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18 votes
4 answers
621 views

What are immediate applications of the classification of connected reductive groups?

After years of putting it off, I finally sat down, read, and understood the classification of connected reductive groups via root data. That's a non-trivial theory! I'm hoping that now that I am done ...
Tim Phalange's user avatar
18 votes
1 answer
631 views

Best texts on Lie groups for number theorists

What are the most comprehensive textbooks on the structure of Lie groups and their infinite-dimensional representations if one is interested in their applications to number theory (so covering ...
user163784's user avatar
17 votes
1 answer
502 views

Irreducibility of root-height generating polynomial

The height $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum_{i=1}^n c_i$ where $\alpha = \sum_{i=1}^n c_i \alpha_i$ is its decomposition as a sum of ...
Christian Gaetz's user avatar
15 votes
3 answers
1k views

orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices. QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not necessarily ...
Misha Verbitsky's user avatar
12 votes
1 answer
2k views

sub-tori of a torus, generated by 1-dimensional subgroup

Ok the question is pretty dumb: suppose you have a torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$ and a vector $\bar{v}=(v_1,\ldots,v_n)\in\mathbb{R}^n$. Consider the torus $T_{\bar{v}}$ given by the closure ...
CuriousUser's user avatar
  • 1,452
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 ...
Spencer Leslie's user avatar
11 votes
4 answers
2k views

Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$. An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...
Marc Palm's user avatar
  • 11.2k
11 votes
2 answers
478 views

Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Let $A,B$ be two rational rotations: $$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\ -\frac{4}{5} & \frac{3}{5} & 0 \\ 0 & 0 & 1 \end{array}\right] \quad\...
john mangual's user avatar
  • 22.8k
11 votes
1 answer
259 views

Algorithmic Borel finiteness for hyperbolic manifolds

It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of ...
Igor Rivin's user avatar
  • 96.4k
11 votes
2 answers
972 views

Rational orthogonal matrices

``everybody knows'' that an integral orthogonal matrix is a signed permutation matrix, so there are exactly $2^n n!$ such matrices in $O(n).$ Now, what if we ask for the enumeration of elements of $O(...
Igor Rivin's user avatar
  • 96.4k
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 ...
Daniel Miller's user avatar
10 votes
1 answer
719 views

what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
Will Chen's user avatar
  • 10.7k
10 votes
1 answer
459 views

Why are root data a natural candidate for classifying connected reductive groups?

For the purpose of this question, you may assume that we are working over the complex numbers. Given a connected reductive group $G$, one can choose a maximal torus $T$, and then let $T$ act on the ...
Tim Phalange's user avatar
9 votes
1 answer
639 views

Borel's Paris Lectures

I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or ...
admissiblecycle's user avatar
9 votes
1 answer
903 views

Principal congruence subgroups in higher rank

I don't seem to have seen any explicit generators for the principal congruence subgroups of $SL(n, \mathbb{Z}),$ for $n>2,$ although it is known (Sury+Venkataramana) is that the number of ...
Igor Rivin's user avatar
  • 96.4k
9 votes
1 answer
461 views

What is the structure of the group of rational points of an abelian variety over a Laurent series field?

Let $K = \mathbb{F}_q((t))$, and let $A_{/K}$ be a nontrivial abelian variety. Then $A(K)$ is a compact $K$-adic Lie group. What can be said about its structure? By way of comparison, if $K/\mathbb{...
Pete L. Clark's user avatar
9 votes
1 answer
335 views

Triple product formula on $K = \mathrm{SU}(2)$

Let $K = \mathrm{SU}(2) = \{ k[\alpha ,\beta] \mid \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with $$ k [ \alpha , \beta ] = \begin{pmatrix} \alpha & \beta \\ - \...
Misaka 16559's user avatar
9 votes
1 answer
553 views

What matrix groups can be embedded in $Sp_4$?

In a joint paper with Yifan Yang we constructed an "exotic" embedding of $SL_2(\mathbb R)$ in $Sp_4(\mathbb R)$ (in fact, of $PSL_2(\mathbb R)$ in $PSp_4(\mathbb R)$), namely, $$ \iota\colon\begin{...
Wadim Zudilin's user avatar
9 votes
0 answers
99 views

Derived subgroups of 2-adic congruence subgroups of $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be a prime, and let $\Gamma_r$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^r\mathbb{Z}_p)$. Explicit formulas with formal ...
stupid_question_bot's user avatar
8 votes
2 answers
1k views

number of irreducible representations over general fields

For a finite group, there are finitely many irreducible representations of complex numbers. What if the field is changed to some other fields? Like real numbers, p-adic field, finite field? In ...
natura's user avatar
  • 1,503
8 votes
1 answer
1k views

Multiplicity one theorem

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) (http://www.cambridge.org/us/academic/subjects/mathematics/number-theory/automorphic-forms-and-l-functions-...
Wiener Schmidt's user avatar
8 votes
1 answer
617 views

$\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$

Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an ...
user avatar
7 votes
2 answers
353 views

Relation between different $E_8$ matrices

There are several rank-8 square matrices known to be related to $E_8$: Cartan $E_8$ matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix $$M_1=\left [\begin{array}{rr} 2 & -1 &...
Марина Marina S's user avatar
7 votes
1 answer
270 views

Is $\Gamma(p) := \text{Ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{F}_p)$ a "standard" subgroup?

Let $\Gamma(p) := \text{ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{Z}_p/p))$. Viewing $SL_2(\mathbb{Z}_p)$ as an analytic group, is there a formal group law $F$ in three variables, defined over $...
stupid_question_bot's user avatar
7 votes
1 answer
456 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 $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
Alex's user avatar
  • 454
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 ...
Anant Atyam's user avatar
7 votes
0 answers
107 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 ...
Zhiyu's user avatar
  • 6,622
7 votes
0 answers
172 views

Subgroups of $\mathbb{Z}^{n}$ with rotational symmetries

Schmidt (https://projecteuclid.org/euclid.dmj/1077377618) showed that the number of $m$-dimensional subgroups of $\mathbb{Z}^{n}$ of covolume $\leq X$ is $$c_{1}\left(m,n\right)X^{n}+O\left(X^{n-\...
Tal H's user avatar
  • 273
6 votes
1 answer
549 views

Computing a Commutator Subgroup

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I’m studying the group $\O(5,5,\mathbb{Z})$, the indefinite orthogonal matrices with integer entries. In particular, I ...
Noah B's user avatar
  • 545
6 votes
1 answer
551 views

Two definitions of automorphic forms on Lie groups

My question is the about the equivalence of two definitions of automorphic forms on a semisimple Lie group. The most common definition of automorphic forms on a semisimple Lie group $G$ with respect ...
Jun Yang's user avatar
  • 391
6 votes
1 answer
765 views

What is a map for the representation theory of reductive groups?

I have finished learning about linear algebraic groups (minus their representation theory) and the associated algebraic structures (root data, root systems, etc.), and will next attempt to summarize ...
Andrew NC's user avatar
  • 2,071
5 votes
1 answer
202 views

Schur multiplier of a Chevalley group of type $D_5$

$\DeclareMathOperator\EO{EO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\St{St}\DeclareMathOperator\Sp{Sp}$This is sort of a follow up question to my post here regarding the commutator subgroup of $...
Noah B's user avatar
  • 545
5 votes
1 answer
445 views

An easier reference than "On the Functional Equations Satisfied by Eisenstein Series"?

I'd like to learn about Eisenstein series so I started reading "On the Functional Equations Satisfied by Eisenstein Series"by Langlands. http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/...
Johnny T.'s user avatar
  • 3,625
5 votes
1 answer
404 views

determining symplecticity (if that's a word)

Suppose you have a matrix $M$ in $SL(n, \mathbb{Z}).$ Question: is there a necessary and sufficient condition for $M$ to be conjugate to $N \in Sp(n, \mathbb{Z}).$ It is clearly necessary that the ...
Igor Rivin's user avatar
  • 96.4k
4 votes
1 answer
410 views

power log distance between matrices

In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of ...
Igor Rivin's user avatar
  • 96.4k
4 votes
1 answer
183 views

What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?

Let $S_1, \ldots, S_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota_i$ be the inversion in $S_i$. Viewing the conformal group ...
Sami Douba's user avatar
4 votes
1 answer
212 views

Subgroups of $Sp_{2g}$ giving rise to Shimura data

Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...
user avatar
4 votes
0 answers
289 views

Formal integration (?) in Chabauty’s method

In Mccallum, Poonen’s paper “The method of Chabauty and Coleman”, the authors define, for the Jacobian $J$ of a geometrically connected smooth proper curve over the rational field and for $\omega \in ...
k.j.'s user avatar
  • 1,364
3 votes
1 answer
355 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 ...
Subhajit Jana's user avatar
3 votes
0 answers
160 views

Orbit representatives for the action of the maximal compact subgroup

Let $F$ be a non-Archimedean local field and $O$ be the ring of integers in $F$. Take $G=GL(2,F)$ and $K=GL(2,O)$. Consider the action of $K$ on $G$ by conjugation. Is it possible to explicitly write ...
user8974's user avatar
  • 185
3 votes
0 answers
87 views

Recovering a $G$-valued representation/parameter

Number theoretic phrasing Let $G$ be a connected reductive group over a characteristic $0$ field $F$. Associated to $G$ is its Langlands dual group $^{L}G$. For every dominant cocharacter $\mu$ of $...
Alexander's user avatar
  • 953
3 votes
0 answers
204 views

Miraculous Parahorics

Let $G$ be a connected simple group over a local field $k$. Let $I\subset G(\mathcal{O})$ denote an Iwahori subgroup of $G(k)$ with Lie algebra $\mathfrak{i}$. Let $P\supseteq I$ be any other ...
Dr. Evil's user avatar
  • 2,751
3 votes
0 answers
386 views

Lie algebra of an elliptic curve

This might be a silly question, and if it has been asked somewhere else, I would appreciate a link; however, I was unable to find it myself. In this paper by Lauter-Viray (arXiv link), in the proof ...
Laarz's user avatar
  • 153
3 votes
0 answers
204 views

Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation: $$ g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F). $$ Here, $\...
Dr. Evil's user avatar
  • 2,751
2 votes
3 answers
181 views

Stabilizers of the action of Levi on abelianization of nilpotent radical

$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
Zhiyu's user avatar
  • 6,622
2 votes
1 answer
417 views

A question regarding Lie group actions

Can you give me an example of a Lie group acting on a compact metric connected space transitively so that it has a closed finite index subgroup which does not act transitively?
Sean Cole's user avatar
2 votes
1 answer
185 views

Fields of definition of parabolically induced representations of $\mathrm{SL}(2,q)$

Let $\alpha_0$ be the unique non-trivial character satisfying $\alpha_0^2=1$ of the split torus $\mathrm{T} \subset \mathrm{SL}(2,q)$ and denote by $\mathrm{R}(\alpha_0)$ the character of $\mathrm{SL}(...
M L's user avatar
  • 381
2 votes
0 answers
98 views

Sublattices in the standard integral symplectic lattice

Let $V$ denote $\mathbb{Z}^{2g}$ with its standard integral symplectic form $\omega = \sum_{i=0}^{g-1}dx_{2i} \wedge dx_{2i+1}$ (or, the homology lattice of a genus $g$ surface with its intersection ...
Rodion N. Déev's user avatar
2 votes
0 answers
74 views

generalization of the discreteness of Hecke groups to general reductive groups

Consider the subgroup $G_{\lambda}$ of $SL_2(\mathbb R)$ generated by $N_{\lambda} = \begin{bmatrix} 1 & \lambda \\ 0 & 1 \end{bmatrix}$ and $S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{...
Zhiyu's user avatar
  • 6,622
1 vote
1 answer
106 views

Actions of torsionfree discrete subgroups on hermitian symmetric domains

Let $D$ be a bounded hermitian symmetric domain with automorphism group $G(\mathbb R)$. In the example I have in mind, $D$ is Siegel upper half-space of degree $g$ and $G(\mathbb R) = \mathrm{Sp}(2g,\...
John's user avatar
  • 13