# Tagged Questions

**-5**

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90 views

### Could it be possible to use Selberg trace formula to prove that the irreducible characters of representation form an orthonormal basis? [closed]

G is a finite group.H is the space of class functions on G;the irreducible characters χ1,χ2...χn belong to H form an orthonormal basis.
That is an basic representation theorem,and you can easily prove ...

**1**

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47 views

### limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals (for example, in Gelbart-Jacquet, 1979):
$$\int_{Z_v N_v\backslash ...

**2**

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**0**answers

61 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 ...

**11**

votes

**1**answer

272 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 ...

**5**

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**0**answers

251 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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**0**answers

89 views

### Constant terms of Eisenstein series and Gindikin-Karpelevich formula

Let $G$ be a split reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of G. Let $\chi$ be an unramified character of $M$ and $f_\chi$ be the spherical section of the ...

**1**

vote

**1**answer

321 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 ...

**8**

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**0**answers

110 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 ...

**9**

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**0**answers

323 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 ...

**2**

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**1**answer

175 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 ...

**2**

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**1**answer

184 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 ...

**4**

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**0**answers

189 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**

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**2**answers

229 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 ...

**3**

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**1**answer

93 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 ...

**0**

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**0**answers

83 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 $F$ ...

**3**

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**3**answers

422 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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**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 ...

**3**

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**1**answer

213 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 ...

**6**

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**5**answers

489 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**

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**1**answer

211 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

107 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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**0**answers

118 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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**1**answer

685 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 ...

**4**

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**1**answer

267 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 ...

**5**

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**1**answer

313 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 $G(\mathbf{A}_F^\infty)$ ...

**2**

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**1**answer

189 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

132 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 ...

**3**

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**1**answer

334 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 ...

**2**

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**1**answer

220 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 ...

**3**

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**0**answers

189 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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**0**answers

197 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 ...

**4**

votes

**1**answer

227 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 ...

**1**

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**1**answer

157 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 ...

**3**

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**1**answer

222 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 ...

**1**

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**1**answer

348 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 ...

**2**

votes

**0**answers

209 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 ...

**7**

votes

**1**answer

302 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

159 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 ...

**0**

votes

**1**answer

233 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.

**4**

votes

**0**answers

153 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 ...

**3**

votes

**1**answer

229 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 ...

**6**

votes

**1**answer

411 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 ...

**1**

vote

**1**answer

279 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 ...

**14**

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 ...

**2**

votes

**1**answer

244 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 $\pi$, ...

**13**

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 ...

**2**

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**1**answer

434 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 ...

**8**

votes

**1**answer

297 views

### Atkin–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 $\Gamma_0(N)$ ...

**5**

votes

**1**answer

237 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 ...

**3**

votes

**2**answers

308 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 ...