All Questions
47 questions
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98
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Are there known effective bounds on the number of semisimple Galois representations?
In continuation to my question here, are there known effective bounds on the total number of semisimple $p$-adic Galois representations unramified outside a finite set of primes $S$, of dimension $d$, ...
- 2,907
5
votes
1
answer
187
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Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases
Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ ...
- 639
1
vote
1
answer
155
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Iwahori action on the $p$-ordinary line of a principal series representation
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
- 639
1
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1
answer
232
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Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.2k
1
vote
0
answers
126
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Reference request: unfolding of Integral representation of an L-function
Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or ...
- 43
2
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0
answers
228
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Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 43.2k
0
votes
0
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171
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Total sum of characters over partitions with distinct parts
In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
- 43.2k
4
votes
1
answer
700
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Total sum of characters of the symmetric group $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.2k
8
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0
answers
481
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Formal degree of discrete series representations
Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \...
- 5,424
7
votes
2
answers
946
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Reference of J.L. Waldspurger's paper on Shimura correspondence
I want to find reference of Waldspurger's paper referred at "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (...
- 73
5
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1
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386
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Arthur's Simple Trace Formula
In Deligne–Kazhdan–Vigneras's "Représentations des groupes réductifs sur un corps local," they use the Simple Trace Formula to prove cases of the local Jacquet–Langlands correspondence ...
- 181
3
votes
1
answer
212
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Seeking a combinatorial proof for the invariance of a $q$-series
Start with some notations: $(a,q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$, shortened by $(a)_n$, and $(a)_{\infty}=\prod_{k=0}^{\infty}(1-aq^k)$.
It's easy to verify (using algebraic means) that, for each $...
- 43.2k
0
votes
0
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91
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Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 739
2
votes
2
answers
474
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For what automorphic representations is Ramanujan-Petersson known?
I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that
If an automorphic representation on $GL(2)$ is ramified at a ...
- 609
6
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1
answer
361
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Decay of matrix coefficients of non-tempered representation
A theorem of Cowling--Haagerup--Howe gives an effective decay rate of the matrix coefficients of a tempered representation $\pi$ of a semi-simple algebraic $G$ in terms of Harish-Chandra $\Xi$ ...
- 1,692
6
votes
2
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453
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Reduction to Lie algebra version of fundamental lemma?
Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration.
For the purposes of the trace formula, one actually needs the fundamental ...
- 2,516
4
votes
1
answer
226
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Simple trace formula with different spectral footprint?
A standard idea when dealing with the Arthur-Selberg trace formula (or a relative trace formula, for that matter) is to impose local conditions on the test function $f=\prod_vf_v$ to obtain a simple ...
- 2,516
3
votes
0
answers
101
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Hermitian sublattices of a given type
Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-...
- 2,516
4
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0
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200
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When does a continuous function's "Fourier series" converge pointwise almost everywhere to the function?
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...
- 6,180
3
votes
1
answer
270
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Eisenstein series for discrete subgroups of SL(2,C)?
I am looking for a reference for Eisenstein series for discrete subgroups of $SL(2,\mathbb C)$, in particular, finite index subgroups of $SL(2,\mathcal O_K)$ where $K$ is an imaginary quadratic field.
...
- 3,799
13
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1
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358
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Cartography of the duals of GL, PGL, SL, etc
A short version of this question could be
What are the duals of $PGL(2,\mathbf{Q}_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?
I should obviously add some precisions.
there are different ...
- 5,424
11
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1
answer
620
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Gauss, Jacobi, Kloosterman sums and representation theory in the $\mathbb F_1$-world
This question is inspired by Why are Bessel function and Kloosterman sum similar? - it developed in me desire to understand Kloosterman sums better.
There seems to be common knowledge that Gauss, ...
- 18.4k
3
votes
1
answer
225
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$f^{\lambda}$: asymptotics and analytic continuations
Let $\mathbb{Y}_n$ denote the set of all partitions of $n\in\mathbb{N}$ and $\mathbb{Y}$ Young's lattice of all partitions. The partition function $g_0(n)=\sum_{\lambda\in\mathbb{Y}_n}1$ has an ...
- 43.2k
9
votes
0
answers
409
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The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)
When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
- 181
8
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1
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476
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How does Jacquet's "Generic Representations" classify tempered representations?
Let $L$ be a $p$-adic field $G = GL_n(L)$. Let $P$ be a standard parabolic subgroup with Levi decomposition $P = MU$, where $M \cong G_1\times \ldots \times G_r$, for $G_i \cong GL_{n_i}(L)$.
The ...
- 1,453
25
votes
3
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1k
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what else is in $\prod_{j=1}^n(1+q^j)$?
From time to time, I run into the finite product $\prod_{j=1}^n(1+q^j)$. And, the more it happens, the more fascinated I've become. So, herein, I wish to get help in collecting such results. To give ...
- 43.2k
0
votes
0
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197
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'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799
6
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0
answers
261
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Local character expansion for discrete series representations of $GL_n(F)$
I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field.
First, some notation: let $G$ be a ...
- 1,453
4
votes
1
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421
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Birch's conjecture from Representation Theory
Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...
- 2,429
4
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1
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260
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Reference request: normalization of intertwining operators for GL(2, C)
Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral
$$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ 1&...
- 590
12
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1
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472
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Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations
Let $\pi$ be a generic irreducible admissible representation of $GL_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The conductor of $\pi$ ...
- 1,453
6
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1
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476
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Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations
Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...
- 1,453
5
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0
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359
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Examples of Rankin-Selberg L-functions from Eisenstein series
I've been digging for awhile to not much success, so I figure I would try here:
I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...
- 3,799
3
votes
0
answers
102
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Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
- 14.2k
13
votes
0
answers
523
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Euler Subgroups and Automorphic L-functions
Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
- 2,516
3
votes
0
answers
168
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Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
7
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2
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595
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Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 327
15
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1
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954
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Funktorialität in der Theorie der automorphen Formen
In 2010 Langlands wrote an article with the title Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele. On the IAS website, he says that
This note ... was written ...
- 18.7k
26
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3
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5k
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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 Moeglin-...
- 809
20
votes
1
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786
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Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$
This question is out of plain curiosity. The first sentence of Deligne's
Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique $0$ (1984) reads (in rough translation) as ...
- 18.7k
18
votes
6
answers
2k
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Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic.
Let $K$ be a non-arch local field (I'm only interested in the char 0 case), let $\mathbb{G}$ be a connected reductive group over $K$ and let $G=\mathbb{G}(K)$. If $V$ is a smooth irreducible complex ...
- 41.4k
10
votes
3
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1k
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p-adic representations of a quaternion algebra over a local field
How to determine a complete set of isomorphism class representatives of the irreducible algebraic representations of $D^{\times}/F$ (where $D$ is a quaternion algebra over a local field $F/\mathbb{Q} ...
10
votes
2
answers
955
views
Semisimplicity of étale cohomology representations
Let $K$ be a number field and $G=Gal(\overline{K}/K)$ the absolute Galois group of $K$. Let $\ell$ be a prime number.
Let $A/K$ be an abelian variety. Then the representation of $G$ on $V_\ell(A)$ is ...
- 1,673
5
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0
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219
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Character tables of the p-core of the binary modular congruence group of p-power level
Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the
American Mathematical Society. 79 (1973), no. 4.), ...
- 2,396
5
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1
answer
854
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Rallis inner product formula for U(2,2) and U(3)
Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are:
"A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998.
"An ...
- 467
6
votes
2
answers
1k
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Does anyone have an electronic copy of Waldspurger's "Sur les coefficients de Fourier des formes modulaires de poids demi-entier"?
Is there an electronic copy of Waldspurger's paper "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" floating around the internet somewhere? This appeared in J. Pures Math. ...
- 13.1k
11
votes
1
answer
875
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An arithmetic highest weight theory?
I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps ...
- 436