X
Chris Jantzen
We remark that Bx depends only on Xu, and not on s. The proof that Bx has
this property is based on a similar statement in [Gus] for the case where x is
unramified. This, in turn, is based on the results of Borel and Casselman that state
that a subquotient of an unramified principal series is generated by its Iwahori-
fixed vectors. Let H H(G,BX) denote the algebra of compactly supported Bx-
biinvariant functions on G. Associated to the induced representation (TT, G, V) is
the representation (TT, H, VB*), where H acts on the l?x-fixed vectors by
*(./ = J f(g)*(9)vdg.
G
It is known (cf. [Gus]) that *) implies the map W —• WB* gives a bijective
correspondence between subquotients of (7r, (7, V) and subquotients of (7r, H, VBX).
As VB* is a finite-dimensional space, we can explicitly compute certain operators
7r(/i1),7r(/i2), with h\,h2 H and show that they can admit no common invariant
subspace except at a finite number of potential reducibility points. For order of
Xu 2, there are no potential reducibility points-rc is always irreducible. If order
of Xu 2, reducibility can only occur if s = 0 or i7r/\nq. For order of Xu 1, the
points where reducibility is possible are s = ±1,0, iw/lnq. The next problem is to
show that there is reducibility at the potential reducibility points. We note that in
the unramified case, if s = ± 1 , the trivial representation is a subquotient, so there
is reducibility.
This leaves us with the potential reducibility points where x is unitary.
Now, if x is unitary, then IT is unitary as well. We show that TT is reducible by
showing that dim Horn^Ti", TT)=2, where 7r=contragredient of it =IndpX - 1 - To show
this, we observe that by Frobenius reciprocity,
HomG(7r,7f) = HoniMCTri/^"1),
where wu is the Jacquet module of TT taken with respect to P = MU. If we let
Mx M fl Bx, we get a representation (7r^,iJ(M//M
x
), (Vt/)Mx). We can show
that
H o m M ^ X "
1
) = H o m t f ^ M ^ T r ^ x "
1
) .
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