Wigner transform and pseudodifferential operators on symmetric spaces of non-compact type

0.21 MB·English

by  S. Twareque Ali

WIGNER TRANSFORM AND PSEUDODIFFERENTIAL
OPERATORS ON SYMMETRIC SPACES
OF NON-COMPACT TYPE
1
1
0 S.TWAREQUEALIANDMIROSLAVENGLIS
2

n
a
J
Abstract. We obtain a general expression for a Wigner transform (Wigner
6 function) on symmetric spaces of non-compact type and study the Weyl cal-
1 culusofpseudodifferential operatorsonthem.

]
h 1. Introduction
p
- The Wigner transform and the Weyl calculus of pseudodifferential operators
h
have long played prominent roles in PDE theory [11] [15], time-frequency analysis
t
a [7] [21] [5] and mathematical physics [19]. As their definition relies on the Fourier
m transform, it is not surprising that they have been studied most extensively in the
[ contextoftheEuclideann-space. Theaimofthispaperistoextendthesenotionsto

amoregeneralcontextwhereaversionoftheFouriertransformisavailable: namely,
1
to symmetric spaces of non-compact type, with the Fourier-Helgasontransform.
v
4 There have been several efforts in this direction before in the literature. First
8 ofall,thereisanextensivetheoryofWeylcalculiforwhichthesymmetricdomains
0 are the phase spaces; these are special cases of the so-called “invariant operator
3
calculi” developed recently by Arazy and Upmeier [4]. (It should be noted that
.
1 these calculi seem not to involve any analogue of the Wigner transform.) Our goal
0 here is different in that we have the symmetric domains only as the configuration
1
space, i.e. the Wigner transform and symbols of the Weyl operators are functions
1
on the cotangent bundles of the symmetric domains (or, more precisely, on the
:
v products Ω×Ω∗, where Ω∗ is the Fourier-Helgasondualof the symmetric space Ω;
Xi the latter product is essentially isomorphic to the cotangent bundle T∗Ω). In this
r direction, Tate [16] studied the situation for the simplest complex bounded sym-
a metric domain, the unit disc; generalization to the unit ball of Rn (realized as
one-sheeted hyperboloid in Rn+1) has then been carried out by Bertola and the
firstauthor[1]. We alsomention thatapparentlyyetanother kindof the Weylcal-
culus for the disc,for whichthe symbols alsolive onthe tangentbundle ofthe disc
and which ultimately leads to the occurrence of Bessel functions, was introduced
byTerras[17]andstudiedbyTrimeche[18]orPengandZhao[14];itseemsunclear
whetherthiscalculusisinanywayrelatedtoTate’sandours. (Wepausetoremark
that the Bessel-function Weyl calculus, however,seems to have rather complicated
behaviour under holomorphic transformations of the unit disc.)
In the physical literature there have been several different generalizations of
the original Wigner function [20] to non-flat configuration spaces and their phase
ThesecondauthorwassupportedbyGACˇRgrantno.201/09/0473andAVCˇRresearchplan
no.AV0Z10190503.
1
2 S.TWAREQUEALIANDMIROSLAVENGLIS
spaces. Oneapproachtowardsageneralizationexploitsthefactthatthattheorigi-
nalWignerfunctionlivesonacoadjointorbitoftheWeyl-Heisenberggroupandcan
beobtainedusingthesquare-integrabilitypropertyofitsrepresentations. Ageneral
descriptionofthismethod,exploitingsquare-integrablegrouprepresentations,may
be foundin[13]andearlierreferencescitedtherein. Anapproachthat is veryclose
to the one adopted in the present paper has been used in [2, 3] to obtain Wigner
functions on hyperboloids and spheres. However, the results obtained there were
onacasebycasebasis,whilewe presenthereageneraltheory. Another suggestion
for a generaliztion, using the entire dual space of the Weyl-Heisenberg group has
been givenin[12]. The virtue of our presentapproachlies in its generality andthe
fact that our construction preserves both the marginality and unitarity properties
thatallowedthe originalWignerfunctionto beinterpretedasapseudo-probability
distribution on phase space.
The Wigner transform is constructed in Section 3 below, after reviewing the
necessaryprerequisitesonsymmetric spacesin Section2. The non-EuclideanWeyl
calculus of pseudodifferential operators is introduced in Section 4. The invertibil-
ity of the Wigner transform and its unitarity are discussed in Sections 5 and 6,
respectively. The final Section 7 contains miscellaneous concluding remarks, open
problems, etc. For the most part, our approach parallels fairly directly that of
Tate’s in [16]; however, Theorem 12 and Corollary 10 seem to be new even for his
situation of the unit disc.
Acknowledgement. Largepartofthisworkwasdonewhilethesecondauthor
was visiting the first; the hospitality of the mathematics department of Concordia
University on this occasion is gratefully acknowledged.
2. Bounded symmetric domains
Recall that a connected Riemannian manifold Ω of dimension d is called a sym-
metric space if for any x ∈ Ω there exists a (necessarily unique) element s ∈ G,
x
the group of isometries of Ω, which is involutive (i.e. s ◦s =id) and has x as an
x x
isolatedfixed-point. Onecallss thegeodesicsymmetryatx. Thesymmetricspace
x
is called irreducible if it is not isomorphic to a Cartesian product of another two
symmetric spaces. Irreducible symmetric spaces come in three types: Euclidean
(these are just Rd and its quotients), of compact type (the compact ones) and of
non-compact type. Any symmetric space of non-compact type can be realized as
(i.e. is isomorphic to) a domain in Rd which is circular with respect to the origin
andconvex(the so-calledHarish-Chandrarealization). Throughoutthe restofthis
paper, we will thus assume that Ω is of the latter form, i.e. a symmetric space of
non-compact type in its Harish-Chandra realization.
Itturnsoutthatthe geodesicsymmetriess infactacttransitivelyonΩ,i.e.for
x
any y,z ∈ Ω there exists an x ∈ Ω such that s y = z; denoting by K = {g ∈
x
G : g(0) = 0} the stabilizer in G of the origin 0 ∈ Ω, it therefore follows that
Ω is isomorphic to the coset space G/K. (It is also true that elements of K are
orthogonalmapsonRdthatpreserveΩ,andthatK isamaximalcompactsubgroup
of G.) There exists a unique (up to constant multiples) G-invariant measure on Ω
(obtained as the projectionof the Haar measure on G); we will denote it by dµ(z).
(Thus dµ(z)=dµ(g(z)) for any g ∈G.)
WIGNER TRANSFORM 3
For x∈Ω, there exists a unique geodesic symmetry φ ∈G which interchanges
x
x and the origin, i.e.
(1) φ ◦φ =id, φ (0)=x, φ (x)=0,
x x x x
and φ has only isolated fixed-points. In fact, φ has only one fixed point, namely
x x
thegeodesicmid-pointbetween0andx;wewilldenote,quitegenerally,thegeodesic
mid-point between some given x,y ∈ Ω by m or m . (Thus the fixed point of
x,y xy
φ is precisely m , and φ =s .)
x x,0 x mx,0
Employing the standard notation, let G=NAK be the Iwasawadecomposition
of G, a the Lie algebra of the maximal Abelian part A, a∗ its dual, r = dimRa
its dimension (known as the rank of Ω), ρ = (ρ ,...,ρ ) ∈ a∗ the sum of positive
1 r
roots, M and M′ the centralizer and the normalizer of A in K, respectively, and
W = M′/M the Weyl group. For any λ ∈ a∗ ∼= Rr and b in the coset space
B :=K/M =G/MAN, one defines the “plane waves” on Ω by
e (x):=e(iλ+ρ)(A(x,b)), x∈Ω,
λ,b
where A(x,b) is the unique element of a satisfying, if b=kM and x=gK,
k−1g ∈NexpA(x,b)K
under the Iwasawa decomposition G=NAK.
The Helgason-Fourier transform of f ∈ C∞(Ω) is a function on Ω∗ := a∗ ×B
0
(∼=Rr×K/M) given by
f˜(λ,b):= f(x)e−λ,b(x)dµ(x).
ZΩ
For any f ∈C∞(Ω) we then have the Fourier inversion formula
0
f(x)= f˜(λ,b)e (x)dρ(λ,b)
λ,b
Za∗ZB
and the Plancherel theorem
|f(x)|2dµ(x)= |f˜(λ,b)|2dρ(λ,b).
ZΩ Za∗ZB
Here
dρ(λ,b):=|c(λ)|−2dbdλ,
where db is the unique K-invariant probability measure on K/M, dλ is a suitably
normalized Lebesgue measure on a∗ ∼= Rr, and c(λ) is a certain meromorphic
function on the complexification a∗C ∼= Cr of a∗ (the Harish-Chandra c-function).
From the Plancherel theorem it can be deduced, in particular, that f 7→f˜extends
to a Hilbert space isomorphism of L2(dµ) into L2(Ω∗,dρ) whose image consists of
functions F(λ,b) which satisfy a certain symmetry condition (relating the values
F(λ,b) and F(sλ,b) for s in the Weyl group; see Corollary VI.3.9 in [10].)
A (linear) differential operator L on Ω is called G-invariant if
L(f ◦g)=(Lf)◦g
for any f ∈ C∞(Ω) and any g ∈ G. For any such L, it is known that the “plane
waves” are eigenfunctions of L:
Le =L˜(λ)e
λ,b λ,b
whereL˜(λ)isapolynomialinrvariables;thatis,eachsuchLisaFouriermultiplier
with respect to the Helgason-Fourier transform. The correspondence L 7→ L˜ sets
4 S.TWAREQUEALIANDMIROSLAVENGLIS
up an isomorphism between the ring of all G-invariant differential operators on Ω
and the ring of all polynomials on Rr ∼=a∗ invariant under the Weyl group W.
The“planewaves”e obeythefollowingtransformationruleundercomposition
λ,b
with elements of G:
(2) eλ,b◦g =eλ,b(g0)eλ,g−1b.
(We will often write g0,gz, etc. instead of g(0),g(z) etc.) It follows from here that
e (g0)e (g−10)=1
λ,gb λ,b
and
dρ(λ,gb)=|e (g−10)|2dρ(λ,b),
λ,b
(3)
dρ(λ,b)=|e (g0)|2dρ(λ,gb).
λ,gb
Indeed, from the formula for the Helgason-Fourier transform and (2) we have
f˜(λ,gb)= f(z)e−λ,gb(z)dµ(z)
ZΩ
e−λ,b(g−1z)
= f(z) dµ(z)
ZΩ e−λ,b(g−10)
1
= e−λ,b(g−10)ZΩf(gz)e−λ,b(z)dµ(z)
(f ◦g)∼(λ,b)
= ,
e−λ,b(g−10)
whence from
f(z)= f˜(λ,b)e (z)dρ(λ,b)
λ,b
ZΩ∗
= f˜(λ,gb)e (z)dρ(λ,gb)
λ,gb
ZΩ∗
(f ◦g)∼(λ,b) e (g−1z)
λ,b
= dρ(λ,gb)
ZΩ∗ e−λ,b(g−10) eλ,b(g−10)
we get, upon replacing f by f ◦g−1 and z by gz,
f˜(λ,b) e (z)
λ,b
f(z)= dρ(λ,gb),
ZΩ∗ e−λ,b(g−10) eλ,b(g−10)
proving the claim. (Note that e−λ,b =eλ,b.)
Since |e (x)|2 =e2ρ(A(x,b)) does not depend on λ, (3) in fact implies that
λ,b
d(gb)=|e (g−10)|2db,
λ,b
(4)
db=|e (g0)|2d(gb).
λ,gb
Afunctionf onΩiscalledK-invariant iff(kx)=f(x)forallx∈Ωandk∈K.
For such functions, the Helgason-Fourier transform f˜(λ,b) does not depend on b,
and reduces to the spherical transform
f˜(λ)= f(z)Φ−λ(z)dµ(z),
ZΩ
where Φ are the spherical functions
λ
Φ (z):= e (kz)dk = e (z)dk.
λ λ,b λ,kb
ZK ZK
WIGNER TRANSFORM 5
One has Φ = Φ for all s in the Weyl group, i.e. f˜is W-invariant. The Fourier
λ sλ
inversion formula and the Plancherel theorem assume the form
(5) f(z)= f˜(λ)Φ (z)dρ(λ),
λ
a∗
Z
|f(z)|2dµ(z)= |f˜(λ)|2dρ(λ),
ZΩ Za∗
respectively, where (abusing notation a little)
dρ(λ):=|c(λ)|−2dλ.
Some examples. 1. The absolutely simplest example of the type of symmetric
space studied here could be the unit interval Ω = (−1,1) ⊂ R, on which G =
O(1,1)/R acts by
ax+b a b
gx= , x∈Ω, g = ∈O(1,1),
cx+d c d
(cid:18) (cid:19)
that is,
xcosht+sinht
(6) gx=ǫ , x∈Ω, t∈R, ǫ∈{±1}.
xsinht+cosht
In particular,
a−x
φ x= , x,a∈Ω.
a
1−ax
The stabilizer of the origin is K =O(1)={±1}, the invariant measure is dµ(x)=
dx , and N ={1}, A=G, M =K, B ={1}. The Lie algebra g can be identified
1−x2
with R, and the exponential map g→G is
(7) ξ 7−→tanhξ.
It follows that
1+x iλ/2
e (x)= , λ∈R.
λ,b
1−x
(cid:16) (cid:17)
The invariant differential operators on Ω are precisely the polynomials in ∆ :=
((1−x2) ∂ )2, and
∂x
∆e =−λ2e . e
λ,b λ,b
However,thisexampleisnotreallyasymmetricspaceofnoncompacttype,since,by
e
dimensional reasons,the Lie algebrag is necessarily abelian and thus Ω is actually
a Euclidean space. In fact, the exponential map (7) gives an isomorphism of R
ontoΩ underwhichthe action(6)becomesjustthe Euclideanmotionξ 7→ǫ(ξ+t),
dµ(x) becomes the Lebesgue measure dξ, ∆ becomes ∂2/∂ξ2, and e (x) reduces
λ,b
to the ordinary exponential eiλξ. Since the Weyl group is just W = {±1} while
ρ = 0 and c(λ) ≡ 1, the Helgason-Fourier teransform on Ω thus reduces just to the
ordinary Fourier transform on R.
2. The simplest genuine example is thus the unit disc Ω = {z ∈ C ∼= R2 : |z| <
1}, considered by Tate [16]. In this case Ω = G/K with G = U(1,1)/C acting
again by
az+b a b
gz = , z ∈Ω, g = ∈U(1,1),
cz+d c d
(cid:18) (cid:19)
6 S.TWAREQUEALIANDMIROSLAVENGLIS
cosht sinht
and K = U(1), A = { : t ∈ R} is the same as in the preceding
sinht cosht
(cid:18) (cid:19)
example, M ={1}, W ={±1} and ρ=1. The geodesic symmetries are given by
a−z
φ z = .
a
1−az
The quotient space B =K/M can be identified with the unit circle T, and
1−|z|2 1+iλ
e (z)= 2 , λ∈R, b∈T, z ∈Ω.
λ,b |z−b|2
(cid:16) (cid:17)
The invariant measure is dµ(z) = (1 −|z|2)−2dz ∧dz, the invariant differential
operatorsarepreciselythepolynomialsin∆:=(1−|z|2)2∆,where∆istheordinary
Laplace operator, and
∆e =−(λe2+1)e .
λ,b λ,b
ThePlancherelmeasuredρisgivenbydρ(λ)= λ tanhπλdλ,yieldingthesimplest
4π 2
nontrivial example of the Helgeason-Fourier transform.
3. The real hyperbolic n-space, modelled in [1] as one-sheeted hyperboloid,
canberealizedastheunitballΩ={x∈Rn :|x|<1}=G/K withG=O(n,1)/R,
K =O(n). The geodesic symmetries are the Moebius maps
(1−2ha,xi+|x|2)a−(1−|a|2)x
φ x= , x,a∈Ω;
a 1−2ha,xi+|a|2|x|2
the maximalabelian subgroupA can be identified with {τ :a=re , −1<r <1}
a 1
where τ (x) := φ (−x) and e = (1,0,0,...,0); and M = {k ∈ K : ke = e } ∼=
a a 1 1 1
O(n−1),sothatB =K/M canagainbeidentifiedwiththeunitsphere∂Ω=Sn−1.
The Weyl group W is again just {±1},the sum of positive roots is ρ=n−1, and
the “plane waves” are
1−|x|2 n−1+iλ
e (x)= 2 , x∈Ω, b∈∂Ω, λ∈R.
λ,b |x−b|2
(cid:16) (cid:17)
The invariant differential operators are precisely the polynomials in
n ∂2 n ∂
∆:=(1−|x|2) (1−|x|2) +(2n−4) x ,
∂x2 j∂x
h Xj=1 j Xj=1 ji
e
and ∆e = −(λ2 +(n−1)2)e . Note that for n = 1 and n = 2, this example
λ,b λ,b
recoversthe previous two as special cases.
4.eAll three examples above are in turn special cases of the unit ball of real
n×m matrices
Ω={z ∈Rn×m :I−ztz is positive definite}
(or, equivalently, kzk<1 when z is viewed as an operator z :Rm →Rn). One has
Ω=G/K with G=O(n,m)/R acting by
a b
gz =(az+b)(cz+d)−1, z ∈Ω, g = ∈O(n,m)
c d
(cid:18) (cid:19)
(with a ∈ Rn×n, b ∈ Rn×m, etc.). The stabilizer subgroup K consists of all
block-diagonal (b = c = 0) elements in G, while A can be taken as {τ : a =
a
r e , −1<r <1}, where τ (x):=φ (−x) and e is the n×m matrix with 1
j j j j a a j
on the (j,j)-position and 0 everywhere else, 1≤j ≤min(m,n). In particular, the
P
WIGNER TRANSFORM 7
rankofΩisr=min(m,n). (Thepreviousthreeexamples,correspondingtom=1,
were thus of rank 1.)
5. General symmetric spaces of non-compact type include, in addition to anal-
ogous unit balls of symmetric or anti-symmetric matrices, also some other infinite
series of matrix domains, as well as so-called “exceptional” symmetric domains
related to (some) exceptional Lie groups.
Formoredetailsandthe proofsofalltheabove,aswellasforthe complete clas-
sification (up to isomorphism) of all symmetric spaces, we refer e.g. to Helgason’s
books [10], [9], [8].
3. Wigner transform
Recall that m stands for the geodesic midpoint between two points x,y of Ω.
x,y
We begin by establishing a few properties of the Jacobian J(x,y) of this map,
defined by the following equality
(8) f(m )dµ(z)= f(x)J(x,y)dµ(x).
z,y
ZΩ ZΩ
Proposition 1. For any g ∈G, J(gx,gy)=J(x,y).
Proof. From the definition of J and invariance of dµ we get
f(x)J(gx,gy)dµ(x)= f(x)J(gx,gy)dµ(gx)
ZΩ ZΩ
= f ◦g−1(x)J(x,gy)dµ(x)
ZΩ
= f ◦g−1(m )dµ(z)
z,gy
ZΩ
= f ◦g−1(gmg−1z,y)dµ(z)
ZΩ
= f(mg−1z,y)dµ(z)
ZΩ
= f(m )dµ(z)
z,y
ZΩ
= f(x)J(x,y)dµ(x),
ZΩ
where the fourth equality follows from the fact that m =gm . (cid:3)
gx,gy x,y
Corollary 2. J(x,y)=J(y,x).
Proof. Take for g the geodesic symmetry interchanging x and y. (cid:3)
Now let F be a function on Ω×Ω. The Wigner transform W :Ω×Ω∗ →C of
F
F is defined by
WF(x;λ,b):=|eλ,b(x)|−2 eλ,b(y)e−λ,b(sxy)F(sxy,y)J(x,y)dµ(y)
ZΩ
=|eλ,b(x)|−2 eλ,b(sxy)e−λ,b(y)F(y,sxy)J(x,y)dµ(y).
ZΩ
The second expression follows from the first upon the change of variable y 7→ s y
x
and noting that J(x,y)=J(s x,s y)=J(x,s y) by the preceding proposition.
x x x
8 S.TWAREQUEALIANDMIROSLAVENGLIS
Note that the quantity |e (x)|−2 is, in fact, independent of λ.
λ,b
The nextthree theoremsshow that our Wigner transformretains the properties
we expect from the Euclidean case.
Theorem 3. (Invariance) For any g ∈G,
WF◦g(x;λ,b)=WF(gx;λ,gb),
where F ◦g(x,y):=F(gx,gy).
Proof. Note that for any x,y ∈Ω and g ∈G,
s gy =gs y.
gx x
Using the definition of W, the invariance of dµ and J, and (2), we therefore have
WF◦g(g−1x;λ,b)
=|eλ,b(g−1x)|−2 eλ,b(y)e−λ,b(sg−1xy)F(gsg−1xy,gy)J(g−1x,y)dµ(y)
ZΩ
=|eλ,b(g−1x)|−2 eλ,b(g−1y)e−λ,b(sg−1xg−1y)F(gsg−1xg−1y,y)
ZΩ
J(g−1x,g−1y)dµ(y)
=|eλ,b(g−1x)|−2 eλ,b(g−1y)e−λ,b(g−1sxy)F(sxy,y)J(x,y)dµ(y)
ZΩ
=|eλ,b(g−10)eλ,gb(x)|−2 eλ,b(g−10)eλ,gb(y)e−λ,b(g−10)e−λ,gb(sxy)
ZΩ
F(s y,y)J(x,y)dµ(y)
x
=|eλ,gb(x)|−2 eλ,gb(y)e−λ,gb(sxy)F(sxy,y)J(x,y)dµ(y)
ZΩ
=W (x;λ,gb),
F
as asserted. (cid:3)
Theorem 4. (Marginality) For F of the form F(x,y) = f(x)g(y), with f,g ∈
L2(Ω,dµ), we have the marginality relations
W (x;λ,b)|e (x)|2dµ(x)=f˜(λ,b)g˜(λ,b);
F λ,b
ZΩ
W (x;λ,b)|e (x)|2dρ(λ,b)=f(x)g(x).
F λ,b
ZΩ∗
Proof. For the first, use the defining property (8) of the Jacobian:
W (x;λ,b)|e (x)|2dµ(x)
F λ,b
ZΩ
= eλ,b(y)e−λ,b(sxy)f(sxy)g(y)J(x,y)dµ(x)dµ(y)
ZΩZΩ
= eλ,b(y)e−λ,b(smz,yy)f(smz,yy)g(y)dµ(z)dµ(y)
ZΩZΩ
= eλ,b(y)e−λ,b(z)f(z)g(y)dµ(z)dµ(y) (since smz,yy =z)
ZΩZΩ
=f˜(λ,b)g˜(λ,b).
WIGNER TRANSFORM 9
For the second, note that by Plancherel
(9) eλ,b(y)e−λ,b(z)dρ(λ,b)=δyz.
ZΩ∗
Thus
W (x;λ,b)|e (x)|2dρ(λ,b)
F λ,b
ZΩ∗
= eλ,b(y)e−λ,b(sxy)f(sxy)g(y)J(x,y)dρ(λ,b)dµ(y)
ZΩZΩ∗
= δ f(s y)g(y)J(x,y)dµ(y)
y,sxy x
ZΩ
=f(s x)g(x)J(x,x)
x
=f(x)g(x)J(x,x).
Ontheotherhand,bytheinvarianceofJ wehaveJ(x,x)=J(φ x,φ x)=J(0,0),
x x
and taking in the defining property for J
f(x)J(x,0)dµ(x)= f(m )dµ(z)
z,0
ZΩ ZΩ
forf anapproximateidentity(i.e.lettingf tendtothedeltafunctionattheorigin),
we get J(0,0)=1. Thus the second part of the theorem follows. (cid:3)
Remark. In addition to the Iwasawa decomposition G = NAK, one also has the
Bruhat decomposition G = KA+K, where A+ is a certain “positive” subset of A
and the bar stands for closure. It can be deduced from the latter that the ambient
space Rd ⊃ Ω = G/K admits a “polar decomposition” as Rd ∼= K/M ×a+ —
more precisely, any x ∈ Rd can be written in the form x = ka with a lying in a
fixed subspace isomorphic to a ∼= a∗ ∼= Rr; and if we set a+ = {t e +···+t e :
1 1 r r
t > t > ··· > t > 0}, where e ,...,e is an appropriate basis for a, then the
1 2 r 1 r
correspondence Rd ∋ x ←→ (kM,a) ∈ K/M ×a+ is one-to-one except for the set
of measure zero where t = t or t = 0 for some j (then the t ,...,t are still
j j+1 j 1 r
determined uniquely, but kM is not). (The r-tuple d(x) := (t ,...,t ) is called
1 r
the “complex distance” of x from the origin.) In this way, the cotangent space
T∗Ω ∼= Rd at any point x ∈ Ω can essentially be identified with K/M ×a∗, and
x
we can thus think of the Fourier-Helgason transform f˜ : Ω∗ → C as living on
the cotangent space T∗Ω. Similarly, the Wigner transform W : Ω × Ω∗ → C
x F
can be envisaged as living in fact on the cotangent bundle T∗Ω. In a way, this
is reminiscent of viewing the ordinary Fourier transform f˜(ξ) on R2 ∼= C in the
polar coordinates as f˜(ξ) ≡ f˜(r,θ) where ξ = reiθ; the subtle difference is that
instead of the simple symmetry relation f˜(r,θ) = f˜(−r,θ +π), for the Fourier-
Helgason transform one has the more complicated symmetry relations, mentioned
in Section 2, relating f˜(λ,b) and f˜(sλ,b) for s in the Weyl group. (cid:3)
4. Pseudodifferential operators
InanalogywiththeEuclideancase,theWignerfunctioncanbeusedtodefinethe
Weyl calculus of pseudodifferential operators by assigning to a “symbol” function
10 S.TWAREQUEALIANDMIROSLAVENGLIS
a on Ω×Ω∗ the operator Ψ on L2(Ω,dµ) defined by
a
hΨau,vi= Wu⊗v(x;λ,b)a(x,λ,b)|eλ,b(x)|2dρ(λ,b)dµ(x),
ZΩZΩ∗
where (u⊗v)(x,y):=u(x)v(y). In other words,
Ψau(y)= a(mz,y;λ,b)eλ,b(y)e−λ,b(z)u(z)dµ(z)dρ(λ,b)
ZΩ∗ZΩ
= a(y,z)u(z)dµ(z),
ZΩ
where a is the integralekernel
(10) e a(y,z):= a(mz,y;λ,b)eλ,b(y)e−λ,b(z)dρ(λ,b).
ZΩ∗
Note that for a(ex;λ,b) = a(x) depending only on the space variable, Ψa reduces
just to a multiplication operator: indeed, by Plancherel’s formula (9),
Ψ u(y)= a(m )δ u(z)dµ(z)=a(m )u(y)=a(y)u(y).
a z,y y,z y,y
ZΩ
Similarly, for a(x;λ,b)=a(λ) depending only on λ the operatorΨ reduces to the
a
corresponding Fourier multiplier:
Ψ u(y)= a(λ)e (y)u˜(λ,b)dρ(λ,b)= a(λ)u˜(λ,b) ∧,
a λ,b
ZΩ∗ (cid:16) (cid:17)
where ∧ stands for the inverse Fourier-Helgason transform. This shows, in par-
ticular, that all invariant differential operators on Ω arise as Ψ for a = a(λ) an
a
appropriate W-invariant polynomial on a∗.
The invariance properties of the Wigner transform are reflected in the corre-
sponding invariance properties for the Weyl pseudodifferential operators Ψ and
a
their integral kernels a.
Theorem 5. For any g ∈G, we have
e
a(gy,gz)=ag(y,z),
i.e. a◦g =ag, where ag(x;λ,b):=a(gx;λ,gb).
e e
Proof. From (2),
e e
a(gy,gz)= a(gmz,y;λ,b)eλ,b(gy)e−λ,b(gz)dρ(λ,b)
ZΩ∗
e = |eλ,b(g0)|2a(gmz,y;λ,b)eλ,g−1b(y)e−λ,g−1b(z)dρ(λ,b)
ZΩ∗
= a(gmz,y;λ,gb)eλ,b(y)e−λ,b(z)|eλ,gb(g0)|2dρ(λ,gb)
ZΩ∗
= a(gmz,y;λ,gb)eλ,b(y)e−λ,b(z)dρ(λ,b)
ZΩ∗
=ag(y,z),
where the penultimate equality used (3). (cid:3)
e