Historical L-group and Satake-parameter bridge

The letter packages two now-separate notions. For a connected reductive group $G$ , the $L$ -group is an extension

{}^LG=\widehat G\rtimes W_F

after a pinning and a Weil-group action on the dual group through a finite quotient have been chosen. At an unramified local place, a spherical irreducible representation determines a Satake parameter, a semisimple $\widehat G$ -conjugacy class in the Frobenius fiber of ${}^LG$ .

Unramified class

Let $F_v$ be a nonarchimedean local field, let $G/F_v$ be unramified, and choose a hyperspecial subgroup $K_v$ . If $\pi_v^{K_v}\neq0$ , the normalized Satake isomorphism sends the Hecke character on this one-dimensional space to a class

[c(\pi_v)]\subset \widehat G\rtimes\operatorname{Frob}_v.

For split $G$ , the Weil action is trivial, so this is commonly recorded as a semisimple conjugacy class $s(\pi_v)\subset\widehat G$ . Arithmetic and geometric Frobenius conventions invert the Weil element and must be coordinated with the Satake normalization.

Local factor

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

L_v(s,\pi_v,r) = \det\!\left( 1-r(c(\pi_v))q_v^{-s}\mid V_r \right)^{-1}.

A unitary versus arithmetic normalization can shift $s$ by a half-integer; the definition is complete only after that convention is stated.

Modern placement

The $L$ -group exists independently of a representation. The Satake parameter is the unramified case of a local $L$ -parameter. Ramified parameters require Weil or Weil–Deligne data and cannot be recovered from the spherical Hecke algebra.

References

A. Borel, “Automorphic $L$ -functions,” Proc. Sympos. Pure Math. 33, part 2, 1979, §§2–6.
Kevin Buzzard and Toby Gee, “The conjectural connections between automorphic representations and Galois representations,” §2. arXiv.