All Questions
47 questions
26
votes
3
answers
5k
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 Moeglin-...
- 809
25
votes
3
answers
1k
views
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
20
votes
1
answer
786
views
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
views
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
15
votes
1
answer
954
views
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
13
votes
1
answer
358
views
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
13
votes
0
answers
523
views
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
12
votes
1
answer
472
views
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
11
votes
1
answer
875
views
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
11
votes
1
answer
620
views
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
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
10
votes
3
answers
1k
views
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} ...
9
votes
0
answers
409
views
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
votes
1
answer
476
views
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
8
votes
0
answers
481
views
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
views
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
7
votes
2
answers
595
views
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
6
votes
2
answers
1k
views
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
6
votes
2
answers
453
views
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
6
votes
1
answer
476
views
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
6
votes
1
answer
361
views
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
0
answers
261
views
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
5
votes
1
answer
187
views
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
5
votes
1
answer
386
views
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
5
votes
1
answer
854
views
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
5
votes
0
answers
359
views
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
5
votes
0
answers
219
views
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
4
votes
1
answer
700
views
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
$$\sum_{\lambda\vdash n}\...
- 43.2k
4
votes
1
answer
260
views
Reference request: normalization of intertwining operators for GL(2, C)
Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral
$$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ 1&...
- 590
4
votes
1
answer
421
views
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
votes
1
answer
226
views
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
4
votes
0
answers
200
views
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
212
views
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
3
votes
1
answer
270
views
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
3
votes
1
answer
225
views
$f^{\lambda}$: 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
3
votes
0
answers
101
views
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
3
votes
0
answers
102
views
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, ...
- 14.1k
3
votes
0
answers
168
views
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
2
votes
2
answers
473
views
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
2
votes
0
answers
228
views
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
1
vote
1
answer
232
views
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\...
- 14.1k
1
vote
1
answer
155
views
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
vote
0
answers
98
views
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
1
vote
0
answers
126
views
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
0
votes
0
answers
171
views
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
0
votes
0
answers
91
views
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
0
votes
0
answers
197
views
'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