1**answer

275 views

### What is the support of the Whittaker function of a new vector on GL(2)?

Let $W$ be the normalized Whittaker function associated to a new vector in an irreducible generic representation $\pi$ of $G=GL_2(k)$, where $k$ is a $p$-adic field. Let $c$ be the conductor of $\pi$, ...

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**0**answers

155 views

### On the Weil representation of U(1) and U(3).

Let $E/F$ quadratic extention number fields.
Let $V$ be the $m$-dimension hermition vector space over $E$.
Let $W$ be the $2n$-dimension skew hermitian vector space over $E$ and $Y_n + Y_n^*=W$ be ...

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votes

**1**answer

202 views

### Reference on Casselman-Shalika formula for GL(n) and PGL(n)?

I am looking for reference on Casselman-Shalika formula for GL(n) and PGL(n) at finite place p.

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votes

**0**answers

147 views

### Restriction of representations from $SO_{2n}$ to $SO_{2n-1}$ and $K$-fixed vectors

Let $F$ be a $p$-adic field, $G=SO_{2n}(F)$ the split special orthogonal group and $H=SO_{2n-1}(F)$, taken as a subgroup of $G$. Assume that we have an irreducible, admissible representation $\pi$ of ...

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votes

**1**answer

214 views

### Writing a basis of a representation for $GL_2(\mathbb Q_p)$ in terms of the new vector

For an irreducible smooth (generic) representation $\pi$ of $G=GL_2(k)$ with central character $\omega$, where $k$ is a $p$-adic field, we define the conductor of a vector $v\in\pi$ as follows. Let ...

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**1**answer

390 views

### Difference between automorphic forms for SL(2) and GL(2)?

Hi,
Let $A$ denote the adeles of $Q$.
I know how to decompose $L^2(SL(2,A)/SL(2,Q))$ into irreducible $SL(2,A)$-representations. What is the difference between this decomposition and the ...

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vote

**1**answer

267 views

### What is an automorphic representation of CM type ?

In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a ...

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votes

**3**answers

1k views

### Are there Maass forms where the expected Galois representation is $\ell$-adic?

Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy:
Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations ...

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votes

**1**answer

224 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 $\pi$, ...

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votes

**3**answers

1k views

### Questions about the Bernstein center of a $p$-adic reductive group

Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of ...

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votes

**1**answer

414 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 ...

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**1**answer

277 views

### Atkinâ€“Lehner operator for GL(3)?

Let $f$ be an automorphic form for $\Gamma_0(N)\subset SL(3,\mathbb{Z})$.
$\Gamma_0(N)=(a,b,c;d,e,f;g,h,i)\in SL(3,\mathbb{Z})|g=h=0(mod N)$
Is there any Atkin-Lehner operator for $\Gamma_0(N)$ ...

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votes

**1**answer

218 views

### Double coset decomposition of symplectic group over a quadratic extension

I'm trying to understand the double coset decomposition of $G(F)\setminus G(E)/K_E$ , where $G = \mathrm{GSp}_{2n}$ is the rank $n$ group of symplectic similitudes, $E/F$ is a quadratic extension of ...

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votes

**2**answers

301 views

### What is the dual of principal series of GL(3,R)?

It is common to construct principal series by induction from Borel subgroup. Say $H_1$ and $H_2$ are dual representations. Both are induced representation from Borel subgroups.
Is the integration ...

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votes

**3**answers

850 views

### Automorphic Forms on product of groups $G\times H$

Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups.
Let $G$, $H$ be two reductive groups defined over a number field
$F$. Let $\mathcal{A}(G)$ be ...

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votes

**4**answers

1k views

### Where do the real analytic Eisenstein series live?

In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...

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votes

**1**answer

467 views

### Different cuspidal automorphic representations with same representations at infinity

Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$).
Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$.
$$K=\Pi_{v<\infty}K_v$$
$K_v$ is ...

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votes

**1**answer

423 views

### classification of irreducible admissible (g,K)-module for GL(3,R)

classification of irreducible admissible (g,K)-module for GL(3,R)
Is there a classification of irreducible admissible (g,K)-module for GL(3,R)?
For GL(2,R) we have principal series, discrete series ...

**2**

votes

**2**answers

459 views

### What is the relationship between (g,K)-module and Maass forms?

What is the relationship between (g,K)-module and Maass forms for GL(2)?
(g,K)-modules are defined in chapter 2 of Bump, Automorphic forms and representations.
There is a classification of ...