Xl l
Chris Jantzen
until we end up with rAG(^o)ss = ^AGi^ss- Therefore, we conclude that 7r must
have been irreducible.
We now describe how Jacquet modules are used in pursuing reducibility.
In this case, one of the rM{G^ will have a reducible composition factor which would
"normally" (i.e., for generic x) be irreducible. Suppose the composition factor has
components 7i,cr2 with ^ £ rAM+(vi)ss^2 TAMS*7*)**, where P+ = MJJ* is the
appropriate parabolic subgroup. Then, let 7r,=Indp^t- We get ipx 6 TAG(^I)SS^I £
rAG('*2)ss, and similarly for x/2- If TT were irreducible, then ir would have to be
a subquotient of either 7Ti or 7r2. Therefore, rAG(n)ss C rAG(^i)ss or rAG(n)Ss C
fAG{^2)sS' But, this is not the case-just look at i/i, ip2- Thus, we have that 7r is
reducible.
The main theorem in chapter 3 is Theorem 3.1.2, which is based on a
generalization of the argument above. The theorem gives necessary and sufficient
conditions for reducibility of 7r = IGMP, where p is an irreducible admissible rep-
resentation of M such that TAMP ^ 0 and the characters appearing in TAMP are
regular. First, we associate a graph to 7r as follows:
vertices: the vertices are the elements of TAG^^SS
edges: two vertices ^i» ^2 are connected by an edge is there is some Levi N and
some r 6 r^oi^ss such that the following hold:
1. r is an irreducible representation of N.
2. ipi, ^2 rAN(r)ss-
where rAN(T)ss denotes the semisimplification of r. Under these conditions, Theo-
rem 3.1.2 states that the following are equivalent:
1. 7r is irreducible.
2. the graph of n is connected
3. the composition factors of r^c^ as computed using the results of Bernstein-
Zelevinsky/Casselman (cf. Theorem 1.2.4) are all irreducible. In particular, it
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