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

) .