Marc Palm

6,956
Reputation
6516 views
Is this your account?

Registered User 

Name Marc Palm
Member for 2 years
Seen 2 days ago
Website
Location Uni Hamburg
Age 27
Postdoc at Uni Hamburg -- Fall 2012-Fall 2014
Phd studies at Uni Göttingen -- 2009-2012
Master studies at Virginia Tech -- 2008-2009
Diploma studies at Uni Trier -- 2005-2008
1d
 awarded ● Fanatic
May
22
 revised Langlands product
added 8 characters in body
May
22
 revised Langlands product
added 136 characters in body
May
22
 answered Langlands product
May
22
 answered Why don’t more mathematicians improve Wikipedia articles?
May
17
 comment Plancherel formula for non-second-countable (non-unimodular) groups
Ah okay, first countability is necessary and sufficient for having a metric in a locally compact group. So correction: first countable implies second countable for lc groups if seperable:(
May
17
 comment Plancherel formula for non-second-countable (non-unimodular) groups
...for non-type 1 groups, then one could argue something. So far I am only saying you get a vNa decomposition into factors (not unique), and of course a state decomposition into extremal states by Chocquet's theorem(not unique though).
May
17
 comment Plancherel formula for non-second-countable (non-unimodular) groups
Any second-countable space is separable: if is a countable base, choosing any from the non-empty gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf. Note that locally compact, second countable groups are always seperable. But that was not the point: I am pointing out that the Hilbert space $L^2(G)$ is seperable iff $G$ is second countable. I assume implicitly (wlog) that locally compact implies Hausdorff by definition. If you make a precise statement what you mean by a Plancherel
May
17
 comment Plancherel formula for non-second-countable (non-unimodular) groups
Ignore no. I don't remember why my impulse was to say no. Seperable and second countable are often the same thing in special situation.
May
17
 comment Plancherel formula for non-second-countable (non-unimodular) groups
No, I am saying the Hilbert space $L^2(G)$ is seperable in the sense that they have a countable orthonormal basis iff $G$ is second countable. Now, how do you define a state on say $C_c^\infty(G)$ or $C_c(G)$ from a representation $\pi$ if $\pi(\phi)$ is not Hilbert Schmidt, trace class or something analogous. What is the suggested analogon you have in mind? Sure, the integral decomposition exists, but is not unique and the unitary dual is not a nice space anymore. For type 1 e.g. it will be almost Hausdorff.
May
17
 answered Plancherel formula for non-second-countable (non-unimodular) groups
May
14
 accepted Finding spherical representations of $GL(n, \mathbb{C})$.
May
14
 revised Finding spherical representations of $GL(n, \mathbb{C})$.
added 91 characters in body
May
14
 revised Finding spherical representations of $GL(n, \mathbb{C})$.
added 34 characters in body; added 39 characters in body
May
14
 comment Finding spherical representations of $GL(n, \mathbb{C})$.
I also would claim that the subquotients are never spherical, but I am not sure in the generality I have stated the results.
May
14
 comment Finding spherical representations of $GL(n, \mathbb{C})$.
Note, that I don't know which one of the parabolically induced ones have irreducible subquotients or are unitarizabile, though. I am only saying a classification of the former gives pretty easily a classification of the latter.
May
14
 revised Finding spherical representations of $GL(n, \mathbb{C})$.
added 34 characters in body; added 35 characters in body
May
14
 answered Finding spherical representations of $GL(n, \mathbb{C})$.
May
14
 comment Finding spherical representations of $GL(n, \mathbb{C})$.
No. What you state is simply the fact when $(G,K)$ is a Gelfand pair. The OP is search for a set of unitary representation, e.g., for $GL_2(\mathbb{C})$, it would be all unitary unramified continuous series representation and the $| \det |^s$ with $\Re s =1$.
May
14
 comment Finding spherical representations of $GL(n, \mathbb{C})$.
I understand the OP is interested in a classification of the unitary (or smooth, admissible) representation, which are irreducible and have a invariant vector under the maximal compact subgroup, or equivalently the trivial representation is contained in the restriction to it. I think in his context, he wants to consider either the $\mathbb{R}$- or $\mathbb{C}$-points of these classical algebraic group, $U(n)$ making no sense over $\mathbb{R}$, though, and having a trivial answer over $\mathbb{C}$. Similarly, for $SO(n)$ over $\mathbb{R}$.
May
13
 comment Why are modular forms (usually) defined only for congruence subgroups?
Actually, it is a theorem that every unitary or even smoth, admissible repreentation of $GL_2(F_v)$ has a $\Gamma_0(p_v^N)$-invariant vector for $N$-sufficiently large. All modular forms turn up earlier or later for some $\Gamma_0(K)$ for $K$ large. Whether it makes always sense computational to go to large $K$, is a different one.
May
13
 comment Local Langlands conjecture for GL(2)
Dear Matthew, I am totally satisfied with your explanation. Would you like to copy and paste it as an answer? Thank you=)
May
10
 comment How do these two Haar measures on SL(2,R) compare?
The modular function is $\Delta$ is constant one for $SL_2(\mathbb{R})$ and any other reductive group over a local field.
May
10
 comment Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
Thanks Matthew for both your comments:-)
May
8
 answered Modular Forms of Weight One
May
8
 comment Orders in number fields
Who voted to close? And why?
May
8
 revised Elementary proof of algebraicity of Hecke eigenvalues in weight 1
deleted 91 characters in body
May
8
 revised Elementary proof of algebraicity of Hecke eigenvalues in weight 1
added 318 characters in body; deleted 13 characters in body
May
8
 answered Elementary proof of algebraicity of Hecke eigenvalues in weight 1
May
8
 comment Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
Excellent. Many thanks!
May
8
 comment Weyl law for arithmetic Fuchsian groups known?
I decided to ask the following follow-up question: mathoverflow.net/questions/130062/… ... Perhaps you have an example at hand?
May
8
 comment Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
This is my motivation: mathoverflow.net/questions/129637/…
May
8
 asked Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
May
7
 revised Weyl law for arithmetic Fuchsian groups known?
deleted 76 characters in body
May
7
 answered Weyl law for arithmetic Fuchsian groups known?
May
6
 accepted Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.
May
6
 revised Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.
added 10 characters in body
May
6
 revised Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.
added 1 characters in body
May
6
 revised Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.
added 1631 characters in body
May
5
 comment Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.
I will edit my answer on Monday with references. Short reply: unitarizabile reps are unitarizable in only one way and additionally to the K decomposition you remember things like the Laplace eigenvalues and Hecke eigenvalues;-)
May
4
 answered Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.
May
3
 accepted Harish-Chandra Modules of PSL_2($\mathbb{R})$)
May
3
 revised Harish-Chandra Modules of PSL_2($\mathbb{R})$)
deleted 74 characters in body
May
3
 revised Harish-Chandra Modules of PSL_2($\mathbb{R})$)
added 232 characters in body
May
3
 revised Harish-Chandra Modules of PSL_2($\mathbb{R})$)
added 433 characters in body
May
3
 answered Harish-Chandra Modules of PSL_2($\mathbb{R})$)
May
2
 comment Gap between first two nonzero Laplacian eigenvalues on closed compact surface?
Sorry, but I suspect that you can't distinguish between close distinct eigenvalues and eigenvalues with higher mutiplicity than one.
May
2
 comment Local Langlands conjecture for GL(2)
Thanks for making me aware of that. The question was local in spirit, so I was able to rephrase the question. Is it safe now?
May
2
 revised Local Langlands conjecture for GL(2)
deleted 511 characters in body; edited title
May
2
 comment Local Langlands conjecture for GL(2)
Joël: I assume you are talking about Maass forms?