All Questions
Tagged with nt.number-theory rt.representation-theory
163 questions with no upvoted or accepted answers
29
votes
0
answers
1k
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Linking formulas by Euler, Pólya, Nekrasov-Okounkov
Consider the formal product
$$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$
(a) If $z=2$ then on the one hand we get Euler's
$$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$
on the ...
- 43.2k
16
votes
0
answers
605
views
Division fields of abelian varieties over function fields
Let $k$ be a finitely generated field (for example a finite field or a number field) and $K/k$ a finitely generated regular extension with $trdeg(K/k)=1$. Let $A/K$ be a principally polarized abelian ...
- 1,673
13
votes
0
answers
266
views
Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$
Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...
- 408
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
0
answers
967
views
What is miraculous about the mirabolic subgroup?
I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...
- 2,516
11
votes
0
answers
283
views
Why are there so few irreducible admissible representations of $\text{GL}(n,\mathbb{R})$ (up to infinitesimal equivalence)?
Studying Langlands's classification of irreducible admissible representations, I have been rather stunned by the following:
Theorem
Up to infinitesimal equivalence, all irreducible admissible ...
- 511
11
votes
0
answers
359
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Representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$
Is there a nice reference for the finite dimensional (characteristic 0) representation theory of $\operatorname{GL}_2(\mathbb Z/n\mathbb Z)$ and $\operatorname{PGL}_2(\mathbb Z/n\mathbb Z)$ for ...
- 7,746
11
votes
0
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332
views
Fourier Transforms of Convolutions
Straightforward computations lead to the following standard property of Fourier transformation: it transforms convolutions into products, i.e. for functions $f$ and $g$ Schwartz class we have
$$\...
- 5,424
11
votes
0
answers
194
views
Explicit L-factor for supercuspidals
I am interested in L-functions for the quasi-split unitary group $U$ in three variables, following the construction by zeta integrals of Gelbart-Piatetski-Shapiro. My aim is to understand the ...
- 5,424
10
votes
0
answers
1k
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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 ...
- 806
10
votes
0
answers
687
views
An analogue of Deligne-Lusztig theory for positive depth representations?
Deligne-Lusztig theory is an important tool in understanding the depth zero representations of $p$-adic groups. Is there an analogue of Deligne-Lusztig theory that helps in understanding positive ...
- 1,000
9
votes
0
answers
233
views
A detail in I. Piatetski-Shapiro and S. Rallis's "Doubling paper": computing the integral on negligible orbits
I'm currently reading the paper "L-functions for the classical groups" by I. Piatetski-Shapiro and S. Rallis, where they introduced the doubling method over classical groups.
I'm confused at ...
- 639
9
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0
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133
views
Is there a reason nice coset representatives exist for Leech or E_8 lattice modulo 2?
Let $\Lambda$ be the Leech lattice. There is a nice set of coset representatives for $\Lambda/2 \Lambda$ given by short vectors [Conway and Sloane, Ch. 10, Theorem 28 or Ch. 23, Theorem 3]. The proof ...
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
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
8
votes
0
answers
1k
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Ramified Geometric Langlands
Is the following a reasonable formulation of (a part of) the geometric Langlands conjecture for $\mathrm{GL}_n$ over a curve $X$?
(*) Let $\mathcal{L}$ be an irreducible rank $n$ $\ell$-adic local ...
- 2,751
8
votes
0
answers
265
views
$L^2$ norms of Whittaker vectors and zeros of Intertwining operators
For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard ...
- 1,692
8
votes
0
answers
366
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 ...
- 26k
8
votes
0
answers
440
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 ...
- 1,453
7
votes
0
answers
102
views
Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights
$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
- 6,622
7
votes
0
answers
389
views
Certain Fourier transforms involving Whittaker function and Bessel functions
I recently meet the following two weird "Fourier transform" questions.
(I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ ...
- 960
7
votes
0
answers
261
views
Invariant lattices of group representations over a $p$-adic field
Let $G$ be a finite group, $K$ be a $p$-adic field with an uniformizer $\pi$ and residue field $k \cong \Bbb F_q$, and $V$ be an irreducible representation of $G$ over $K$.
Let $X_{V}^G$ be the set ...
- 6,622
7
votes
0
answers
642
views
Automorphisms of semistable $G$-bundles
Let $X$ be a projective smooth curve over an algebraically closed field $\mathbb k$ of any characteristic. Let also $G$ be a reductive group over $\mathbb k$. (Probably) following Ramanathan, a $G$-...
- 790
7
votes
0
answers
277
views
Branching laws for orthogonal groups
Let $G = SO_5(\mathbf{Q}_p)$ be the split special orthogonal group over $\mathbf{Q}_p$, and $H \subset G$ the group $SO_4(\mathbf{Q}_p)$, embedded in the usual way as the stabiliser of an anisotropic ...
- 37k
7
votes
0
answers
266
views
Closed formula for some dimension
This question has a background from representation theory/homological algebra, but I state everything in elementary terms here:
Call an n-CNak an n-tupel $[c_0,c_1,...,c_{n-1}]$ where $c_0=c_1=...=c_{...
- 26.5k
7
votes
0
answers
492
views
mod $p$ Jacquet-Langlands correspondence
Let $F$ be a local field of characteristic $0$. Let $D$ be division algebra over $F$ of dimension $n^2$. The construction of irreducible complex representations of $D^*$ is known by Howe, Zink, and ...
- 255
7
votes
0
answers
570
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 ...
- 13.1k
7
votes
0
answers
241
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 also)...
- 22.8k
7
votes
0
answers
309
views
Computing Jacquet modules of irreducible admissible representations of GL(n,F)
Let $F$ be a $p$-adic field, $n>2$ be an integer, $G=GL(n,F)$, and $Z\subset G$ be the unipotent subgroup of $G$ that consists of matrices of the form $I_n+tE_{1,n},\ t\in F$, where $E_{1,n}$ ...
- 955
7
votes
0
answers
249
views
Embeddings between $p$-adic linear groups?
Let $p$ be a prime and let $\mathbb Z_p$ denote the $p$-adic integers.
If n<m, then what are the embeddings $SL_n(\mathbb Z_p)\rightarrow SL_m(\mathbb Z_p)$? I am particularly interested in those ...
- 391
6
votes
0
answers
177
views
Permanent of the symmetric group
Let $A$ be the algebra corresponding to a representation-finite block of a Schur algebra.
See for example 6.1. of https://arxiv.org/pdf/1607.05965.pdf for quiver and relations and some relevance of ...
- 26.5k
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
6
votes
0
answers
354
views
Sporadic and Exceptional
I have been reading this recent paper of J.McKay and YH. He (they've written a number of papers recently, including a fun and joking one on 42 which overflow commented on) called "Sporadic and ...
- 151
6
votes
0
answers
165
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 semi-...
- 61
6
votes
0
answers
460
views
Semistable reduction theorem over higher dimensional schemes
Let $k$ be a field, $S/k$ a smooth variety with function field $K$ and $U$ a nonempty open subscheme of $S$. For every finite separable extension $E/K$ we denote by $S^E$ (resp. $U^E$) the ...
- 1,673
5
votes
0
answers
169
views
Question About Page 11 of Cornell, Silverman, Stevens, "Modular Forms and Fermat's Last Theorem"
Looking at page 11 of the text, consider a Galois representation $\rho: G_{\mathbb Q} \to \operatorname{GL}_2(A)$, where $A$ is a coefficient ring (i.e. complete Noetherian local ring with finite ...
- 283
5
votes
0
answers
185
views
Finite coefficients Langlands for function fields
Do we have a finite coefficients Langlands correspondence for function fields? By which I mean a bijection between Galois representations $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)...
- 4,408
5
votes
0
answers
213
views
Truncation and weighted orbital integrals in hyperbolic term of trace formula for $\mathrm{GL}(2)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I am looking at Gelbart--Jacquet's article in the first Corvallis volume (the article entitled Forms of $\GL(2)$ from an analytic point of ...
- 241
5
votes
0
answers
192
views
Globalizable Galois representations
Let $\rho$ be a $p$-adic representation of $G=\text{Gal}(\bar{\mathbb{Q}_p}/\mathbb{Q}_p)$.
When does $\rho$ extend to a representation of the global galois group? What can be said about the locus ...
5
votes
0
answers
132
views
Field of definition of compatible system of Galois representations
Let $K,F$ be number fields and suppose that there is a compatible system of Galois representations
$$(\rho_{\lambda} : \text{Gal}(\overline{K}/K) \longrightarrow \text{GL}_n(\overline{F}_\lambda) )$$
...
- 173
5
votes
0
answers
158
views
Maass-Saito-Kurokawa Lift of Weak Jacobi Forms
Given a holomorphic Jacobi form $\varphi_{k,1} \in \mathbb{J}_{k,1}$ of weight $k$ and index 1, we know we can use the Hecke operators $V_{m}$ to lift $\varphi_{k,1}$ to a Siegel modular form
$$\...
- 1,701
5
votes
0
answers
97
views
Is there a composite-order generalization of the homomorphism on Rep(Z/p) giving total dimension of Tate cohomology?
Let $p$ be a prime, let $\mathbb{Z}_p$ be the ring of $p$-adic integers, and let $G$ be a cyclic group of order $p$. It is rather well-known that finite rank $\mathbb{Z}_p$-free representations of $G$...
- 45.7k
5
votes
0
answers
193
views
reduction mod $p$ of Weyl modules
Let $G$ be a reductive algebraic group defined over a non-Archimedean field $F$.
Let $k_F$ be its residue field, of characteristic $p$.
Assume $G$ is unramified over $F$, then it admits a hyperspecial ...
- 736
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
163
views
Uniqueness of cohomological holomorphic discrete series representation
In Claus Sorenson's PhD thesis, he proves a theorem about level lifting of paramodular forms whose associated automorphic representation has component $\pi_{\infty}$ that is the "cohomological ...
- 2,429
5
votes
0
answers
258
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 ...
- 15.4k
5
votes
0
answers
338
views
Orbital integrals and cohomology of compactified Jacobians
I have heard it said that the backdrop to the Laumon-Ngo proof of the fundamental lemma was a series of reductions in which orbital integrals were replaced by their analogues over p-adic fields which ...
- 8,723
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
0
answers
81
views
Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 6,622
4
votes
0
answers
204
views
To what extent are Langlands conjectural global L-functions unique?
Question
To what extent do the properties that are conjectured of L-functions determine them?
Explanation
Following Shahidi: So Langlands defines local L functions associated to unramified ...
- 383