Spherical Hecke algebra and Satake isomorphism

Let $F$ be a nonarchimedean local field, let $G/F$ be unramified, and let $K$ be a hyperspecial subgroup. With Haar measure normalized by $\operatorname{vol}(K)=1$ , the spherical Hecke algebra

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

has convolution product and unit $\mathbf 1_K$ . It is commutative.

Split Satake isomorphism

If $G$ is split with split maximal torus $T$ and Weyl group $W$ , the normalized Satake transform gives

\mathcal H(G(F),K) \otimes_\mathbb Z\mathbb C \cong \mathbb C[X_*(T)]^W \cong \mathbb C[X^*(\widehat T)]^W \cong R(\widehat G)\otimes_\mathbb Z\mathbb C.

The lattice is $X_*(T)=X^*(\widehat T)$ , not $X_*(\widehat T)$ . The normalization includes the square root of the modulus character, equivalently a $\rho$ -shift.

For an unramified nonsplit group, Frobenius acts on the dual root datum and the invariant algebra is described through the corresponding $L$ -group fiber.

Eigencharacters

If $\pi$ is an irreducible unramified representation, then $\dim\pi^K=1$ . Its Hecke character corresponds under Satake to a semisimple Satake parameter. Conversely, that class determines the spherical irreducible representation.

Integral and normalization issues

The normalized isomorphism can require adjoining $q^{1/2}$ . An unnormalized Satake transform is defined over a smaller coefficient ring but shifts the recorded parameter. Arithmetic applications must specify this choice before comparing Galois Frobenius eigenvalues.

Relation to the letter

The letter describes the target as invariants in the group algebra of its dual lattice. Modern dual-group language identifies that invariant algebra with the representation ring of $\widehat G$ .

References

Ichirō Satake, “Theory of spherical functions on reductive algebraic groups over $p$ -adic fields,” PMIHÉS 18 (1963), 5–69. Numdam.
A. Borel, “Automorphic $L$ -functions,” Proc. Sympos. Pure Math. 33, part 2, 1979.