Hecke algebra of a locally compact group pair

Definition

Let $G$ be a locally compact Hausdorff group, let $K\leq G$ be a compact subgroup, and fix a left Haar measure $dg$ . The Hecke algebra of the pair $(G,K)$ is

\mathcal H(G,K)=C_c(K\backslash G/K),

the continuous compactly supported functions $f:G\to\mathbb C$ satisfying $f(k_1gk_2)=f(g)$ , with multiplication given by convolution

(f_1*f_2)(x)=\int_G f_1(y)f_2(y^{-1}x)\,dy.

Compact support is understood on $G$ , equivalently on the double-coset space because $K$ is compact.

Algebraic structure

Bi- $K$ -invariance is preserved by convolution, and associativity follows from associativity of group convolution. When $K$ is compact open, one may normalize $dg(K)=1$ ; then the characteristic function $1_K$ belongs to $\mathcal H(G,K)$ and is its identity. With the group-convolution involution, $\mathcal H(G,K)$ is a star-algebra.

Compact-open and spherical cases

If $G$ is totally disconnected and $K$ is compact open, the double cosets $KgK$ are open and compact. Their characteristic functions span $\mathcal H(G,K)$ , and multiplication is encoded by finite double-coset decompositions. For a reductive group over a nonarchimedean local field and a hyperspecial maximal compact subgroup, this is the usual spherical Hecke algebra treated in the Satake theory Bump, Chapter 4.

Representation-theoretic action

If $\pi$ is a continuous unitary representation of $G$ , integration gives operators

\pi(f)=\int_G f(g)\pi(g)\,dg.

For bi- $K$ -invariant $f$ , the operator $\pi(f)$ preserves the subspace of $K$ -fixed vectors. Thus $\mathcal H(G,K)$ packages the part of representation theory visible to $K$ -spherical vectors.

References

D. Bump, Automorphic Forms and Representations, Cambridge University Press, 1997. DOI record. Relevant: spherical functions and spherical Hecke algebras.
C. J. Bushnell and P. C. Kutzko, The Admissible Dual of GL(N) via Compact Open Subgroups, Princeton University Press, 1993. Publisher record. Relevant: compact-open subgroup Hecke algebras and their representation-theoretic modules.