# Marc Palm

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## 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 GĂ¶ttingen -- 2009-2012
Master studies at Virginia Tech -- 2008-2009
Diploma studies at Uni Trier -- 2005-2008
 1d awarded ● Fanatic May22 revised Langlands productadded 8 characters in body May22 revised Langlands productadded 136 characters in body May22 answered Langlands product May22 answered Why don’t more mathematicians improve Wikipedia articles? May17 comment Plancherel formula for non-second-countable (non-unimodular) groupsAh 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:( May17 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). May17 comment Plancherel formula for non-second-countable (non-unimodular) groupsAny 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 May17 comment Plancherel formula for non-second-countable (non-unimodular) groupsIgnore no. I don't remember why my impulse was to say no. Seperable and second countable are often the same thing in special situation. May17 comment Plancherel formula for non-second-countable (non-unimodular) groupsNo, 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. May17 answered Plancherel formula for non-second-countable (non-unimodular) groups May14 accepted Finding spherical representations of $GL(n, \mathbb{C})$. May14 revised Finding spherical representations of $GL(n, \mathbb{C})$.added 91 characters in body May14 revised Finding spherical representations of $GL(n, \mathbb{C})$.added 34 characters in body; added 39 characters in body May14 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. May14 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. May14 revised Finding spherical representations of $GL(n, \mathbb{C})$.added 34 characters in body; added 35 characters in body May14 answered Finding spherical representations of $GL(n, \mathbb{C})$. May14 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 $\Re s =1$. May14 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 $\mathbb{R}$. May13 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. May13 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=) May10 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. May10 comment Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)Thanks Matthew for both your comments:-) May8 answered Modular Forms of Weight One May8 comment Orders in number fieldsWho voted to close? And why? May8 revised Elementary proof of algebraicity of Hecke eigenvalues in weight 1deleted 91 characters in body May8 revised Elementary proof of algebraicity of Hecke eigenvalues in weight 1added 318 characters in body; deleted 13 characters in body May8 answered Elementary proof of algebraicity of Hecke eigenvalues in weight 1 May8 comment Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)Excellent. Many thanks! May8 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? May8 comment Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)This is my motivation: mathoverflow.net/questions/129637/… May8 asked Concrete examples of noncongruence, arithmetic subgroups of SL(2,R) May7 revised Weyl law for arithmetic Fuchsian groups known?deleted 76 characters in body May7 answered Weyl law for arithmetic Fuchsian groups known? May6 accepted Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation. May6 revised Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.added 10 characters in body May6 revised Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.added 1 characters in body May6 revised Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation.added 1631 characters in body May5 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;-) May4 answered Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation. May3 accepted Harish-Chandra Modules of PSL_2($\mathbb{R})$) May3 revised Harish-Chandra Modules of PSL_2($\mathbb{R})$)deleted 74 characters in body May3 revised Harish-Chandra Modules of PSL_2($\mathbb{R})$)added 232 characters in body May3 revised Harish-Chandra Modules of PSL_2($\mathbb{R})$)added 433 characters in body May3 answered Harish-Chandra Modules of PSL_2($\mathbb{R})$) May2 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. May2 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? May2 revised Local Langlands conjecture for GL(2)deleted 511 characters in body; edited title May2 comment Local Langlands conjecture for GL(2)Joël: I assume you are talking about Maass forms?