All Questions
10 questions
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
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(...
- 96.4k
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 ...
- 10.7k
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 ...
- 96.4k
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 ...
- 1,503
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 ...
user515519
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 ...
- 1,692
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 ...
- 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 $...
- 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 ...
- 2,751