Group Algebra of a Lattice and Multiplicative Basis

Let $L$ be a free abelian group (a lattice), e.g. $L=X^*(T)$ (see $X^*(T)$ ).

The group algebra $\mathbb{C}[L]$ is the $\mathbb{C}$-vector space with basis $\{\xi_\lambda\}_{\lambda\in L}$ and multiplication $\xi_\lambda\cdot \xi_\mu=\xi_{\lambda+\mu}.$

If $t\in T(\mathbb{C})$, then $\lambda(t)\in\mathbb{C}^\times$ and one “evaluates” by $\xi_\lambda(t):=\lambda(t)$.

In the letter: Satake identifies the spherical Hecke algebra with invariants in such a group algebra on a dual lattice.