9
votes
3answers
420 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 ...
9
votes
0answers
267 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 ...
19
votes
2answers
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 ...
10
votes
1answer
325 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 ...
3
votes
0answers
157 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 ...
3
votes
2answers
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 ...
6
votes
0answers
81 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 ...
0
votes
0answers
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 $F$ ...
2
votes
1answer
121 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 ...
6
votes
1answer
219 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 $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
12
votes
2answers
418 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 ...
3
votes
3answers
379 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 ...
2
votes
1answer
204 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 ...
0
votes
1answer
218 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 ...
1
vote
2answers
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 ...
5
votes
2answers
309 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 ...
0
votes
2answers
178 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
1answer
195 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
votes
1answer
182 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
votes
5answers
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 ...
4
votes
1answer
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 ...
0
votes
0answers
140 views

special values of L-functions cohen-lenstra heuristic

I found some lecture notes on links between number theory and random permutations. It was difficult to follow: The notes start with an interesting fact, whose proof I've asked on Math.StackExchange: ...
7
votes
0answers
219 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 ...
5
votes
0answers
214 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 ...
3
votes
0answers
81 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 ...
7
votes
0answers
165 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 ...
1
vote
1answer
122 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 ...
3
votes
0answers
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 ...
5
votes
0answers
234 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 ...
1
vote
0answers
368 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
1answer
241 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 ...
11
votes
0answers
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 ...
1
vote
0answers
101 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 ...
7
votes
0answers
228 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 ...
5
votes
1answer
158 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
1answer
172 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
1answer
372 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 ...
5
votes
0answers
200 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 ...
1
vote
0answers
267 views

Automorphisms of S and representations

EDIT July 22nd 2013: I add further details in bolded sentences: Assuming Selberg's orthonormality conjecture and following Automorphisms of the Selberg class, I define automorphisms of the Selberg ...
5
votes
1answer
161 views

Casselman-Shalika formula for split reductive groups

In the paper of Casselman and Shalika they give an explicit formula for the spherical Whittaker function of an unramified principal series. Apparently, upon combining their formula with the Weyl ...
3
votes
0answers
82 views

Decompositions of representations of pro-p groups

Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, ...
21
votes
1answer
1k 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 = ...
15
votes
2answers
875 views

Status of (Global) Langlands Conjecture for $GL_2$ over $\mathbb{Q}$

Apologies if this question has already been dealt with on MO. I am wondering about the status of the global Langlands conjectures for $GL_2$ over the rational numbers. How close is humanity to the ...
7
votes
2answers
274 views

When is Ad(pi) an irreducible representation ?

For a finite group $G$ with a representation $\pi:G\to GL_n(\mathbb C)$ one can define the adjoint representation $Ad$ as the non-trivial summand in $End(\pi)$, i.e. $End(\pi)=\pi\otimes ...
3
votes
1answer
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 ...
17
votes
4answers
909 views

Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?

Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
5
votes
1answer
372 views

On local Rankin-Selberg L-functions for Steinberg representations

This is a basic question concerning the local Langlands correspondence for $GL_2(\mathbb{Q}_p)$, in particular the compatibility with tensor products . I apologize if this is too basic but I hope ...
4
votes
1answer
301 views

Is strong multiplicity one (obviously) stronger than multiplicity one?

In the theory of automorphic representations one says that G satisfies a "multiplicity one" property if every cuspidal representation occurs with multiplicity one in $L^2(G(F)\backslash G(A))$. One ...
1
vote
1answer
200 views

unramified base change in characteristic p > 0?

Hi, Suppose that $E/F$ is a unramified extension of local fields of characteristic zero. Let $G = GL_n$. Then it is well-known (due to Clozel?) that base change of tempered representations from ...
3
votes
0answers
175 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 ...