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?
Best Answer
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 \[\pi(f) \cdot v = \int_{\mathrm{GL}_n(F)} f(g) \pi(g) \cdot v \, dg.\]