Satake parameter - Knowlpedia

Definition

Let $G$ be an unramified connected reductive group over a nonarchimedean local field $F$ , choose a hyperspecial subgroup $K\leq G(F)$ , and let $\pi$ be an irreducible unramified representation. Its Satake parameter is the semisimple $\widehat G$ -conjugacy class

s(\pi)=[g\rtimes\operatorname{Fr}_F] \subset \widehat G\rtimes\operatorname{Fr}_F \subset{}^LG

corresponding, under the normalized Satake isomorphism, to the character by which the spherical Hecke algebra acts on the line $\pi^K$ . If $G$ is split, this is commonly recorded simply as a semisimple conjugacy class $[g]$ in $\widehat G$ .

As a local parameter

The associated local $L$ -parameter is trivial on inertia and on the Deligne $\mathrm{SL}_2$ , and sends geometric Frobenius to $s(\pi)$ . Thus Satake parameters are precisely the unramified part of the local Langlands correspondence.

Normalization warning

The unnormalized Satake isomorphism differs from the normalized one by a $q_F^\rho$ -shift. Arithmetic instead of geometric Frobenius also inverts the Frobenius convention. A local-factor formula must use the same normalization as its Satake parameter.

Local factors

For an algebraic representation $r:{}^LG\to\operatorname{GL}(V)$ , the unramified local factor is

L(s,\pi,r)= \det\!\left(1-r(s(\pi))q_F^{-s}\mid V\right)^{-1}.

References

Armand Borel, “Automorphic $L$ -functions,” in Automorphic Forms, Representations and $L$ -Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979, §§3–4.
Jayce R. Getz, An Introduction to Automorphic Representations, §§9–10. Author notes.