1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 75

We let the reader formulate an analogue of Conjecture 7. We now have the

following remarkable consequence of Conjecture 8. Assume p =/=-q E S. (One could

simply consider the case when S = {p, q} ).

If 1fp1Hp = {~ mp(T)dμp(T) and 7rq1Hq = {~ mq(cr)crdμq(cr) (Theorem 3.3),

Jiip Jiiq

the decomposition of 7rp 0 7rqlHpxHq is given by the obvious double integral. In

particular all tensor products T 0 er associated to T , er in the supports of the p-

restriction and the q-restriction occur. (As always, we consider only unramified

representations). Assume then that T, T^1 occur in 7r P IH p. Choose er in 7r q I H q •

According to Conjecture 8, T 0 er E Hr,Ar where T = {p, q}. So T , er have the same

81(2)-type. But this also applies to T^1 and er. As a consequence we have:

Conjecture 9. - Assume 1fp E Gp,Ar · If Tp, T; ( unramified representations of Hp)

occur in 1fp1Hp, then Tp and T; have the same 81(2)-type.

3.5. The case of S1(n).
It follows from Theorem 3.2 that for S1(n) the 81(2)-type of a representation
in G s,aut is well-defined. We consider only unramified representations, but this

could be extended to all representations in Gs,aut· In the following discussion we
will pass freely from representations of S1(n) to representations of G1(n). This is
justified by the fact that an automorphic representation of S1(n, As) extends to
one of G1(n, As) - see [39], and that the "type" does not change amongst unitary
extensions to G1(n) of the same representation of S1(n).

Write G = S1(n). A representation 81(2) ±.. G = PG1(n, q lifts to S1(n, q
and is then defined by a partition - still denoted by 'ljJ - n = n 1 + · · · + nr of n.
The ni are the dimensions of the irreducible summands. The type is the set of ni
up to permutation. Assume first that 1fp, a representation of G1(n, Qp), is a local
component of a representation 7r of G1(n, A) in the discrete spectrum. According
to Moeglin and Waldspurger, the Hecke matrix trr,p can be written

(3.10) trr,p = ta ,p 0 sp(b)

where n =ab, er is a cuspidal unitary representations of G1(a, A), ta,p is its Hecke
matrix and

(3.11) sp(b) = (p';' p';'. ).

p-2-1-b

For a partition 'ljJ set

(3.12)

(block-diagonal matrices).