Archimedean Hecke Algebra of $\operatorname{GL}_1(F)$ for $F = \mathbb{R}$ or $\mathbb{C}$ Suppose  we have a $\operatorname{GL}_1$ over a number field $F$. I am interested in a description for the archimedean Hecke algebra (always taking the maximal compact subgroup). We know will be the tensor product of the archimedean Hecke algebras on the real places and the complex places. So that reduces the problem to a description of the Hecke algebras for $\mathbb{R}^{\times}$ and $\mathbb{C}^{\times}$. We have for $\mathbb{R}^{\times}$ that $K=\{1,-1\}$. That should mean that the Hecke algebra should be given by derivatives of all orders at $\{1,-1\}$, but I can not get a good sense of the convolution product.Is this isomorphic to something more familiar? The case for $\mathbb{C}^{\times}$ is even more confusing. A reference for the archimedean Hecke algebra will be also good. I only know the book Cohomological Induction and Unitary Representations by Knapp and Vogan, but there is no copy in my library.
 A: Let $G = \operatorname{GL}_1(F)$ for $F = \mathbb{R}$ or $\mathbb{C}$, and fix a maximal compact subgroup $K$ of $G$.
The archimedean Hecke algebra $H(G,K)$ is the algebra of right- (or left-) $K$-finite distributions on $G$ supported in $K$.
Case 1: $F = \mathbb{R}$
Here, as you say, a maximal compact subgroup is $K = \{\pm 1\}$. Accordingly, the $K$-finiteness condition is unnecessary, and the space $H(G,K)$ consists of all distributions supported on $\pm 1$.
The distributions supported at a point $g \in G$ are spanned by the Dirac delta $\delta_g$ and all of its derivatives, so in this case $H(G,K)$ takes the form $\mathbb{C}[d/dx]\delta_1 \oplus \mathbb{C}[d/dx]\delta_{-1} \cong U(\mathfrak{g}) \oplus U(\mathfrak{g})$.
Case 2: $F = \mathbb{C}$
This case is a bit more complicated since both $K = S^1 = \{z \in \mathbb{C} : |z| = 1\}$ and derivatives on $G$ (that is, the Lie algebra of $G$) is less trivial. I can leave it as an exercise or come back and provide a revision when time permits.
Instead let me write up a brief summary of a process by which you can analyze $H(G,K)$.
Take $\mathfrak{g}_\mathbb{R}$ for $\operatorname{Lie}(G)$ as a real Lie algebra and $\mathfrak{g}$ for the complexification $\mathfrak{g}_\mathbb{R} \oplus i\mathfrak{g}_\mathbb{R}$ of $\mathfrak{g}_\mathbb{R}$.
Similarly define $\mathfrak{k}$ as a complexification of the real vector space $\mathfrak{k}_\mathbb{R}$ corresponding to $\operatorname{Lie}(K)$ by forgetting complex structure.
Let $U(\mathfrak{g})$ and $U(\mathfrak{k})$ denote the universal enveloping algebras of $\mathfrak{g}$ and $\mathfrak{k}$, respectively.
Finally, let $H(K)$ denote the space of right- (or left-) $K$-finite distributions on $K$.
(Left- and right-$K$-finiteness are equivalent for distributions on $K$.)
By fixing a Haar measure $dk$ on $K$, the space $H(K)$ may be identified with the convolution algebra of measures $C^\infty(K) \cdot dk$.
The algebra $H(K)$ can be embedded into $H(G,K)$, where the action of $T_f := f \cdot dk \in H(K)$ on $F \in C^\infty(G)$ is given by
$$
    T_f(F) = \int_K f(k) F(k)\,dk.
$$
In fact, for each $u \in U(\mathfrak{g})$ and each $T_f \in H(K)$, we may form the tensor $u \otimes T_f$ and let it act as a distribution on $G$ by setting
$$
    (u \otimes T_f)(F) = \int_K f(k) (\partial(u)F)(k)\,dk.
$$
Here $\partial(u)F \in C^\infty(G)$ is the $u$-th right derivative of $F$ (that is, the image of $F$ under the canonical extension to $U(\mathfrak{g})$ of the action of $\mathfrak{g}$ on $C^\infty(G)$ by right-differential operators).
Extending to linear combinations of these tensors, it turns out we exhaust $H(G,K)$; one has an isomorphism of algebras
$$
    H(G,K) \cong U(\mathfrak{g}) \otimes_{U(\mathfrak{k})} H(K),
$$
where the tensor product is over $U(\mathfrak{k})$ rather than, say, $\mathbb{C}$ due to relationship between the actions of the two factors.
If you look closely, our characterization in the real case is a decomposition of exactly this form, and you can use the same decomposition to compute $H(G,K)$ for $F = \mathbb{C}$ (and in general).
