LECTURE 2. THE SPECTRAL DECOMPOSITION OF L^2 (G(IQl)\G(A)) 63The existence of (2.8) also has consequences if we impose conditions on 7rp,
even when it is ramified. For example for p finite these is a tempered parameter
<p: w~---> G associated to the Steinberg (or special) representation.
It is easily defined: take <plw,, = 1; recall that any reductive group G contains
regular unipotent elements, which form a unique orbit under conjugation. Fi-
nally, recall that the Jacobson-Morozov theorem associates a representation (mod-
ulo G-conjugation) SL(2, <C) ---> G to a unipotent orbit. This is <plsL{ 2 ,q.
Because <plsL{ 2 ,q is very large, a map SL(2, <C) ---> G commuting with it must
be trivial. So we deduce (see Blasius and Rogawski [7a]):
Conjecture 5. - If 7r = ®7rv occurs in Aa and 7rp is a Steinberg representation,
all components of 7r are tempered.
Finally, we only mention the most delicate ingredient of Arthur's conjectures.
Assume 'If; : L Q x SL(2, <C) ---> G is a (candidate) parameter for a representation
occurring in the discrete spectrum Aa,dis, and 7r = ®7rv with 7rv E II('l/Jv)· When
does 7r actually occur? The answer depends, in some cases, on the sign of the
functional equation of certain L-functions associated to 7r, cf. [4b, §8]. This has
actually been checked in some cases, for instance for U(3) by Rogawski [49b].^6
6In the paper [4a] where Arthur stated a first form of his conjecture, he contends that it is
confirmed by the multiplicity formulas in [4b, Thm. 13.3.6]. These formulas, as was noticed by
Harder, are false! The correct formulas given by Rogawski in [49b] do corroborate the conjecture.