sing

8. c-function

The purpose of this section is to obtain an explicit formula for the c-function µ(χ). We also obtain a result which plays a crucial role in proving that IndGB(χ) is irreducible for singular

If s 6= 0 let

Aw(s) : I(s) ! I(¡s)

be the intertwining operator which, via the Frobenius reciprocity, corresponds to the functional αw. Recall the definition of αw. On the subspace of functions f such that f(1) = 0, it

and, if s 6= 0, then αw is the unique extension to I(s) . In order to calculate Aw(¡s) ± Aw(s), we need to obtain an explict formula for αw on the whole I(s). In other words, our task is to regularize this integral for any f in I(s).

To do so, we shall now present some key technical ingredients. Let dx be the Haar measure

on F , and for every s in C, let λs be the functional on C0∞(F ×) defined by

Z

λs(f) =

F ×

Let π denote the (right) regular representation of F × on C0∞(F ×). The functional λs is equivariant with respect to π in the sense that

λs(π(a)f) = jaj−sλs(f).

Proposition 8.1. The functional λs extends to an equivariant functional on C0∞(F ) if and only if s 6= 0. The extension is unique if s 6= 0.

On the other hand, since Λs is equivariant, we have Λs(f ¡ π($)f) = (1 ¡ j$j−s)Λs(f). Putting things together,

Z

(1 ¡ qs)Λs(f) = [f(x) ¡ f($x)]jxjs dx.

F× jxj

This formula shows that the extension exists and is unique if 1 ¡ qs is not zero. Otherwise, which happens precisely when s = 0, the left hand side is always zero, while the right hand side can be easily arranged to be non-zero (take, for example, the characteristic function of O). In particular, the functional λ0 does not extend. The proposition is proved. ¤

We shall now apply this ideas to our situation. For any f in I(s) define

µ ¶

Next note that f ¡f0 vanishes at 1, so it is supported on the big double coset NwB. Arguing exactly as in the proof of the previous proposition we see that an extension of αw to I(s)

exists if and only if s 6= 0, in which case it is given by

Z

αw(f) = (1 ¡ q−s)−1 (f ¡ f0)(wn) dn.

N

Corollary 8.2. HomG(I(0), I(0)) = C.

This corollary follows from the Frobenius reciprocity, and the fact that αw does not extend to I(0). Combined with the fact that I(0) is completely decomposable, which will be shown in the next section, it will imply that I(0) is irreducible.

Proposition 8.3. Let L(s) = 1/(1 ¡ q−s). Then

L(¡s)L(s) c(s) = L(1 ¡ s)L(1 + s).

In particular, c(s) = 0 if and only if s = §1.

Proof. In order to calculate Aw(¡s) ± Aw(s), we need to make the formula for αw(f) more explicit. Clearly, f is Ki-invariant for some i, and we shall show that f(1) determines f(wn)

Let j · ¡i, and assume that x is in $jO n $jO. Since f is Ki-invariant, the relation implies that

In particular, it follows that f ¡ f0 is supported on N−i.

Next, note that the Iwasawa decomposition implies that I(s) has a unique element fs such that fs(gk) = fs(g) for all k in K, and fs(1) = 1. The above calculation can be made very precise for fs:

Exercise. Show that

αw(fs) = L(s)/L(1 + s).

Since Aw(fs) is K-invariant, it must be a multiple of f−s. The above exercise shows that

the multiple is L(s)/L(1 + s). The proposition follows.

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14 GORDAN SAVIN

be proportional to the form hf, gi. Since hfs, fsi = 1, it follows that they are equal, so our form is hermitian, after all.

Next, note that Aw(0) is the identity operator on I(0). Indeed, we know that it is a multiple of the identity operator, but the formula for the spherical vector shows that it is in fact equal to the identity operator. It follows that the form is positive definite on I(0). We shall use this observation to prove the following proposition.

Proposition 9.2. If 0 < s < 1, then the G-invariant hermitian form h¢, ¢i on I(s) is positive definite.

Proof. Since I(s) is irreducible for ¡1 < s < 1 the G-invariant form must be non-degenerate (otherwise the kernel of the form would be an invariant subspace). Moreover, as the following exercise shows, the restriction of this form to every finite-dimensional subspace of Ki-invariants is also non-degenerate.

Exercise. Let V be a smooth G-module with a non-degenerate G-invariant hermitian form. Then the restriction of the form to V Ki is also non-degenerate. Hint: Use the Ki-invariant projection operator P from V onto V Ki .

Every element in I(s) is completely determined by it restriction to K. In this way, we can identify, in a K-invariant fashion, I(s)Ki with the (finite-dimensional) space Ei of functions on B \KnK/Ki. By the above exercise, the invariant form on I(s) induces a non-degenerate form h¢, ¢is on Ei, which is positive definite for s = 0. The proposition follows from the following exercise.

Exercise. Let E be a finite dimensional vector space, and h¢, ¢is a family of non degenerate hermitian symmetric forms, depending continuously on the parameter s in a connected open

10. Steinberg representation

By Proposition 7.3 we now know that the principal series I(s) decomposes if ond only if s = §1, in which case it has a composition series of length 2. One subquotient is the trivial representation, and the other is the Steinberg representation V . Recall that

VN = δ.

We shall show that the Steinberg representation is square integrable, which means that the matrix coe cients of V restricted to G0 are square integrable. The results of this section are, in large part, due to Casselman.

Proposition 10.1. Let V be the Steinberg representation of G. Then the matrix coe cients of V are square integrable, when restricted to G0.

Proof. To prove this proposition, we need a result on the asymptotics of matrix coe cients

furnishes a canonical injection from HomG0 (V, V ) to H²(G0//K). Conversely, an element T in H²(G0//K) defines an element A in HomG0 (V, V ) by

Lemma 11.3. The algebra H²(G0//K) is one-dimensional.

Proof. Recall the Cartan decomposition

G0 = [∞a=0KλaK

Let f be in H²(G0//K). We shall show that f is a multiple of eK. Clearly, f is determined by its values at all λa. Let

If a ¸ 1, then n is in K1 and ²(n) = 1. On the other hand, recall that ²(n0) = ¡1. It follows that

f(λa) = f(nλa) = f(λan0) = ¡f(λa).

Thus, f(λa) = 0 if a ¸ 1, and f must be a multiple of eK. The lemma is proved. ¤

We can now finish the proof of proposition. Since V is generated by eK, Proposition 4.2 implies that there exists an irreducible quotient V 0 of V . Let P be the projection from V onto V 0. Since V 0 is cuspidal, by Proposition 6.3 there is a splitting s : V 0 ! V such that P ± s = IdV 0 . But

s ± P 2 HomG0 (V, V ) = C ¢ IdV

so s ± P must be equal to IdV . It follows that V = V 0, and the proposition is proved.

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12. The root system for GLn(F )

The purpose of this section is to introduce the root system for GLn. It is a combinatorial object which provides us with a language to study representations of GLn(F ). (For more information on root systems see [2]).

Let R be the field of real numbers, and consider the space Rn with the standard basis e1, . . . , en. With respect to this basis, we shall identify every element x in Rn with an n-tuple (x1, . . . , xn). The group Sn acts on Rn by permuting the entries of (x1, . . . , xn). This action

preserves the inner product

Xn

hx, yi = xiyi.

i=1

18 GORDAN SAVIN

Note that the group Sn preserves Ω. In fact, it is an irreducible representation. We shall now describe the action of Sn by means of euclidean reflections. Let

Ψ = fei ¡ ej j i 6= jg,

be a finite set in Ω. Its elements are called roots. Any root α defines a reflection w by

w(x) = x ¡ hα, xi ¢ x.

Exercise. Let α = ei ¡ ej. Show that the corresponding w is simply the permutation of i and j.

It follows from the exercise that the reflections w define the representation of Sn on Ω. Clearly, the fixed points of the reflection corresponding to the root α = ei¡ej is the hyperplane fxi = xjg, defined by the equation xi = xj. Consider the open subset obtained by removing all hyperplanes fxi = xjg

Ω◦ = Ω n ([fxi = xjg).

The connected components of Ω◦ are called chambers. We shall point out a particular chamber given by

C+ = fx 2 Ω j x1 > . . . > xng.

Next, note that the entries of any x in Ω◦ are pair-wise di erent, so there exists a unique permutation which puts them in a decreasing order. This means that Ω◦ is a disjoint union of w(C+) as w runs thru Sn. In particular, every chamber C is equal to

C = fx 2 Ω j xi1 > . . . > xin g

for some permutation (i1, . . . , in).

The choice of C+ gives us a set of generators of Sn as follows. Note that the closure of C+ is obtained by adding parts (called walls) of hyperplanes fxi = xi+1g. The corresponding roots

fe1 ¡ e2, . . . , en−1 ¡ eng

are called simple. Note that the simple roots form a basis of Ω. Let wi be the (simple) reflection defined by ei ¡ ei+1. We claim that the reflections wi generate Sn. Of course, since wi is nothing but the permutation (i, i + 1) this is a well known fact. We shall give a proof based on the fact that Sn acts simply transitively on the set of chambers. Let w be an element in Sn. Pick x+ in C+ and x in w(C+) such that the segment between them avoids the singularities of ΩnΩ◦. Since those singularities form a codimension 2 subvariety, a generic choice of x+ and x will do. The number of hyperplanes intersected by the segment is clearly independent of the particular choices, is the length `(w) of w. The chambers containing the segment form a gallery

C+, C1, . . . C`(w) = w(C+).

Any two consecutive chambers in this gallery share a wall. In particular, there exists a

we can obtain wi1 , . . . , wi`(w) such that

wi`(w) ¢ . . . ¢ wi1 (C+) = w(C+).

Exercise. Make a picture of the root system for GL3(F ). The roots form a regular hexagon. Let w be the element in W such that w(C+) = C−. Express w as a product of simple reflections by picking a gallery between C+ and C−.

13. Induced representations for GLn(F )

Let e1, e2, . . . , en be the standard basis of F n. Let Vr be the span of e1, . . . , er. Consider a partial flag

Vi1 ½ Vi2 ½ . . . ½ Vim

and let P be its stabilizer in G. Then P is called a parabolic subgroup of G, and it admits the Levi decomposition P = MN, where M is isomorphic to the product of GL(Vi/Vi−1), and U is the unipotent radical of P . Note that the minimal parabolic subgroup B corresponds to the full flag.

If (τ, E) is a smooth representation of M, then we can induce it to G by

IndGP (τ) = ff : G ! τ j f(umg) = δP1/2τ(m)f(g)g∞.

The character δP of M is defined so that for every locally constant, compactly supported function h on U we have Z Z

h(mum−1) du = δP (m) h(u) du.

U U

As a particular case of interest, consider the minimal parabolic subgroup B = T U. Note

w W

It turns out that VN 6= 0 for any irreducible subquotient, just as in the case of GL2(F ). In particular, each irreducible subquotient is completely determined by VN . Let S be the set of all fi, jg such that χi/χj = j ¢ j±1. Let

Ωχ = Ω n ([Sfxi = xjg)

where fxi = xjg is the hyperplane in Ω defined, of course, by the given equation. A result of Rodier states that the irreducible subquotients corresponds to the connected components of Ωχ. We shall here give a special case of that result, describing VN for the unique irreducible

20 GORDAN SAVIN

submodule V . (Since any irreducible subquotient of IndGB(χ) is a submodule of IndGB(χw) for some w in W , the general case easily follows.)

Let Ω+χ be the connected component containing the positive chamber C+. Define Wχ to be the set of all Weyl group elements w such that

w(C+) ½ Ω+χ .

Proposition 13.1. Let χ be a regular character. Let V be the unique irreducible submodule of IndGB(χ) (normalized induction). Then

Proof. The proof of this statement is combinatorial, and based on the special case of GL2(F ), proved in Proposition 7.3. Let Pi be the parabolic subgroup corresponding to the flag with Vi ommited. Note that its Levi factor Mi has exactly one factor isomorphic to GL2, and let wi (= w) be the permutation matrix in that factor. Assume that χi/χi+1 6= j ¢ j±1, so wi is in Wχ. Then, by Proposition 7.3

IndPi (χ) » IndPi (χwi )

B = B

Inducing both representation further up to G, this implies IndG(χ) » IndG(χwi ). In par-

B = B

ticular, χwi is a summand of VN . For a general element w of length m in Wχ we have to consider a gallery

C+, C1, . . . , Cm = w(C+)

stages (first from B to Pi), and using the second part of Proposition 7.3 (GL2(F )-reducibility) we get an inclusion

IndGPi (η) ½ IndGB(χ0),

where η is a character of Mi, which on the GL2(F )-factor is given by composing the determinant with the character

χij ¢ j1/2 = χi+1j ¢ j−1/2.

A more general form of the Bruhat-Tits decomposition implies that

M