For any $f\in H(\mathbb{C})$, we say $f$ belongs to the Fock space $F_{\alpha}^{p}$ if
$$\int_{\mathbb{C}}|f(z)|^{p}e^{-\frac{p\alpha}{2}|z|^{2}}dA(z)<\infty$$
Most papers and books I read trivially state that every complex polynomial is contained in $F_{\alpha}^{p}$. Can anybody show me why this...
well, you can show that admissable representations of a locally profinite group $G$ correspond to admissable modules over the Hecke algebra $\mathcal{H}(G)$. That's pretty strong evidence that it's the right analogue
you probably also defined the relative Hecke algebra for a compact open subgroup $K\subset G$, right? One thing that's surprising is that some relative Hecke algebras have a reasonable explicit description. For example $\mathcal{H}(\operatorname{GL}_2(K),\operatorname{GL}_2(\mathcal{O}_K)) \cong \Bbb C[x,y,y^{-1}]$
that's the so-called Satake isomorphism (well, a special case at least)
we didn't define it at all, the book just mentions that the group algebra doesn't hack it for smooth reps of locally profinite groups and that the Hecke algebra is "righter" than the group algebra in this setting
@Tobias for a totally disconnected locally compact Hausdorff group $G$, the Hecke algebra $\mathcal{H}(G)$ is the space of locally constant and compactly supported functions to a fixed coefficient field $k$, which becomes a non-unital non-commutative ring under convolution wrt a Haar measure
@Tobias I think these might actually be related to one another. For a compact open subgroup $K$ of $G$, there's the relative Hecke algebra $\mathcal{H}(G,K)=e_K * \mathcal{H}(G) * e_K$, where $e_K$ is the indicator function of $K$, which is an idempotent in the Hecke algebra. Now if $G$ is a reductive group over a local field, then $G$ has subgroups called Iwahori subgroups (basically generalizations of Borel subgroups I think). For such a Iwahori subgroup $B$, one can show that $G$ has a Tits system $(B,N)$ where $N$ is an affine Coxeter group (that's a generalization of the Bruhat decompo…
I mainly know two definitions. One using generators and relations and the other as the endormorphism algebra of an induced module for a finite group of Lie type
can anybody enlighten me about that microstate stuff in thermodynamics. Why do assumptions such as this are reasonable? "In particular, if each microstate x had a total energy H(x), and one had a law of conservation of energy which meant that microstates could only transition to other microstates with the same energy"?
Most of the work I did with these was as motivating examples using that they have some very nice properties when considered as positively based algebras in the KL basis
I can understand that the whole system doesn't lose energy, but individual microstates? Is it because in our intuition, only one particle can fit into any microstate, and transitioning between microstates means that the energy moves with it?
@Edward unramified representations of $\mathrm{GL}_2$ as I defined them above correspond to unramified representations of the Galois group, i.e. those on which the inertia group $I$ acts trivially. Now a nice thing is that if you have a global automorphic representation of $\mathrm{GL}_2$, the corresponding local representations are all unramified for all the primes except for the finitely many ones which devide the level
because we can understand the unramified representations of $\mathrm{GL}_2$ fairly well (they're just modules over $\Bbb C[x,y,y^{-1}]$ after all), one can actually check fairly easily that if we consider Deligne's construction of a Galois representation associated to a modular form as a form of global Langlands for $\mathrm{GL}_2$, then there's local-global compatibility at all primes not dividing the level
there's actually local-global compatibility for all primes, but for those dividing the level, it's much harder because you can't use that unramified reps are really nice
that's the reason why p-adic reps are harder to study than $\ell$-adic reps right? lmao noob questions
tbh I'm still trying to figure out what I'm really interested in hahaha. I find the classical Iwasawa theory really interesting (though I still haven't read it in detail) and would like to work on something along those lines (which is why I want to work with Venjakob obviously) and study p-adic L-functions and stuff, but at the same time I'm pretty interested in the p-adic representation junk and I know that's a hotter topic than Iwasawa theory of p-adic Lie extensions
no, those reps I am taking about are all $\ell$-adic (well except for one prime $\ell$, but we can always take some $\ell$ distinct from $p$ if we want to look at the $p$-local component)
it's just that for primes dividing the level, the local $\ell$-adic rep of the $p$-adic field is not unramified
Tbf the professor was a physicist for the lecture I took, so he probably knows more than enough math, but it also explains the horrendous level of vagueness in the "notes"
Hm, $T$ be a linear self-isomorphism of an inner product space. $\langle v, w \rangle = 0$ implies $\langle Tv, Tw \rangle = 0$. This implies $T$ is actually orthogonal, I think.
Yeah, it clearly sends a orthobasis to an orthobasis, but different scaling on each vector in the basis will screw things up: If $v, w$ are unit vectors, $v + w$ and $v - w$ are orthogonal, hence so are $T(v) + T(w)$ and $T(v) - T(w)$ but that means $T(v)$ and $T(w)$ have equal norms.
I had not noticed this before! I just realized I can start by explaining a conformal map $U \to V$ as something which sends a fine square-grid on $U$ to an orthogonal curvilinear grid on $V$
@copper.hat I actually have a good amount of tolerance for that, but if they're handling expressions that could be well-defined with just, two lines more, that's extremely annoying
I'm trying to prove that there is an exact sequence of C-modules $\Omega^1_{B/A}\otimes_B C\to \Omega^1_{C/A}\to \Omega^1_{C/B}\to 0$, where $\Omega^1_{B/A}$ is the module of relative differentials of B on A (or whatever the correct terminology is in english). So far I've proven that there's an exact sequence in the opposite direction for the corresponding modules of derivations onto a module M $Der_B(A, M)\equiv Hom(\Omega^1_{B/A}, M)$
So in some sense I want to get rid of Hom(-, M) and preserve exact sequences
@user193319 substitute $x=\frac{1}{\sqrt{|q|}}y$ so that we may assume $q=-1$. If we can show $\int_0^\infty x^{2m} e^{-x^2} \mathrm{d}x$ is finite for all $m \in \Bbb N$, that would solve the problem because $x^{2m}e^{-x^2}$ dominates $x^{np+1}e^{-x^2}$ for $2m\geq np+1$ Introduce a new parameter $a>0$ we start from $\int_0^\infty e^{-ax^2}\mathrm{d}x$, do a substitution $y=\frac{1}{\sqrt{a}} x$ to obtain $$\int_0^\infty e^{-ax^2}\mathrm{d}x=\frac{1}{\sqrt{a}} \int_0^\infty e^{-x^2}\mathrm{d}x=\sqrt{\frac{\pi}{a}}$$
i feel though that there's a lot more than film/series that i find is well within this genre i have in mind, which i haven't really defined, but stuff that kind of gives you this sense of walking through a portal into a totally different realm of stuff which only makes half-sense
@user193319 I should be more careful: $x^{2m}e^{-x}$ dominates $x^{np+1}e^{-x}$ on $[1,\infty)$, but that's enough since the integral over $[0,1]$ is trivially finite as the integrand is continuous
@Astyx $X' \to X\to X'' \to 0$ is exact iff for all $M$ the sequence $0 \to \mathrm{Hom}(X'',M) \to \mathrm{Hom}(X,M) \to \mathrm{Hom}(X',M)$ is exact. This means you're basically done
Oh yes, what I said is only true if you assume that the arrows in $0 \to \mathrm{Hom}(X'',M) \to \mathrm{Hom}(X,M) \to \mathrm{Hom}(X',M)$ are induced by arrows $X' \to X \to X''$
@Astyx the reason, from a high-level point of view of why that works is that the $\mathbf{Ab}$-enriched contravariant Yoneda-embedding $\mathbf{C}^{op} \to [\mathbf{C},\mathbf{Ab}]$ that sends $X$ to $\mathrm{Hom}(X,-)$ is a left-exact fully faithful functor, so it reflects kernels
$X' \to X \to X'' \to 0$ being exact just says that $X''$ is the cokernel of $X' \to X$
which is the same as saying that $X''$ is the kernel of $X \leftarrow X'$ in $\mathbf{C}^{op}$
whereas $0 \to \mathrm{Hom}(X'',M) \to \mathrm{Hom}(X,M) \to \mathrm{Hom}(X',M)$ being exact for all $M$ says that $\mathrm{Hom}(X'',-)$ is the kernel of $\mathrm{Hom}(X,-) \to \mathrm{Hom}(X',-)$
But for the energy of the whole system to remain constant, one could in theory have more configurations than equal energy at every state, or energy remaining equal at every transition
@Astyx energy of the microstates = energy of each individual microstate, or the sum of the energies?
In the context of abelian categories, the Hom-sets are actually abelian groups. In general a $\mathbf{Ab}$-enriched category is more general than an abelian category, we just have that all Hom-sets are abelian groups and composition is $\Bbb Z$-bilinear. For usual categories, the Yoneda embedding is a functor $\mathbf{C} \to [\mathbf{C}^{op},\mathrm{Set}]$. For an $\mathbf{Ab}$-enriched cartegory, the Yoneda embedding is from $\mathbf{C}$ to additive functors $\mathbf{C}^{op} \to \mathbf{Ab}$
@Balarka well I mean, some of the scenes in Cabinet of dr Caligari are weird and distorted and some are fairly normal, and I think the same thing applies
Yes, its very annoying to be in the process of answering and have the question disappear.
On the flip side (which is what I thought you were asking initially) I sometimes add an answer which I subsequently discover is incorrect, it gets accepted and then I cannot delete it, which is frustrating as I then need to scramble to solve properly :-)
