t1
. ..
tn
, z =
z1
. ..
zn
,
for notational convenience. If z ∈ Tˆ then we have a quasicharacter χz of T(F) defined by χz(t) =Q
zord(ti i). Every unramified quasicharacter is of this sort.
In general, let X∗(T) be the group of rational characters of T, and X∗(T) the group of one-parameter subgroups. As we explained in Section 12, both groups are isomorphic toZr and come equipped with a dual pairingX∗(T)×X∗(T)−→Zthat makesV =R⊗X∗(T) and V∗ =R⊗X∗(T) into dual spaces.
The exercises below will show that the torus ˆT may be chosen so that X∗( ˆT)∼= X∗(T) and X∗( ˆT)∼=X∗(T). Let Xnr(T(F)) denote the group of unramified charac-ters of T(F).
Exercise 18 Show that there is a natural isomorphism Hom(X∗(T),C×) ∼= Xnr(T(F)).
Indeed, givenχ ∈ Xnr(T(F)) associate with χ the homomorphism X∗(T)−→ C× be the map that sends the one-parameter subgroupi:Gm−→T toχ(i($)) for prime element$.
Exercise 19 Show that there is a natural isomorphism Hom(X∗( ˆT),C×)∼= ˆT(C). Indeed, givez∈Tˆ(C), associate withz the homomorphismX∗( ˆT)−→C× be the map that sends the rational characterη ∈X∗( ˆT) to η(z)∈C×.
Since X∗(T) and X∗( ˆT) are identified, it is clear that the unramified characters of T(F) are parametrized by the elements of ˆT(C).
20 Intertwining integrals: nonarchimedean fields
The intertwining integrals appear very naturally in the theory of Eisenstein series.
They were introduced into the theory of induced representations of Lie and p-adic groups by Bruhat around 1956.
Some results, particularly on analytic continuation of the integrals will be stated without proof. References:
• Casselman,Introduction to Admissible Representations ofp-adic Groups, linked from the class web page. Section 6.4 contains the fundamental results about the intertwining operators.
• Casselman, The unramified principal series of p-adic groups. I. The spherical function. Compositio Mathematica, 40 no. 3 (1980), p. 387-406.
• Bump, Automorphic Forms and Representations only treats GL2 completely, but many of the important ideas can be understood from that special case.
LetF be a nonarchimedean local field. LetGbe aF-split reductive group. Let T be a maximal F-split torus, and let B be a Borel subgroup containing T. The derived groupG0 is semisimple, and we may let xα :Gm −→G0 be as in Section 16. We will denote by K◦ = G0(o) the group generated by the xα(o). It is a maximal compact subgroup of G(F). We have the Iwasawa decomposition
G(F) = B(F)G(o).
We still have functors of parabolic induction and the Jacquet module. However parabolic induction requires a “shift” byδ1/2, whereδ is the modular quasicharacter of B(F). Thus δ : B(F) −→ C is defined so that if dµL(b) is a left Haar-measure then δ(b)dµL(b) is a right Haar-measure. We have
δ(tu) = Y
Now letχbe an unramified quasicharacter of T(F). We defineV(χ) to be the space of smooth (locally constant) functions f :G(F)−→C that satisfy
f(bg) = (δ1/2χ)(b)f(g), b ∈B(F). (77) The group Gacts on V(χ) by right translation:
π(g)f(x) = f(xg). (78)
The condition that f is smooth is equivalent to assuming that π(k)f = f for k in some open subgroup ofG(F). Thus
V(χ) =[
K
V(χ)K
as K runs through the open subgroups of G(F). We recall that V(χ)K is a HK -module, whereHK is the convolution ring of K-biinvariant functions.
In view of (3) and (78), if φ∈ HJ and f ∈V(χ)K then (φ·f)(x) =π(φ)f(x) =
Z
G
φ(g)f(xg)dg. (79) We will be particularly interested inV(χ)J whereJ is the Iwahori subgroup.
One reason for including the factor δ1/2 in the definition of the induced repre-sentation is that if χ is unitary, then π(χ) is a unitary representation. Indeed if IndG(FB(F))(δ) is the space of functions that satisfy
f(bg) = δ(b)f(g)
then by Lemma 2.6.1 of Bump, Automorphic Forms and Representations we may define a linear functionalI on IndG(FB(F))(δ) that is invariant under right-translation by
I(f) = Z
K◦
f(k)dk.
Thus if f1, f2 ∈V(χ), and if χis unitary, then f1f2 is in the space IndG(FB(F))(δ) and so hf1, f2i=
Z
K◦
f1(k)f2(k)dk
is an inner product, making the representationV(χ) unitary. It is possible for V(χ) to be unitary even ifχ is not, owing to the existence of complementary series.
Another reason for the normalization factor is so that the intertwining integrals map V(χ)−→ V(wχ). The intertwining integrals may be defined by the analogs of either (75) or (76). That is:
M(w)f(g) = Z
U(F)∩wU−(F)w−1
f(w−1ug)du = Z
(U(F)∩wU(F)w−1)\U(F)
f(w−1ug)du. (80) The two formulas are equivalent due to the fact that
U(F) = (U(F)∩wU−(F)w)(U(F)∩wU(F)w−1) by Proposition 30.
Proposition 63 The integral (80) is convergent if is formally similar to the finite field case: one simply replaces the summations by integrations. Using this, the convergence statement reduces to the case wherew=sα for a simple rootα. In that case,U(F)∩wU(F)w−1 =iα(F), and the integral is
The factor|v|−1 is fromδ1/2. If v is sufficiently large, the value of f is constant since f is locally constant. Therefore absolute convergence depends on the convergence of
Z
where C is a nonzero constant. The absolute value of χ is constant on the sets
$−ko×, which have volume qk(1−q−1). The factor |v|−1 = q−k on this set, so we
The convergence of this geometric series follows from (81).
Lemma 26 If t∈T(F) and w∈W then the Jacobian of the transformation
Proof In order to show that one proceeds as follows. If t ∈T(F) we can write M(w)f(tg) =
We make a variable change u7→tut−1 and by the Lemma the Jacobian of this map isδ1/2(t)/δ1/2(w−1tw). Therefore we obtain (δ1/2·wχ)(t)M(w)f(g), as required.
Although the intertwining integrals are not convergent for allχ, even when they are not convergent we may make sense of the integrals more generally as follows.
Let us organize the quasicharacters of T(F) into a complex analytic manifold X as follows. First, the unramified characters, we have seen, are, by Exercise 18, Xnr(T(F)) ∼= Hom(X∗(T),C×) which is a product of copies of C×. This is thus a connected complex manifold Xur. If we fix a (not necessarily unramified) character χ0, then we may consider χχ0 with χ ∈ Xnr(T(F)) to vary though a copy of Xur. ThusX is a union (over cosets of the unramified characters) of copies of Xur, and is therefore a complex analytic manifold.
Let Xreg be the set of regular characters in X. It is an open set, the regular set.
The complement Xsing is the singular set. Now we may consider the disjoint union
V= [
χ∈X
V(χ).
By asection we mean a function X−→V, to be denoted χ7→fχ such that ifχ∈X thenfχ ∈V(χ). We would like to define the notion of ananalytic section. First, let us say that the section is flat if fχ|K◦ is constant on each connected component of X. The flat sections are analytic. Moreover sinceX is a complex analytic manifold, we have a notion of analytic and meromorphic functions onX. We say that a section is analytic (resp. meromorphic) if it is a linear combination of flat sections with analytic coefficients.
The intertwining operators do not necessarily take flat sections to flat sections, but they do take analytic sections to meromorphic sections. The poles in the com-plement of the regular set: that is, if fχ is an analytic section, then M(w)fχ is meromorphic, and analytic onXreg.
We now come to a major difference between the finite field case and the local field case. Whereas in the finite field case, the reducible places of the principal series were when χ was not regular, regularity in the local field case happens at shifts of the singular set. Nevertheless something interesting does happen where regularity fails.
Let us see this at work when G= GL2. Let χ
y1
y2
=χ1(y1)χ2(y2)
and consider the intertwining integral M(w0) : V(χ) −→ V(wχ), where w0 = 1
−1
is the long Weyl group element. If χ1 = χ2, then M(w0) has a pole.
In this case, V(χ) = V(wχ), and this module is irreducible. There is only one in-tertwining operator V(χ)−→V(wχ) = V(χ), and this is M(1). If both M(w0) and M(1) were analytic, there would be two, and one is “not needed” so it has a pole.
However the Jacquet module at this special value shows some interesting behav-ior. The Jacquet module J(V) of a smooth G(F)-module V is
J(V) =V /VU, VU =hv−π(u)v|v ∈V, u∈U(F)i. It is an T(F)-module, and if V =V(χ) whereχ is regular, then
J(V(χ)) = M
w∈W
δ1/2·wχ.
The Jacquet module is an exact functor and is an important tool in the representation theory ofp-adic groups. We refer to the references of Casselman and Bump for further information.
In the GL2 case, if χ is regular, the last formula reduces to J(V(χ)) = δ1/2χ⊕δ1/2·w0χ.
When χ = w0χ, the Jacquet module V(χ) becomes indecomposable. It has two isomorphic composition factors, bothδ1/2χ, and sits in a short exact sequence:
0−→δ1/2χ−→J(V(χ))−→δ1/2χ−→0.
However the irreducible submodule is not a direct summand, and this exact sequence does not split.
The reducibility of the principal series is whenχ1χ−12 (t) =|t|orχ1χ−12 (t) =|t|−1. In the general case, V(χ) will be reducible if χ(hα∨($)) = q±1 for some coroot α∨, and will be maximally reducible if χ is in theW-orbit of δ1/2.