The Bochner measure and holomorphic extension of elemenrary spherical functions Текст научной статьи по специальности «Математика»

THE BOCHNER MEASURE AND HOLOMORPHIC EXTENSION OF ELEMENRARY SPHERICAL FUNCTIONS

B. Krôtz *, R. A. Kunze and R. J. Stanton 2

Max Planck Institute University of Georgia (ret.),

Ohio State University

Introduction

Let G be a noncompact simple linear Lie group, and let K be a maximal compact subgroup. Let G denote the set of equivalence classes of irreducible unitary representations of G and let Gs denote those containing a trivial K-type. We recall several results from the work of Harish-Chandra. Any (■K,'H7r) G Gs contains the trivial AT-type with multiplicity one; let vq € be one of norm one. Denote by ipn the matrix coefficient

<P*(9) = (n{g)vo,v0).

Then (/^7,- is in CU(G//K), ipn(e) = 1, and cpn is a positive definite function on G. One can give an explicit realization of such (7r, ,H7r) by using induced representations, and in this way enlarge the parameter domain for these functions. So let P = MAN be a minimal parabolic subgroup with M C K. Set n^x = Indp(a ® xa ® 1) acting on H^x where (a, WŒ) G M, and Xa £ Hom(^4,C3:) = ajç. For a = 1, 7Ta := tïi.a contains the trivial K type with multiplicity one. While, for A 6 a£, nx need not in general be irreducible nor unitarizable, we do have Gs {(7TA)^a)|A € a^} and, after incorporating the intertwining operator into the inner product, this gives unitary realizations. The explicit form of the inner product on %x gives

<P\(9) '■= <P*x(9) = f a(gk)x~p dk Jk

which is clearly entire on a^. If we set

(p(x : A) := ipx(x), x E G, AG flj

then tp is real analytic in “x” and entire in “A”.

In [KS-I] we constructed a universal domain in the complexification, Gc, of G and we showed that the matrix coefficients obtained from A^-iinite vectors of any irreducible Banach representation of G have holomorphic extension to this domain. When one of the vectors is the trivial K-type this domain naturally maps to one in Gc/Kc. In [KOS] we showed that this domain, denoted S, is a maximal G'-invariant domain with this property, although not a maximal Ac-invariant one. Consequently </?(■ : A) has holomorphic extension to E. This domain is determined mainly by imposing two constraints, namely the holomorphic continuation of the A component from the Iwasawa decomposition and the construction of non-unitary principal series in conjunction with the subrepresentation theorem. In this note we shall give an

^his research was supported in part by a Heisenberg grant of the DFG

2This research was supported in part by NSF grant DMS 0301133

alternative justification of the holomorphic extension in the specific setting of spherical unitary representations and elementary spherical functions. The novelty of this analysis includes the definition of a spectral version of the Abel transform / -» Ff, and an integral formula which we make explicit for split rank one groups and from which we derive several consequences.

The starting point is the comment above that (pn is a positive definite function. The characterization of continuous positive definite functions on M by means of their Fourier transforms and more importantly their relevance to such areas as spectral theory and probability was discovered by Bochner. While there have been extensions of the characterization theorem to locally compact abelian groups and to some extent non-abelian groups, they have not played a central role in the harmonic analysis on these groups. Yet a central theme in the development of harmonic analysis on semisimple Lie groups has been to examine the objects of harmonic analysis - matrix coefficients and characters - on certain abelian (or almost abelian) subgroups. It seems natural then, following Bochner’s lead, to try to characterize the abelian Fourier transforms of these objects and hopefully to find some uses of them in harmonic analysis. This paper is the result of our first look at this problem.

The setting which seemed most amenable to this investigation is the non-unitary principal series of representations induced from a minimal parabolic subgroup MAN. Here the abelian group A enters in three well-known ways: part of the starting data is a non-unitary character of A; the representation space can be realized on functions on a subgroup V on which A acts in a reasonable way; the growth properties of if-finite matrix coefficients are determined by their restrictions to A.

In order to apply abelian harmonic analysis on A in the most direct way to matrix coefficients of principal series representations, we shall use a minor modification of the “non-compact picture” of non-unitary principal series which is built from an ^-invariant measure on V. The measure is constructed in §3 and the realization in §4. The Fourier transform, i.e., the Bochner measure, of matrix coefficients (not necessarily if-finite) of unitary principal series follows easily from this realization and is included in §4. In §5 we consider the M-rank 1 case and show that the Bochner measure can be computed for non-unitary principal series, if-finite coefficients provided |Re A| < \p\. For the zonal spherical functions, in §6 we compute the Bochner measure explicitly for split rank one groups, obtaining an integral representation which seems new even from a classical special function point of view. In §7 we give four applications of the Bochner measure. In one we derive the Harish-Chandra expansion directly from the integral formula without any differential equations. Another is to define a spectral Ff with properties in the spectral parameter analogous to those of Ff in the group parameter.

§1. Laplace transforms.

In classical abelian harmonic analysis there is a well known relationship, due to Paley and Wiener, between the holomorphic extension of functions to tubular neighborhoods and the exponential decay of their Fourier-Laplace transform. This will be examined in the context of spherical functions. We shall identify spherical functions as Laplace transforms on A = ia* = a*. For this purpose we briefly review some standard facts on Laplace transforms.

Let F be a finite dimensional real vector space and V* its dual. We denote by Vc = V + iV the complexification of V. Let p be a positive Borel measure on V* and set

= {x € V : j e dp(a) < oc}.

Shortly we shall assume that p is admissible, i.e., DtJ ^ 0 (thus, in particular, that p is a 372

Radon measure). The Laplace transform of ¡1 is then defined by

Cjj, : Eq , x f e~a^ dfi(a).

Jv*

Note that Cji automatically extends to a holomorphic function on the tube domain T= iV + intD/( C Vr- We denote the extension also by Cft. We shall need the following results on Laplace transforms.

Proposition 1.1. Let ¡j, be an admissible positive Borel measure on the dual V* of a finite dimensional real vector space V. Then

(i) the domain of definition DM of the Laplace transform Cfi is a convex subset of V;

(ii) suppose that fj, is finite and that there is an open connected set Q C V with 0 G il such that Cfj, has an extension to a holomorphic function on the tube domain Tq = iV + then we have ii C D^; i.e., on Tq the extension is represented by

Tq C, f e~a{z) dfi(a).

Jv*

Proof, (i) [Ne, Prop. V.4.3], (ii) [Ri, p. 311]. ■

Let A = Mn be a simply connected abelian real Lie group. We identify the unitary dual A of A with (Rn)* by means of the isomorphism,

(Rn)* -> A, a^(x^eia^).

Let ¡3(A) denote the Borel cr-algebra on A and for (7x,'H) a unitary representation of A let P(V.) be the set of selfadjoint projections on H. Then there exists a spectral measure

E : ¡3(A) -> V(H)

such that

(Vs G A) n(x) = f eiQM dE(a).

Ja

We are now going to apply the above considerations to spherical functions i.e. as matrix coefficients of the if-fixed vector of spherical principal series representations. Recall from [KS-I] the definition of the set fl C a (or see below) and from Proposition 4.1 therein that ip\ holomorphically extends to Gexp(iCt)Kc, and from Theorem 4.2 to ifcexp(i2i2)ifc-

Lemma 1.2. Let A € ia* and ip\ be the associated spherical function on G/K. Then there exists a Radon probability measure ¡i on a* such that

(Va G exp«2fi) (p\(a^1) =

Proof. We consider the unitary representation (7r,"H) = {tt\ \a,'M\). The probability measure fj, in question is then given by fx = EV0jV0 with vq = Ik- In particular, we get that

(Va G A) (a-1) = [ eia(loga) dfi(a).

J a*

Now the assertion of the Theorem follows from [KS-I Theorem 4.2] and Proposition l.l(ii). ■

I eia(loga) dfx(a^

Remark 1.3. For A 6 ia*, as mentioned before, the spherical function tp\ is a continuous positive definite function with <p\(e) = 1. In particular on the abelian group A = a, tp\ \a has these properties. Hence the existence of a probability measure // on a* such that

^(cT1) = [ e“(loga) dn{a)

J a*

holds for all a G A is also a consequence of Bochner’s Theorem. (In fact, Bochner used his result to give a proof of the spectral theorem.) However, the validity of the extension to exp 2iQ of the Bochner integral representation is far from obvious. For example, the geometry of the tube reflects growth estimates of the Bochner measure and these are usually difficult to compute. All these remarks are illustrated in considerable detail in the case of rank 1 groups in subsequent sections.

In view of Lemma 1.2 and Proposition l.l(ii) every restricted spherical function ip\ o exp^, has a holomorphic extension to a tube domain

^A,max = A +

which is maximal with respect to u\ C a open, convex, 0 £ u\ and ui\ is Wa-invariant. In [K-S I Th. 4.2] we show that 2Q C ui\ and in §5 for all real rank one groups 20, = u)\ where 20 = {X E a : (Va G S) \a(X)\ < tr}.

§2. The Technique.

Let X be a locally compact space and let A be a locally compact group with a continuous proper action A x X —> X, denoted (a, x) —> a ■ x. Let dx be a measure on X and da a left Haar measure on A. We shall make some assumptions to specialize the presentation for the simple applications we have in mind.

First concerning the action, we shall assume that A acts without fixed points so that each orbit is homeomorphic to A; we shall require that the orbit space A\X is paracompact; and we want the measure dx to be A invariant. We will be concerned only with the case in which A is abelian and isomorphic to a finite dimensional real vector space. Under such conditions, there is a unique measure dui on A\X and the usual “double integration” formula for reasonable / ([B] p. 44):

j f(x) dx =

X A\X A

Moreover, if we make A x A\X an yl-space by translation in the A coordinate, then X and A x A\X are homeomorphic ^4-spaces ([B] p. 73).If / is a function on X, by abuse of notation, we will let / also denote the corresponding function on A x A\X. The integration formula then becomes

J f(x) dx =

X AxA\X

Since each orbit type is basically A we can transfer A harmonic analysis to each orbit. Thus, if A denotes the unitary character group of A with dual measure, for reasonable functions / and (j on I we have the familiar abelian results:

i1)' iixx^A /(A,w) = J f{t,u)x\{t)~lda;

A

fnr n a A wiiVi tViP nfinal t.ransnnri; nf ctrnrfiivp T* f(i /.A — /.A flion

/(a,cj) da x du.

f(a ■ x) da du.

(2 Y

374

= x\(a l)f(A,w);

and the Parseval formula

(3)' J f{t,cj)g{t,uj)da= J

A a

For our purposes, we shall need to consider the usual action L*a on functions twisted by a character. This causes minor changes in (1)', (2)' and (3)'. So suppose Xv is a continuous character of A (not necessarily unitary) and define

a ■ = Xv(a)L*af(t, u).

For the function jv defined by fv(t,oj) = Xv{t~l)f{t,uj) we have

(a •/)„(*, w) = X«(i_1)xw(a)L*/(i,w)

= L*afv(t,u>).

We shall call the contragredient to Xv the character Xv' defined by XvXv' = 1- For this ^4-action the modified Fourier transform defined by

(2.1) :=

has the property

(2.2) a ■ f{X,u) = XA(a_1)7(A,w),

and the Parseval formula

(2.3) J f(a,cj)g(a,u))da = j fv(\,u)gvi(\,w)d\

A A

= J f(\,u)g(\,u)d\

A

provided fv and gvi satisfy reasonable growth conditions.

§3. The invariant measure on V.

An interesting example of such an A-space (in fact our motivation) arises in semisimple Lie groups. We shall briefly recall some basic facts about these groups. (See [HI]).

Let G be a real semisimple Lie group with Lie algebra 0. Fix a Cartan involution and let

0 = i®p be the corresponding decomposition. Choose a maximal abelian subspace a C p. The adjoint action of a on g is diagonalizable; let E denote the set of non-zero real eigenvalues and 0Q, a G E, the corresponding eigenvectors. Fix a lexicographic order on E and let E+ be the positive elements. If subalgebras of 0 are defined by

n = S«’ 0 = 9“

o£E+ -ae£+

ana mi = zero eigenspace, then 0 = n © mi © 0. Let V,N,Mi,A denote the corresponding subgroups. Then Mi = MA where M C K and centralizes A. Moreover, V, N and A are homeomorphic to their respective Lie algebras.

Since 0 consists of a-eigenspaces, the group A acts on it and hence on V. Denote this action on V (conjugation) by (a,v) —> a ■ v or Sa(v), and on functions by 5*f(v) = f(Sa-i(v)). There are two aspects of this action which we must examine before using §2: (1) the obvious measure on V, Haar measure dv, is not invariant but only relatively invariant; (2) the action of A on V is not free. We shall remedy the first and define away the second.

The multiplier for the A action on dv is given by the character x~2p, where 2p = ^ag£+ ai i.e. for / G Cc(V)

Sa*dv[f] := dv[S*f] = x~2p{a)dv[f] = %2p(a~l)dv[f].

We will use x~p to construct an invariant measure on V.

The eigenspace decomposition of g has an analog for G, namely the Bruhat big cell, such that except for a closed set of lower dimension G — VMAN with the unique ^.-component denoted a(g). Let w G G represent the Weyl group element that sends £+ to — S+. Then w2 represents the identity, wNw~~l = V and for v ^ e, w~lv G VMAN ([Kn-S]). We extend x~p to a function defined a.e. on G by

:= X-p{a(x))-

Lemma 3.1. If x is in VMAN and a is in A then ax and xa are in VMAN. Also, £_p(ax) = X-p{a)^p{x) and £,-p{xa) = p{x)x~p{a).

Proof. Obvious. ■

Define Xwp{a) = Xp{w~1(lw)- Then Xwp is a character and

Lemma 3.2. Xwp is contragredient to xP-

Proof. Since Xp is M-valued, this follows from wT,+ = ■

Proposition 3.3. The measure £_p(w~1v)dv on V is A-invariant.

Proof. Let / G Cc(V). First, notice that

¿a(i-p(^_1-)/)(^) = (,-P(w-1a~1va)f{a~1 ■ v)

= X-p(w~1a~1w)^-p(w~1v)x^-p(a)S*f{v)

= x-wP{a~1)x-p{a)£-p(w~1v)SZf(v)

= X-2P(a)^-p{w~1v)6*af(v).

Here we have used Lemmas 3.1 and 3.2. Now

■№«)[/] = £-p(™-1-)<M^a/]

= dv[C_p(u;_1-)5o/]

= dv[x2p(a)6*a(^-p{w~l-)f)}

= X2p(a)8a*dv[i-p(w~l-)f]

= X2P{a)x~2P{a)dv[i-p{w~l-)f}

= i-p{w~l)dv{f]. ■

To obtain a space on which A acts freely we shall discard a finite number of lower dimensional subspaces of t> leaving an open dense subset t>' which can be identified with a similar set

T/7 C~ 1/ T?r»r nrrlAr li>f rv. . . ru * p Kocia cimnlo rocfrr»r\+c PY»v ocnYt n lof

Each Wi has a unique complementary subspace which is a sum of eigenspaces, say W-. Let

Wc = U Wf and set o' = o\Wc.

¿=i

Lemma 3.4. A acts freely on o'.

Proof. If for each i, P, denotes projection along Wf of 0 Wi, then o' can be characterized l

as fl {x G o|P{X 7^ 0}. Now each a £ A has all non-zero eigenvalues and commutes with each i~ 1

Pi, therefore o' is A-invariant. Also, a\, • • • ,a.i form a basis for the dual space a', so for each a G A, a ^ 1, there is an i with Xa% (a) ^ 1. If a: G o' were fixed by a then PiX = P^a • x = a ■ PiX. But PiX ^ 0 and on Wi the element a acts by Xa;(a) or Xai(a)2 both of which are not 1. Thus the action is free. ■

We let V' be the corresponding dense open set in V and restrict £,~p(w~1v)dv to V'. This serves as the (X, dx) of §2.

Remarks. (1) If dim a = 1 then o' = t)\{0} and so is maximal. If dim a > 1 the maximal subset on which A acts freely is not necessarily o' but can be constructed. As we have no use for it here and the construction is standard, we omit it.

(2) M also acts on V by (m, x) —> m ■ x = mm“1, One can show that M acts on V' and the measure £,-p{w~lv)dv is M-invariant. However, for the elementary spherical functions we shall not need this.

(3) It is worth noting that all that is done here can also be done for degenerate principal series induced off a maximal parabolic with n a prehomogeneous vector space of parabolic type.

§4. The Bochner measure — unitary induction.

We recall briefly the construction of the principal series representations of G induced from the minimal parabolic subgroup MAN. We shall use as Hilbert space for these representations, the “noncompact picture” consisting of vector valued square integrable functions on V. Since the A-invariant measure is absolutely continuous with respect to Haar measure on V', a standard procedure provides an alternate realization of the principal series.

Let (<j,Wa) G M, and let xa be a character of A, not necessarily unitary. In the “induced picture” one uses

C°°(G; a 0 xa) = {/ : G -*• Wa I / is C°° and f{gman) = a{m)~lx\+P{a)~l f{g)}

and has G act by left translation. From the Iwasawa decomposition, / is determined by its values on K and one takes as norm the L2 norm of / restricted to K. From the Bruhat big cell, / is also determined by its values on V. In this “non-compact picture” the representation space H{a,x) consists of the closure of C^°(V, Wa) in the norm

/ II/WIIJ

where v = x(v) exp H(v)n(v) from the Iwasawa decomposition and e2ReAF^) = |xA(exp H(v))\2. The G action in this realization is given by

U(a, X){g)f(x) = aimig^x^xx+piaig^x^fivig^x))

where g~lx has been expressed in VMAN coordinates.

Let x'\ be the contragredient character to xa and form 7i(a, A'). The mapping / h-> x|(exp H(x))f(x) is an isometry of Hier, A) onto %(a,X'). Moreover T-L(a, A) and 7i(a,X') are naturally dual via the pairing

if,g) = J <f(v),g(v) >adv. v

In addition, for / in %((7, A), g in A') and y in G, the matrix coefficient c/i5(y) is defined by

cf,g(y) = (U(a, A)(y)f,g)

and one has the identity

{U{a, A)(y)f,g) = (/, U(a, X')(y~1)g).

The A-invariant measure on V' is £_p(w~1v)dv. Define a Hilbert space 7î(<j, X) to consist of the closure of C£°(V, Wa) in the norm

/ \\f(v)\\l e^MO-1«)*.

y

The map T : T-L{a, A) -> H(a, A) given by

Tf{x) = ^p{w~lx)~ll2f{x), x^e,

has inverse

T-1f(x)=^p(w-1x)1^f(x)

and establishes an isomorphism between these spaces. There is then an equivalent realization of the principal series U(a, A)(-) on 'H(a, X) which we refer to as the “A-adapted” realization and is given by

U(a,X)(-)=ToU(a,X)(-)oT~1.

A straightforward computation using Lemma 3.1 and Lemma 3.2 gives

(4.1) Ü(a, A)(a)f(x) = x\(a)Kf(x)ia ^ A’

(4.2) Ü{o,X)(m)f(x) = a{m)ô*mf(x),m e M.

Furthermore, 'H{a, A) and H(a, A') are in natural duality via

(4.3) (f,g)= J < f(x),g{x) >a £-p{w~lx)dx,

V'

and one has

(4-4) (Ù(a, X)(y)f, g) = (/, U(a, X')(y~1)g).

The A-adapted realization gives a new integral representation for matrix coefficients with close connections with abelian harmonic analysis. Using the construction from §2 we take fx(t,u>) = Xa(^-1)/(Î) w) with modified Fourier transform f{v,u) = and similarly

with the contragredient x'\ an(ï g we obtain g.

Theorem 4.1. Let x\ be a unitary character of A and let dima = i. Let a be a finite dimensional unitary representation of M. Let f be in ’H(cr,\) and g be in H(a,X'). Then, for a in A

(4.5) Cft9(a) = Í e~lvlogam(X,v)dv.

Here m(A, v) is a function depending on f and g and given by

(4.6) m(X,v) = / < f(v,cj),g(v,uj) >a duj.

A\V'

Remarks. We shall refer to the measure m(X,v)dv as the Bochner measure for cj>g. Since U is unitary the matrix coefficients c¡j are positive definite functions on G, a fortiori on A. Hence Bochner’s theorem assures the existence of a non-negative measure with this property while (4.5) shows it is absolutely continuous with respect to Haar measure and (4.6) gives a formula for it.

Proof. We begin with (4.3)

cf,g(a) = (U{o,\){a)f,g)

= J <Ü{a,X)(a)f(x),g{x)>a^p{w~1x)dx v

< U(a,X)(a)f(h,u)),g(h,u) >a dh x dw

AxA\V'

< dh x dco.

AxA\V’

We continue with a formal application of Parseval’s formula (2.3) and (2.2) to get

cftg{a) = J Xvia)"1 < f{v,uj),g{v,uj) >a dv x du

AxA\V

= Í e-™loga m(X,v)dv.

To justify the use of (2.3), we observe that f,Uf and g have square integrable cr-norms with respect to £,-p(w~lx)dx. Hence, a.e. duj, Uf(-,u) and g{-,uj) have square integrable cr-norms relative to dh. Since a 6 A acts by Xa(o-1) and (2.3) is valid for each component of the vector the result follows. ■

Remarks. The unitarity of %a was used only to justify (2.3). For non-unitary xa- whenever

(2.3) can be used there is such an integral formula. Also, notice that / and g need not be if-finite.

§5. Non-unitary principal series.

We assume in this section that dim o = 1. Then a = cl(a+ U wa+w~l). If A and v are C-valued linear forms on a we say A > v if Re(A — w)(a+) > 0. 379

Theorem 5.1. Let x\ be a character of A with ReA ^ 0 and both p > A and p > —X'. Let a be a finite dimensional unitary representation of M. Let f in %{a,X) and g in 'H(a,X') be smooth vectors. Then

(5.1) Cftg(a) = J e~tvloga m(X,v)dv, a E A,

R

where

(5.2) m( A, v) =

A\V'

Proof. As remarked in §4 it is enough to justify the use of (2.3). Now it is well known that the intertwining operator between the compact and non-compact realization A) is given by

Ih{x) = e-(x+riH(-xh{x(x)), x G V.

Following I with T we may suppose / (and similarly g) is of the form

^p{w^xr1/2e-^+^H^h{x(x))

with h a if-finite valued function on K. In particular, each component of h is bounded.

Now,

U(a, X)(a)f(x) = x\(a)£-p(w~1a~1xa)~1/2e~(x+p')H(a 1 xa^h(n(a~lxa)).

Since h has bounded components, for the purpose of estimates it may be ignored. Choose a cross-section A\V' —> V' and denote a fixed element in the range by to. Recall the definition from §2 of fx and gx>, i.e., fx{tu)t~l) = x\{t)~l f{t,u) and gx'{tLot~l) = x\' {t)~lg{t, w). We shall show that the Parseval formula (2.3) applies to the functions

XA (í)^_p(«j_1ícoí“1)-1/2e-(A+p)jHr(twt_1)

and

Xv(í)f-p(^“1^í"1)"1/2e-(V+p)ff(íut"1).

First notice that these functions are locally bounded in t and so it suffices to consider their behavior for t —> oo in A+ and = wA+w~l. Next, from Lemma 3.1 we have

f\{t,u) ~ e~xlostx-p(w~1tw)~1/2x-p{r1)~l/2í,-p(w~1ui)-1^ x

x e-(A+p)ff(twi_1)

and

g'x{t,uj) ~ e~A<l°stx-p(w~1tw)~1/2£_p(w~lu))~1/2e~(x' and using Lemma 3.2 and ignoring ^p{w^lu)^1!2 we get

f\(t, uj) ~ e_ Alog te~plog*e_(A+/,)ir(iaJi_ 1)

and

gx>(t,cj) ~ e-A'logíe-/,logte-(A'+p)ií(íwi“1).

Now, if t £ A.|_, since uj is in V', lim tujt~l = 1 when t -» oo. Thus for t —> oo, e~(x+p)H(tujt an(j

A+ A+

e-(X'+p)H{tujt are boun(je(j5 while e~^A+p)logi and e_^A,+^logi are integrable on A+ because p > A and also p > —A'.

380

J < f{v,u),g(v,Oj) >a du

Before we consider t in A_ we shall express f\ and gy in another form. For x in V H(x) = H(w~~1x) = H(v(w~lx)) + log a(u;-1 a;),

thus

H{tujt~l) = H{v(w~ltujt~l) + log a(w~ltbjt~l).

Also, we have

log = loga(«;_1ic;) — logi + \og(w~1tw)

and

v(w^1tu>t~l) = w~ltw v(w~loj)w~lt~1w.

After discarding terms independent of t we have

~ e~(A+p)1°síe_(A+p)log^~lííí’)e(A+p)1°sí X

v — (\+p)H{w~1tw v(w~1u))w~1t~1w)

X (3

and

g\i(t,Lo) similarly with AflA1.

Simplifying gives

~ e-(x+P)l°g(w~ltw)e-(x+P)H(w~ltw v(w-1uj)w-1t~lw)

and g\i similarly with A o A'.

Now let í G A_. Then w~Lt/w G A+ and Harish-Chandra [H-C] has shown that if h G A+,

¡j, > 0 and igF then

fi(H{x) — H(hxh~1)) ^ 0.

So, with /1 = Re (p + A) or /j, = Re (p + A') we get

| — (\+p)H(w~1tw v(w~1ui)w~1t~1w)\ ^ g—Re(A+p)H(w~1u>)

and similarly for the A' term.

Thus, ignoring terms bounded, for t in A- we have

|/A(i,o;)| ^ e~M^+p)^(w-Hw)

and

\gx'(t,co)\ < e-W+p^o&iv-1*»).

Both of which are integrable on A_. Hence, the Parseval relationship (2.3) can be applied to /, 9-

Remark. We note that the only use of dim a = 1 was to write A = cl(A+ U A_). Otherwise, the estimates hold in general rank on a+ and the opposite chamber.

§6 Computation of Bochner measure.

The function part of the Bochner measure, m(A, u), contains much information about the matrix coefficients of unitary induced representations. For complementary series the unitary structure on T-L{ 1, A) must be modified by the intertwining operator, however on the if-fixed vector the intertwining operator in the “compact picture” is a scalar. Thus we can use Theorem 5.1 for the spherical complementary series (cr trivial) as well as for the spherical principal series

381

and the zonal spherical function. To get some understanding of m(X,v) we shall compute it, exactly in these cases.

We first recall some calculations from [H2], The set £+ consists of, at most, {a, 2a}. We let p = dimga and q = dim 02a and choose the usual inner product on 0. If X denotes elements of 0_a and Y those from 0-2<*> each x in V can be written uniquely x = exp X exp Y. The relevant formulae are ([H2] p. 59)

(6.1) e-pCur1*) = [c2|X|4 + 4c|y I2]

(6.2) epH(x) = [(1 + c\X\2)2 + 4c|y|2]1/4(p+29) where c~1 =4 (p + 4q).

As remarked in §3, o' = o\{0}. It follows from Lemma 3.1 that the level set ^p{w^lx) = 1 provides a cross-section A\V' —» V'. Also, being M rank 1 we express A = Ap with |Re A| < 1 in order to apply Theorem 5.1.

The if-fixed vector in %(1, A) is

fx(a,co) = e~xp log ae~(xP+p)H(auja~ 1^_p(w~1auja~1)"~1/2

— g- Ap log ae(\p+p)H(auja-1 )Q-p log (^- 1 w) -1/2 _

Using £_p{w~lui) = 1 and Helgason’s formula (6.2) gives

fx(a, u>) = e-(A^)losa[(l + c|e“QlogaX|2)2 + 4cle-2“logay|2]'(p+29)(1+A)/2 We set s = e-“loga and use p = (p + 2q)a/2 to get

s(p+2q)(l+X)/2 ~ [(1 + CS2\X\2)2 + 4C54|y|2](P+2<7)(l+A)/2 ‘

Next setting t = s2 and using 1 = £-p{w~1uj) = e2|X|4 + 4c|Y|2 gives

i(Jj+2?)(l+A)/4

fx(t, w) - ^ + 2i + ^(p+2?)(1+A)/4 •

Finally, since c\X\2 ^ 1 we write it as cos 9. Thus, using (2.1), we have

f{vp,uj) = J XivP{ay1fx{a,^)da

A

r Av^aMl+X-iv)!^ j./,

J " ILL

J [1 + 21 COS 9 + i2](p+29)(l+A)/4 J 0

as the integral to be evaluated for A and similarly A'. Let a = (p + 2q)(l + X — iv)/4 — 1 and b = (p + 2q)(l + A)/4 and note that Rea > —1 and Re6 > 0. We consider, under these circumstances, the integral

OO

r 4a

0

We factor the denominator and treat the integral as a contour integral. Thus,

OO

ta

(etd + t)b{e.~ld + t)b

0

[e~iet]adt

[1 + e~iSt]b[e~2lS + e~t9t]b

= e

-2ib6 AaS

§7. Applications.

In this section, we give four applications of the previous sections.

Positive Definite.

We have already remarked that m(A, v) ^ 0 is necessary for unitarity of U. Conversely, one might ask if it is sufficient, or, in other words, if the restriction to A are positive definite must U be unitary?

Proposition 7.1. Let A = Ap and suppose |Re A| ^ 1. If either (i) Re A = 0 or (ii) Im A = 0, then ip\ is positive definite on A.

Proof. First suppose |Re Aj < 1 for then we may use (6.3). Of course it suffices to show that m(A, v) ^ 0. If Re A = 0, then A' = A and

m( X,v) — / f(v,u))f(v,uj)dw^0.

A\V'

If Im A = 0, then A' = —A. In (6.3) the parameters a,/3 in the hypergeometric function are complex conjugates, while 7 is positive, so the function is positive. Also, the T-functions in the numerator are conjugates while the denominator is positive. Hence, f(v,u) > 0 for A and similarly for A'. If A = p, a careful look at the integral I shows the calculation is valid for A and as just observed is a positive function. But for A' = —p we must compute the Fourier transform of 1, getting a Dirac measure. Thus, the Bochner measure for A = p is positive multiple of a Dirac measure at v = 0. ■

In [K] the unitarizable spherical principal series are determined and shown to be (i) Re A = 0 or (ii) Im A = 0 and |A| ^ p/2 when q = 0 and when g / 0 one also has |A| ^ (p + 2q)/2. Comparison with Proposition 7.1 suggests an extension problem.

Problem 1. Let f be defined on G and suppose f is positive definite on A. Find sufficient conditions for f to be positive definite on G.

If instead we restricted to a larger group there is an easy answer.

Lemma 7.2. Let G be a locally compact group, K a closed subgroup S C G with G = KS. Let f be defined on G and bi-K-invariant. Then, f is positive definite on G if and only if f is positive definite on S.

Proof. It is enough to show / is positive definite on G. Let x\ . ■ ■ • , xn be in G and say Xi = kiSi. Let Ci, ■ ■ ■ ,cn be complex numbers.

T: f{xjx~l)cjcj = /{kisisj1kj1)clcj

= ^fisiSj1)^ ^ 0. ■ ij

In [F-K] the set S = AK was used and the if-type expansion together with Bochner’s theorem on the compact group K gave another proof of Kostant’s theorem. Alternatively

After some elementary manipulations involving deformation of contours one arrives at

l

J = J xa(1 _ ^Tb-a-2^. _

0

With v = z~1 this is found in [W-W], p. 293 to be

J=r(° + 1)r(26)~Q~~ ei(a+1)&F(b’a + 1’2b’v)’

using Gauss’ hypergeometric function, provided |u| < 1. Now |w| = |2sin#| < 1 means 0 ^ 9 < 7r/6. To get the integral for 9 G [0,7r/2] we analytically continue the hypergeometric function by the quadratic transformation ([BP])

________ „fat „ a ,, , 1 z2 \

*-ya,pW,Z) = V-Z) p + -; -^—Y))

or, finally,

r(a + l)r(2b — a — 1) /a + 1 a + 1 , 1 . 2

I — —------------—----------F[-------, b---------; b-1—; sin 0

r(26) V 2 ’ 2 ’ 2’

Recalling how a, b were defined we get

r(?(p + 2g)(l + A - w))r(i(p + 2g)(l + A + it»))

(6.3) /(w,w) =

where

r(i(p + 2g)(l + A))

x F(a, /3; 7; sin2 9)

a = ^(p + 2g)(l + A - iv),

/3 = I(p + 2g)(l + A + ti;),

7 = \{P + 2<l)(t + X) + \-

Similarly, for A'.

Some comments are in order. When there is no 2a root the cos0 appears as 1 in the beginning and then the hypergeometric function is evaluated at zero, hence is 1 (a considerable simplification!). Next, after we had evaluated the integral, we happened across it in a table. But, as similar computations for prehomogeneous vector spaces of parabolic type are a possibility, we think it worth presenting. Finally, compressing the notation, the resultant integral formula for the spherical function ip\,

(6.4) <Pa(g) = [ e_wlogam(A, v)dv

seems new to us even from the special function literature. The only analogous, though definitely different, integrals were obtained by Barnes [Ba],

S = MAN leads to the use of Kirillov theory on N together with a Bochner theorem for nilpotent groups.

Harish-Chandra expansion.

We shall obtain from the integral formula (6.4) another derivation of Harish-Chandra’s expansion for the elementary spherical function tp\(a) but without the use of any differential equations. Fix H £ a of norm one, and set at = exp tH. Recall that the level set Q, =

{£y-p(w~1x = 1} is a cross-section for A\V/. From (5.1)

<P\(at) = J e~lvt J < f(v,u)),g(v,uj) >a du>dv

R Q

here f(v, u>) is given by (6.3). But first, we need

Lemma 7.3. If X = Xp with |Re A| < 1 then the Bochner measure is of the form

m(X,v) = T(X,v)T(—X,v)h(X,v) where h is entire in v G ac and

^ _F{r(l + X-iv))T(r(l + X + iv))

{,V) r(r(2 + 2A))

here r = (p + 2q)/A.

Proof. From (6.3) we define Y(A,u) to be the gamma functions appearing there. Next, simply observe that the complex conjugate of (4.5) for A' simply takes A —> — X. We must consider the integral over A\V’/ of the product of two hypergeometric functions. But, the D-dependence is only in the a and /3 parameters in which the hypergeometric function is entire (note that Re(Y — a — ¡3) >0) and as above, the complex conjugate can be absorbed as a change of sign since A\V' may be viewed as the compact set £-p(w~1x) = 1 we have the resulting integral, h(A, v), entire in v G ac. ■

We illustrate the computation by giving the details for SL(2,R) and A imaginary. In order to compare the result to ones in the literature we change notation slightly. Henceforth A imaginary will be written as iX := iXa, and similarly in the integral we replace v with va. and tH with the co-root tHa. In this case the integral becomes with r = 1/4.

(7.1) ------------r(r(2 + 4A))------------X

m

r(r(i-i(| +j)))r(r(i-,(|-D))

r(r(2 — 4A))

The integral consists of terms of the form T(r ± i(r/2)(A ± v)) and these functions have poles at ±A ± i(2k + 1/2). For A 0 all poles of the integrand are simple and one easily sees that the residues are of r(^) when z — —k. To compute Res^r(z) use the duplication formula

r/V'irh _ — IT / G1 n tt y nnH nnp nht ^ inc

To evaluate the other terms one needs to simplify F(—k + iX). Again, a use of the duplication formula and the fact that

^ (-1)7r Sin7T?A = ' ——

iAr(iA)r(—iX)

gives

rt k I

M k + ,M- r(1 + t_,A) •

We evaluate (7.1) by an application of Cauchy’s theorem and a rectilinear contour from —R to RtoR + iRto —R + iR to —R. It is routine to show the contributions other than along —R to R tend to zero as R —>• oo. The result is

(n] = r(*'A) V r-iA* -f2A:+l/2H r(fc + V2) r(V2 + k-iX) T(1 - iX)

r(i/2 + iX)£^o T(k -f 1) r(l/2 — iX) T(1 + k + iX)

_L T(-iA) ^¿\tn-(2k+i/2)tF(k + 1/2) r(l/2 + k + iX) r(l + iA)

1 r(l/2-tA)j^" " r(Jfe + l) r(l/2 + iX) r(l + k — iX)'

Remarks. (1) It is curious that this derivation uses in no explicit way the differential equation for the spherical function, in contrast to all other derivations of it.

(2) Obviously the same computation works for SO(n, 1) using a different value of “r”. The computation for the other rank one groups is complicated by having to evaluate

IT/2

J |_F(o!,/3;7; sin2(0))\2dd.

o

Our only progress to obtain a clean formula here was to expand one of the hypergeometric functions and use [B] p. 399 to evaluate each of the resulting integrals; thereby obtaining an infinite series in quotients of T functions at various arguments. This was sufficient to obtain the ”c” functions and the leading term.

Holomorphic extension.

Again, we illustrate the method by presenting the details solely for SL(2, ] with (7.1):

T (r(2 + 4A))

T(r(l - + S)))r(r(l - ¿(| - J)))

dv

We begin

T(r(2-4A))

We are interested in the Paley-Wiener phenomenon, i.e. exponential decay of the Fourier transform is equivalent to a holomorphic extension of the function to a tube. The case A = 0 contains all the essential technicalities; so we present that. Hence consider

I =

dv.

Since T(u + iv) = T(ti — iv) the T functions in the integral can be rewritten as

, 2 jlN - - -?)1 12

386

r(i + - i„)

Take the logarithm and use Binet-Stirling to get

1 i

—I—V

4 2

v ¡1 i

+ 2argU“2V

I

4

v /1 i

2arg(i + 2V

i (I 7 ar8

(H“)

- 7arS (4 “ + Oi1)-

Now for v > 0 we have

while for v < 0 we have

/1 i \ 7r 1

^Vi 2 / ~ 2 2v’ V^Ca'

Then

while

Then

arg

/1 * \ 7T 1

&XgKl±2V)~^2±M ^°°-

(i + _ arg(l ~ 5V) ~ ±7F T j^l ’ V < ° ^ r6Sp' V >

(i + +arg(i "" i0) =

arg

logr(l + ^)+logr(i

v I ~ log

log

Consequently

1 I

—I—v 4 2

1 i

—I—V

4 2

~ c

-i/z v ~ 2

-1/2

± 7T =F

+ 0(1)

u|7r/2 + 0(1).

7T|U|

_1______I V2

16 ~r 4

So the spherical function holomorphically extends in “x” to “x + iy”. Moreover, one can detect the nature of the singularity. Indeed, taking x — 0 and writing y = n — e, we obtain the Laplace transform

dv

I

M>.R

J_______, v2

16 'r 4

which is easily seen to be ~ lne. Both these results can be found in [KS-I] with a different proof. In fact, the preceeding computation was used as independent verification of the computations therein.

Remarks. The extension to A ^ 0 presents no difficulties, nor does the extension to SO(n, 1). The remaining rank 1 groups can be done this way; albeit with additional work. As usual, the sticky point is (5.2) which here is the integral over S7 of F(2a, 26; a+6+1/2; sin2(0/2)). However we need only estimate - not evaluate - it. After some transformations on the parameters of the hypergeometric function one obtains (up to easily estimated factors) the Legendre function P„m(cos ip) where n £ C, |n| —» 00. For this the asymptotics in [Wa] p. 291 are adequate to obtain estimates which allow one to detect the size of the holomorphic extension, and the nature of the singularity. We omit the details because there is a detailed proof of the result in [KS-I],

Spectral Ff.

We begin by recalling properties of the Abel transform as defined by Harish-Chandra. For / <E Cc{G//K) define Ff by

Ff(a) = eploga J f(an) dn.

N

The Abel transform A : f —» Ff has several interesting properties of which we highlight only those immediately relevant.

(7.2) A : Cc{G//K) > CY{A)

where C™ (A) denotes the compactly supported Weyl group invariant functions on A.

(7.3) A is continuous in the usual topologies.

(If one uses the Harish-Chandra-Schwarz space, then an analogous version of both (7.2) and

(7.3) are valid.) Hence there exists A' : {A)' —> Cc(G//K)' satisfying for / 6 Cc(G//K)

and T G C™ (A)'

< Af,T >=< f,A'T > .

Let ip\, A pure imaginary, be the elementary spherical function. Then we have that (f\ £ Cc(G//K)'; similarly we have the averaged character e C™{A)'. Using / to

denote the spherical transform of / we recall the familiar identity

(7.4) 7(A) = F}( A)

The duality relationship combined with the fact that A is 1 — 1 give the identity

p.5) ^(-) = -A'i^I2De"'AO-

i 1 w

Indeed this relationship can be read in reverse, so that (7.4) and (7.5) are equivalent. As an aside, in [S-T] the specific form, in the case of real rank 1, of A' as an integral operator in (7.5) was used to derive an expansion for the elementary spherical functions in terms of spherical functions for the associated Cartan motion groups.

Now take / € S(G//K) or just in Cc(G//K) but with sufficient decay of its spherical Fourier transform, /. Recall the Bochner measure m(A,v). Define F~(v) by

(7.6) FZ(v)= [ /(A)m(A, v)- dX

/v ' J ' v ’ |c(A)|2 ‘

in*

Recall the inversion formula for the spherical transform

/(«) = /^A)^(°)j^j2-

ia*

Then an elementary application of the Fubini theorem and the Euclidean Fourier inversion theorem gives

(7.7) f(a) = Fs~wln)

\ - - / «/ V •• /

which is to be contrasted with (7.4).

388

References

[B] Bourbaki, N., Eléments de Mathématique. Intégration, Chap. 7, Hermann, Paris, 1963.

[Ba] Barnes, E.W., A new development of the theory of the hypergeometric functions. Proc.

London Math. Soc., Ser II, V. 6 (1908), p. 141.

[BP] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. Higher transcendental functions (Bateman manuscript project), V. I, McGraw Hill, N. Y., 1953.

[F-K] Flensted-Jensen, M. and Koornwinder, T.H. Positive definite spherical functions on a non-compact, rank one symmetric space. Lecture Notes in Math., V. 739, Springer, Berlin, 1979, 249-282.

[H-C] Harish-Chandra. Spherical functions, I. Amer. J. Math., V. 80 (1958), 241-310.

[HI] Helgason, S. Differential geometry and symmetric spaces. Academic Press, N. Y., 1962.

[H2] Helgason, S. A duality for symmetric spaces with applications to qroup representations.

Adv. in Math., V. 5, No. 1 (1970), 1-154.

[K] Kostant, B. On the existence and irreducibility of certain series of representations. Bull. Amer. Math. Soc., V. 75 (1969), 627-642.

[Kn-S] Knapp, A. W. and Stein, E. M. Intertwining operators for semi-simple groups, II. Inv. Math., V. 60 (1980), 9-84.

[KOS] Krôtz, B., Olafsson, G. and Stanton, R. J. The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type. Int. Math. Res. Not., V. 22 (2005), 1307-1329.

[KS-I] Krôtz, B., and Stanton, R. J. Holomorphic extension of representation (I): automorphic functions. Ann. Math., V. 159, No. 2 (2004), 641-724.

[Ne] Neeb, K.-H. Holomorphy and Convexity in Lie Theory. De Gruyter, Exposition in Mathematics 28, 1999.

[Ri] Richter, H. Wahrscheinlichkeitstheorie, 2nd edn., Springer, Berlin, Heidelberg, New York, 1966.

[S-T] Stanton, R. J. and Tomas, P. A. Expansions for spherical functions on noncompact symmetric spaces. Acta Math., V. 140, Nos. 3-4 (1978), 251-276.

[Wa] Watson, G.N. Asymptotic expansions of hypergeometric functions. Transactions of the Cambridge Philo. Soc., V. 22 (1918), 277-308.

[W-W] Whittaker, E.T. and Watson, G.N. A course of modern analysis. Cambridge Univ. Press, London, 4th ed., 1973.