Group algebra of a lattice

Let $L$ be a finite free abelian group. The group algebra $\mathbb C[L]$ has basis $\{e^\lambda\}_{\lambda\in L}$ and multiplication

e^\lambda e^\mu=e^{\lambda+\mu}.

After choosing a basis $L\simeq\mathbb Z^r$ ,

\mathbb C[L] \simeq \mathbb C[x_1^{\pm1},\ldots,x_r^{\pm1}].

This is a Laurent polynomial ring.

Coordinate ring of a torus

If $T$ is a complex algebraic torus with character lattice $X^*(T)=L$ , then

\mathcal O(T)=\mathbb C[L].

A point $t\in T(\mathbb C)$ defines the evaluation character

\mathbb C[L]\longrightarrow\mathbb C, \qquad e^\lambda\longmapsto\lambda(t).

Conversely, every complex algebra homomorphism to $\mathbb C$ arises from a point of $T$ .

Weyl invariants

If a Weyl group $W$ acts on $L$ , then $\mathbb C[L]^W$ is the coordinate ring of the affine quotient $T/W$ . Its complex points encode semisimple conjugacy classes in a connected reductive group with maximal torus $T$ .

Satake role

For a split group $G$ , the normalized Satake isomorphism identifies the spherical Hecke algebra with

\mathbb C[X_*(T)]^W = \mathbb C[X^*(\widehat T)]^W.

The letter describes this using the group algebra of its dual lattice.

References

Ichirō Satake, “Theory of spherical functions on reductive algebraic groups over $p$ -adic fields,” PMIHÉS 18 (1963), 5–69. Numdam.