The Local Langlands Conjecture for GL(2)
If F is a non-Archimedean local field, local class field theory can be viewed as giving a canonical bijection between the characters of the multiplicative group GL(1,F) of F and the characters of the Weil group of F. If n is a positive integer, the n-dime
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Proof. Let K be a compact open subgroup of G; we denote by K C c (G) the ∞ space of functions in Cc∞ (G) fixed by λ(K). We view K C c (G) as G-space via right translation. It is then identical to the representation of G compactly induced from the trivial representation 1K of K: K
∞
C c (G) = c-IndG K 1K .
Lemma. Viewing C as the trivial G-space, we have ∞
dimC HomG (K C c (G), C) = 1. ∞
There exists a non-zero element IK ∈ HomG (K C c (G), C) such that IK (f ) 0 whenever f 0. If hK is the characteristic function of K, then IK (hK ) > 0. Proof. The first assertion is given by 2.5 Proposition. For g ∈ G, let fg denote the characteristic function of Kg. The set of functions fg , g ∈ K\G, then forms ∞ a C-basis of the space K C c (G) (2.5 Lemma). The functional IK : fg → 1 has the required properties, noting that hK = f1 . We choose a descending sequence {Kn }n1 of compact open subgroups Kn of G such that n Kn = 1. We then have Kn ∞ Cc∞ (G) = C c (G). n1 ∞
For each n 1, there is a unique right G-invariant functional In on Kn C c (G) which maps the characteristic function of Kn to (K1 : Kn )−1 . We have In+1 | Kn ∞ C c (G) = In , and so the family {In } gives a functional on Cc∞ (G) of the required kind. The uniqueness statement is immediate.
3. Measures and Duality
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Remark . The lemma also implies that, if we view Cc∞ (G) as a smooth representation of G under right translation, then dim HomG (Cc∞ (G), C) = 1.
(3.1.2)
The functional I of the Proposition is a right Haar integral on G. One can produce a left Haar integral in exactly the same way. Alternatively, one can proceed as follows. Corollary. For f ∈ Cc∞ (G), define fˇ ∈ Cc∞ (G) by fˇ(g) = f (g −1 ), g ∈ G. The functional I : Cc∞ (G) −→ C, I (f ) = I(fˇ), is a left Haar integral on G. Moreover, any left Haar integral on G is of the form cI , for some c > 0. The uniqueness statement follows from the proposition on observing that, if J is a left Haar integral, then f → J(fˇ) is a right Haar integral. Let I be a left Haar integral on G. Let S = ∅ be a compact open subset of G and let ΓS be its characteristic function. We define µG (S) = I(ΓS ). Then µG (S) > 0 and the measure µG satisfies µG (gS) = µG (S), g ∈ G. One refers to µG as a left Haar measure on G. The relation with the integral is expressed via the traditional notation
f (g) dµG (g), f ∈ Cc∞ (G). I(f ) = G
Further traditional abbreviations are frequently permitted, in particular,
ΓS (x)f (x) dµG (x) = f (x) dµG (x). G
S
Definition. The group G is unimodular if any left Haar integral on G is a right Haar integral. 3.2. One can extend the domain of Haar integration, much as in the classical case of the Lebesgue measure. We outline a few examples of what we will need. First, one can integrate more general functions. For example, let f be a function on G invariant under left translation by a compact open subgroup
28
1 Smooth Representations
K of G. Let µG be a left Haar measure on G. If the series |f (x)| dµG (x) g∈K\G
Kg
converges, so does the series without the ab
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