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Marc Palm

Postdoc in Mathematics

http://www.plusepsilon.de


Mar
20
comment reference help about a result on representation theory
Ah okay, I see imaginary line modulo $x=2 \pi i / log(q)$, $q$ being the residue characteristic, that's isomorphic to $U(1)$, I guess:) I didn't see that before:\ But that's a confusing embedding of $U(1)$ into $\mathbb{C}^\times$ modulo $x$.
Mar
20
comment reference help about a result on representation theory
@WillSawin My issue is that unitary one-dimensional representation live on the imaginary line, not $U(1)$.
Mar
20
comment Simultaneously extending the functionals of a subspace of a Banach space to the whole space
Be careful, there exists Banach spaces which do not admit a Schauder Basis. The first examples are due to Enflo.
Mar
20
comment Simultaneously extending the functionals of a subspace of a Banach space to the whole space
There are results available, when the extension is unique, e.g. in a Hilbert space or more general results can be found here jstor.org/discover/10.2307/…. Of course, Hamel basis are not to be chosen in topological vector space, e.g. for Banach spaces one works with a Schauder basis.
Mar
20
revised reference help about a result on representation theory
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Mar
20
revised reference help about a result on representation theory
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Mar
20
revised reference help about a result on representation theory
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Mar
20
answered reference help about a result on representation theory
Mar
16
comment What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?
There is no Eichler-Selberg type trace formula for SL(3) because of the abscence of discrete series for SL(3,R).
Mar
16
comment What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?
there are certainly not easy ways to study this rep.theory... I meant to say easier ways in my last comment:\
Mar
16
comment How can I find the spectrum of this operator?
Pls use one/two dollar signs for math
Mar
16
revised How can I find the spectrum of this operator?
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Mar
16
comment What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?
I think your question has no obvious/easy answer because automorphic forms fix a specific discrete subgroup $\Gamma$ up front, so you can simply give get information for subgroups containing $\Gamma$, and $SL(n,Z)$ is known to be maximal.
Mar
15
answered Generic topology on a field
Mar
10
comment Adelization of modular forms: What measure on $L^2(G_\mathbb{Q}Z_{\mathbb{R}}\backslash G_{\mathbb{A}})$?
Okay, trivial is a strong word. Sorry, for that. I meant that it follows immediately from the definitions for weight zero plus a coordinate change. Compact subgroups are never an issue in harmonic analysis. The point of my answer was simply that the volume of the adelic quotient is usually considered as a rational number and $\Gamma \backslash H$ to be something else. You have to give $GL_2(Z_p)$ and $O(2)$ the right volume for both measures to coincide. The usual choice to give a compact subgroup having unit volume fails here.
Mar
10
comment Adelization of modular forms: What measure on $L^2(G_\mathbb{Q}Z_{\mathbb{R}}\backslash G_{\mathbb{A}})$?
ask if they are equal instead, so one can give you a definite answer. It depends on the normalization of the measures of the compact subgroups for example.
Mar
10
comment Adelization of modular forms: What measure on $L^2(G_\mathbb{Q}Z_{\mathbb{R}}\backslash G_{\mathbb{A}})$?
Ok. I misread. I think the invariance is trivial though. Working with GL(2) instead of SL(2) works only well for weight zero Maass wave forms. What is tricky is the normalization? There are differences between SL(2) or GL(2), which you seem to ignore.A discrete series of GL(2,R) splits up into two discrete series, when restricted to SL(2,R), holomorphic and antiholomorphic part. The normalization is the only issue in your question. And there is no natural quotient measure on quotient, but a one-parameter family. Your question is pretty long. I suggest you define the measures on both space,
Mar
8
comment Adelization of modular forms: What measure on $L^2(G_\mathbb{Q}Z_{\mathbb{R}}\backslash G_{\mathbb{A}})$?
You have a finite volume Radon measure. It is automatically regular. That's why I am confused.
Mar
7
comment Adelization of modular forms: What measure on $L^2(G_\mathbb{Q}Z_{\mathbb{R}}\backslash G_{\mathbb{A}})$?
The point is that the normal(=Tamagawa) measure does not coincide with the normal measure. The Tamagawa measure is a rational number, the measure on $X$ is a multiple of $1/\pi$. So no if you mean normal=Tamagawa.
Mar
7
comment Adelization of modular forms: What measure on $L^2(G_\mathbb{Q}Z_{\mathbb{R}}\backslash G_{\mathbb{A}})$?
Möbius transformations, where $z$ is equivalent to $-z$. Classically one uses upper plus lower halfplane. What is regularity? You mean normalization?