What is the archimedean Hecke algebra?

Question

Let $\mathbf G$ be a connected, reductive group over $\mathbb Q$. For each nonarchimedean place $v$, let $K_v$ be a maximal compact subgroup of $\mathbf G(\mathbb Q_v)$. The space $\mathscr H(\mathbf G(\mathbb Q_v),K_v)$ of bi $K_v$-invariant compact supported functions $\mathbf G(\mathbb Q_v) \rightarrow \mathbb C$ is a unital algebra over $\mathbb C$ with respect to convolution. This is the non-archimedean Hecke algebra.

I'm not familiar with the archimedean version of this. Jayce Getz's notes define the Hecke algebra of $G = \mathbf G(\mathbb R)$ to be the "convolution algebra of distributions" of $G$ supported on $K_{\infty}$. What is this exactly? If it is difficult to describe, I would appreciate any good references on this. Is there a more general notion of a Hecke algebra associated to a real Lie group and a maximal compact subgroup?

Answer 1

Here is the picture, as I understand it, for $\mathrm{GL}_n$; this is described in chapter 8 of Godement-Jacquet (and see also the archimedean theory in Jacquet-Langlands).

Let $F \in \{\mathbb{R},\mathbb{C}\}$ be an archimedean local field. Let $\mathcal{H}_1$ denote the space of smooth compactly supported functions on $\mathrm{GL}_n(F)$ that are bi-$K$-finite, where $K = \mathrm{U}(n)$ if $F = \mathbb{C}$ and $K = \mathrm{O}(n)$ if $F = \mathbb{R}$. These may be regarded as measures on $\mathrm{GL}_n(F)$, in which case $\mathcal{H}_1$ is an algebra under convolution: for $f_1, f_2 \in \mathcal{H}_1$, \[f_1 \ast f_2(g) = \int_{\mathrm{GL}_n(F)} f_1(gh^{-1}) f_2(h) \, dh.\] Every function $\xi$ on $K$ that is a finite sum of matrix coefficients of irreducible representations $\tau$ of $K$ may be identified with a measure on $K$, and hence on $\mathrm{GL}_n(F)$. Under convolution, these measures form an algebra $\mathcal{H}_2$. We let $\mathcal{H}_F = \mathcal{H}_1 \oplus \mathcal{H}_2$. This is an algebra under convolution of measures: for $f \in \mathcal{H}_1$ and $\xi \in \mathcal{H}_2$, \[\xi \ast f(g) = \int_{K} \xi(k) f(k^{-1} g) \, dk\] and \[f \ast \xi(g) = \int_{K} f(gk^{-1}) \xi(k) \, dk.\]

This is the Hecke algebra of $\mathrm{GL}_n(F)$. Given a representation $(\pi,V)$ of $\mathrm{GL}_n(F)$, we define the action of $f \in \mathcal{H}_F$ on $v \in V$ by \[\pi(f) \cdot v = \int_{\mathrm{GL}_n(F)} f(g) \pi(g) \cdot v \, dg.\]

Answer 2

The terminology is a bit misleading, and the analogy with the non-archimedean situation is a bit forced.

The goal was/is to have a $\mathfrak g,K$-module be a "Hecke algebra module", for some suitable notion of "Hecke algebra". One wanted/wants all differential operators (identified with the universal enveloping algebra of $\mathfrak g$) and also the action of $K$. Well, as a consequence of some old, standard lemmas, to say "convolution algebra of distributions supported on $K$" is equivalent to that $\mathfrak g,K$-module structure. Some work to be done, though.

Also, the relevant vector-valued integral (as in Peter Humphries's answer) needs a bit of shoring-up to be guaranteed to do what we expect. And, indeed, the Gelfand–Pettis "weak" (ironically, "weak" mostly in terms of assumptions, rather than conclusions) integral has been around for a long time and does the job.