LECTURE 3
Local L-functions
If (7r, V7r) is a cuspidal representation of GLn(A) and (7r', v11",) is a cuspidal rep-
resentation of GLm(A) we have associated to the pair (7r, 7r^1 ) a family of Eulerian
integrals {I(s;cp,cp')} (or {J(s;cp,cp',<P)} if m = n) and through the Euler factor-
ization we have for each place v of k a family oflocal integrals {Wv(s; Wv, W~)} (or
{wv(s; Wv, W~, <Pv)}) attached to the pair of local components (7rv, 7r~). In this lec-
ture we would like to attach a local £-function (or local Euler factor) L(s, 1l"v x 7r~)
to such a pair of local representations through the family of local integrals and
analyze its basic properties, including the local functional equation. The paradigm
for such an analysis of local £-functions is Tate's thesis [91]. The mechanics of the
archimedean and non-archimedean theories are slightly different so we will treat
them separately, beginning with the non-archimedean theory.3.1. The non-archimedean local factors
For this section we will let k denote a non-archimedean local field. We will let o
denote the ring of integers of k and p the unique prime ideal of o. Fix a generator
w of p. We let q be the residue degree of k, so q =lo/pl= lw1-^1. We fix a non-
trivial continuous additive character '¢ of k. ( 7r, V11") and ( 1!"^1 , V11"') will now be the
smooth vectors in irreducible admissible unitary generic representations of GLn(k)
and GLm(k) respectively, as is true for local components of cuspidal representations.
We will let w and w' denote their central characters.
The basic reference for this section is the paper of Jacquet, Piatetski-Shapiro,
and Shalika [42].
3.1.1. The local £-functionFor each pair of Whittaker functions WE W(7r,'¢) and W' E W(7r','¢-^1 ) and in
the case n = m each Schwartz-Bruhat function <P E S(kn) we have defined local
integrals
w(s; W, W') = j W (h ) W'(h)I det(h)ls-(n-ml/^2 dh
In-miii(,;W,W') ~ff W (Z In-m-1 ,) dx W'(h)ldet(h)l'-(n-m)/' dh
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