All Questions
13 questions
11
votes
1
answer
688
views
Unitary representations of finite groups over finite fields
I would like to learn the basic theory of unitary representations of finite groups over finite fields.
Here, the unitary group $\operatorname{GU}(n,\mathbb{F}_{q^2})$ consists of all invertible ...
- 111
9
votes
1
answer
521
views
Which group algebras in analysis are "true group algebras"?
Let $G$ be a group, $A$ a unital associative algebra over ${\mathbb C}$, and let us call a representation of $G$ in $A$ an arbitrary map $\pi:G\to A$ such that
$$
\pi(1)=1,\qquad \pi(a\cdot b)=\pi(a)\...
- 7,426
8
votes
2
answers
1k
views
What does the unique mean on weakly almost periodic functions look like?
There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
- 4,432
5
votes
2
answers
452
views
"geometric" description of the algebra of central functions on a Lie group
I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
- 2,201
5
votes
0
answers
298
views
What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?
What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$?
Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\...
- 2,071
3
votes
1
answer
282
views
Inducing from cocompact subgroups
Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of ...
- 11.2k
3
votes
1
answer
213
views
About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$
Let $G$ be a locally compact topological group. A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L_Gf$ of left translates is relatively compact in the (weak) ...
- 185
3
votes
1
answer
197
views
Siegel domains and cuspidal functions
Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$.
Do we have a analog of Siegel subset for the quotient $GL_{n}(\mathbb{A}...
- 3,472
3
votes
0
answers
168
views
Efficient integration over part of a compact group
I am trying to find the matrix coefficients $\{\hat{c}^{\alpha}_{i j}\}$ that minimize the mean squared error against a function $f(g)$ over a compact group to some bandwidth cutoff $\ell$.
$$\...
- 121
2
votes
0
answers
118
views
What are the finite-dimensional irreducible unitary representations of $E(3)$?
Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by
$$E(3)=SO(3)\ltimes T(3)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
- 31
1
vote
0
answers
93
views
Exploring Plancherel measure decay rates linked to a specific $AD(\Gamma)$ range
In this paper on the amenability constant of Fourier algebras Theorem 1.5 presents a formula connecting $AD(\Gamma)$, the anti-diagonal constant of a countable virtually abelian group $\Gamma$, to ...
1
vote
0
answers
297
views
Intersection of thickly syndetic sets
Question: Let $\Gamma$ be a countable group. Is the intersection of two thickly syndetic sets still thickly syndetic?
I've only seen the proof for the group $\mathbb{Z}$ (and I believe this method ...
- 4,432
1
vote
0
answers
311
views
Show that $\mathrm{SL}_2(\mathbb{F}_p)$ is quasi-random
Terry Tao gives this oblique definition of quasirandom group in his notes 3
$G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$ have dimension at least $...
- 22.8k