Harmonic Analysis of Radon Filtrations for S(n) and GL(n,q)

0.21 MB·English

by  M. Francisca Yanez

Harmonic Analysis of Radon Filtrations for S and
n
GL (q)
n
9
0 M. Francisca Y´an˜ez∗
0
2
January 18, 2009
n
a
J

8
1
Abstract

] We present a unified approach to the study of Radon transforms
T
R related to symmetric groups and to general linear groups GLn(q),
regarded as q−analogues of the former. In both cases, we define a
.
h sequence of generalized Radon transforms which are intertwining op-
t
a erators for natural representations associated to Gel’fand spaces for
m our groups. This sequence enables us to decompose in a recursiveway
[ these naturalrepresentationsand to compute explicitly the associated
sphericalfunctions. Ourmethodsandresultsarerelatedbyq−analogy.
1
v
9 Keywords: RadonTransform,SphericalFunction,q-analogue,Gelfand
6 Space, Natural Representation
6
2
. 1 Introduction
1
0
9
In 1917 Radon [7] showed that a function on Euclidean space could be re-
0
: covered from its integrals over affine hyperplanes, solving in this way the
v
inversion problem for the nowadays called “Radon Transform” which asso-
i
X
ciates to a point function f the hyperplane function Rf : H 7→ f.
H
r
a Later, Gel’fand [2] and Helgason [3, 4] studied the Radon TransformRin the
more general setting of homogeneous spaces for a Lie group G.
In 1975 Soto-Andrade [8] used finite Radon Transforms associated to
the geometry of finite symplectic spaces to study the intertwining algebra of
natural representations of the finite similitude symplectic group GSp(4,q)
in function spaces on isotropic flags and so obtained by decomposition the
principal series representations of GSp(4,q).
∗partially supported by Fondecyt Grants 1040444, 1070246 and 7070263 and PICS
CNRS 1514
1
In 1977 Dunkl [1] used finite Radon Transforms to obtain a convenient
basis of the intertwining algebra of the natural representation of GL (q) as-
n
sociatedtotheactionofthisgrouponthelatticeofsubspacesinn−dimensional
space, to decompose this representation and to calculate the corresponding
spherical functions.
In 1983 Grinberg [5] independently took up this viewpoint in the real
case to analyze the Radon Transform on compact two point homogeneous
spaces, considering it as an intertwining operator between representations
of the isometry group G = U(n+1).
Later, in 2002, Marco and Parcet [6] took advantage of finite Radon
transforms to study the interwining algebra of the natural representation
L2(P(Ω)) of the symmetric group S(Ω) on a finite set Ω, associated to the
power set P(Ω) of Ω and to decompose this representation.
On the other, it should be noticed that Stanton [8] also considered the
Gel’fand spaces (S ,P ) and (GL (q),V ) and proved that the spheri-
n s n s
cal functions correspond to discrete orthogonal polynomials, without using
Radon Transforms.
Wedescribenowtheresultsinthispaper,whichwerealreadyannounced
in[11]. ForGbeingthesymmetricgrouporitsq−analogue,thefinitegeneral
linear group, the Radon transforms that we consider are G -intertwining
operators between naturalrepresentations L2(X) associated tothe action
of G on Gel’fand spaces X (i.e., G -spaces whose associated natural
representation is multiplicity free).
We introducehere Gel’fand spaces (S ,P ) for the symmetric group S
n s n
and their “q-analogues” (GL (q),V ), for the finite general linear group
n s
GL (q), where P is the set of all subsets of N = {1,2,...n} with just s
n s
elements, and V isthesetofalls-dimensionalvectorsubspacesof V = Fn,
s q
where F is the finite field with q elements, where q = pn, p being a prime
q
number.
We define for each positive integer s a “generalized Radon Transform”
R , between the natural unitary representations L2(P ) and L2(P ) of
s s s+1
S . This is done in an analogous way to the construction of the classical
n
Radon transform.
Intheorems1and2weprove, applyingtheharmonicanalysistechniques
developped in [6], that the sequence of these operators and the sequence of
their adjoints give a “resolution” of the natural representations L2(P )
s
and L2(Pn−s) , respectively. That is, we show that the subrepresenta-
tions KerRt∗ (resp. KerRn−t) of Sn give all the irreducible components
of L2(Ps) (resp. L2(Pn−s)) . Simultaneously we construct the corre-
sponding spherical functions. Furthermore, we prove in theorem 3 that the
multiplicity-free natural representations L2(Ps) and L2(Pn−s) are
isomorphic via the operator C∗ induced by taking complement s.
s
2
InTheorem4weprovemutatismutandisfortheq−analogue (GL (q),V )
n s
of (S ,P ), theresultscorrespondingtothoseobtainedinTheorems1and
n s
2 for (S ,P ). Finally in Theorem 5 we prove the q−analogue of Theorem
n s
3. However the q−analogue C∗(q) of the operator C∗ and the proof are more
s s
involved in the case of GL (q). Indeed, C∗(q) is given by a summation over
n s
all possible supplementaries to a given subspace.
2 The Gel’fand space (S ,P ).
n s
Let G = S and N = {1,2,....n}. We denote by P the set of all subsets
n s
of N with just s elements. We consider the natural, transitive, action of
S on P .
n s
We endow (S ,P ) with the S −invariant distance d defined by
n s n
d(X,X′)= |X −X′|= |X|−|X ∩X′|= 1|X△X′|).
2
Proposition 1 The distance d is an S −invariant function on P ×P
n s s
which classifies the G−orbits in P ×P . The number of orbits is s+1 .
s s
Proof: It is enough to prove that (Y,Y′) and (Z,Z′) belong to the same
orbit iff |Y ∩Y′| = |Z ∩Z′|. Let σ ∈ S such that σ(Y) = Z and
n
σ(Y′) = Z′. Since σ is bijective, we have σ(Y ∩Y′) = Z ∩Z′. On the
other side, if |Y ∩Y′| =|Z ∩Z′| then we can find a biyection σ : N → N
such that
′ ′ ′ ′ ′ ′ ′ ′
σ(Y−(Y∩Y )) = Z−(Z∩Z ); σ(Y∩Y ) = Z∩Z ; σ(Y −(Y∩Y )) = Z −(Z∩Z )
and σ(N −(Y ∩Y′)) = N −(Z ∩Z′).
Then σ(Y)= Z and σ(Y′) = Z′.
(cid:3)
Definition 1 Let Ω (X ) be the set of all elements X of P such that
i 0 s
d(X,X )= i.
0
Furthermore, for 0 ≤ t ≤ s ≤ n we define the pseudo-distance ℓ on
P ×P by: ℓ(Y,X) = |Y −X| .
t s
s n−s
Remark 1 The number of elements of Ω (X ) is .
i 0 (cid:18) s−i (cid:19)(cid:18) i (cid:19)
3
Corollary 1 The natural unitary representation (L2(P ),τ) associated to
s
the geometric space (S ,P ) is a Gel’fand geometric space, i.e., the unitary
n s
representation (L2(P ),τ) , (where τ is defined by (τ f)(X) = f(σ−1(X))
s σ
for X ∈ P ,f ∈ L2(P ),σ ∈ S ), is multiplicity free. The number of
s s n
irreducible components of L2(P ) is s+1.
s
Ourpurposehereistodecomposethenaturalrepresentation (L2(P ),τ)
s
with the help of certain operators which we denote by R and R∗. Their
s s
definition is completely analogous to the classical Radon Transform in the
continuous case.
Definition 2 For 0 ≤ s ≤ n, Z ∈ P and X ∈ P , let
s s+1
a) R :L2(P ) → L2(P ), be given by
s s s+1
R (f)(X) = f(Z); f ∈ L2(P )
s Z⊂X s
P
and
b) R∗ : L2(P ) → L2(P ), be given by
s s+1 s
R∗(f)(Z)= f(X); f ∈ L2(P ).
s Z⊂X s+1
P
Remark 2 A straightforward calculation shows that R and R∗ are
s s
intertwining operators and
∗
R ◦R = (n−s)Id+M
s s 1
where M = M is the averaging operator associated to the orbit Ω of
k Ωk k
P2/S defined by
s n
′
M (f)(X) = f(X ), (X ∈ P ).
k s
d(XX,X′)=k
Proposition 2 For 1 ≤ s ≤ n , we have
i) L2(Ps) = ImRs−1 ⊥ KerRs∗−1,
ii) L2(Pn−s) = ImRn∗−s ⊥ KerRn−s.
Proof: By computing the Fourier coefficients of a function f ∈ L2(P )
s
such that hf,Rs−1δZ)i = 0, where δZ(Z′) = δZ,Z′ we obtain
⊥ ∗
(ImRs−1) = KerRs−1.
Case ii) is completely analogous .
4
(cid:3)
ThefollowingTheoremleadsustowardsthedecompositionofthemultiplicity-
free natural representation (L2(P ),τ) into its irreducible components
m
Hi (0 ≤ i≤ m) and gives us the spherical functions Φ associated to each
i
Hi.
Theorem 1 Let 0 ≤ m ≤ n. The sequence:
2
L2(P0) →R0 L2(P1) → ···L2(Pm−1) R→m−1 L2(Pm)
isaninductivesystemofmonomorphisms whichprovides allirreducible com-
ponents of L2(P ) as successive orthogonal supplements.
m
Proof: In order to prove that R is injective for 0 ≤ s< n, we will apply
s 2
induction. Let s = 1. Since KerR is trivial we have the decomposition
0
L2(P )= Ho ⊥ H1, where Ho isthetrivialonedimensionalrepresentation
1
and H1 = KerR∗ is the Steinberg representation of dimension n−1 of
0
S . We construct next the spherical functions Φ ,Φ associated to Ho
n 0 1
and H1 , respectively. As we know, Φ will be the spherical function
i
associated to Hi if and only if Φ is invariant under the action of the
i
isotropy group K of the origin X = {1}. In our case this means that
1 0
the values of Φ at X ∈ P depend only on the distance d(X ,X).
i 1 0
Furthermore Φ should take the values 1 at X and Φ must belong to
i 0 i
Hi. These conditions allow us to get:
Φ ≡ 1
0
1 X = X
0
Φ (X) =
1 (cid:26) −n−11 d(X,X0) = 1
Next we have to prove that R is an injective operator. This is however
1
equivalent to proving that R∗ ◦R is an automorphism of L2(P ). Since
1 1 1
R∗◦R = (n−1)Id+M , it is enough to prove that the eigenvalues of M
1 1 1 1
are different from −(n−1). We recall that the eigenvalues λ of M are
i 1
given by M (Φ )(X ). In this way, we obtain: λ = n−1 and λ = 1.
1 i 0 0 1
Then R is injective.
1
Let us suppose that R is injective for 0 ≤ t < s; s < n. We have to
t 2
prove that R injective.
s
Due to the induction hypothesis and using proposition 1 we have:
L2(P ) = H0⊕H1⊕···⊕Hs (orthogonal sum)
s
5
where H0 is the one dimensional trivial representation and
Ht = (Rs−1◦···◦Rt) (KerRt∗−1), (1 ≤ t ≤ s),
n n
with dimension − .
(cid:18) t (cid:19) (cid:18) t−1 (cid:19)
We have decomposed L2(P ) in (s+1)−non isomorphic subrepresen-
s
tations of S . Since the number of S −orbits in P2/S is just s+1,
n n s n
we have proved that the subspaces Ht, 0 ≤ t ≤ s, are all the irreducible
components of S appearing in L2(P ). We are now able to construct the
n s
spherical functions Φ associated to each Ht.
t
We observe that f ∈ Ht if and only if there exists a function h ∈
t
KerR∗ suchthat f(X) = h (Y), X ∈ P . Inordertoobtain
t−1 Y∈Pt,Y⊂X t s
the spherical function Φ ∈ PHt, we need to define a function h on P
t t t
such that Y∈Pt,Z⊂Y ht(Y) = 0 for each Z ∈ Pt−1 and
P
min(j,t)
Φt(X) = ht(Y) =  ht(Y),
YX⊂X kX=k0 Y⊂X,ℓX(Y,X0)=k 
where X = {1,2,··· ,s}, d(X ,X) = j and ℓ(Y,X ) = k where k = 0
0 0 0 0 0
if j ≤ s−t or k = t+j−s if j ≥ s−t. Thevaluesof Φ at X ∈ P must
0 t s
depend only on the distance d(X ,X). We impose the following conditions
0
to the function h :
t
i) h (Y) = 0, (Z ∈ Pt−1)
Y∈Pt,Z⊂Y t
P
ii) h (Y) = αk if ℓ(Y,X ) = k
t t 0
Now let us suppose that we have found one such function h and let us
t
define the following functions Φ on P by:
t t
min(j,t)
Φ (X) = h (Y) = |Ak,j|αk,
t t t t
YX⊂X kX=k0
where
k,j
A = {Y ∈ P :ℓ(Y,X )= k,Y ⊂ X,d(X,X ) = j.
t t 0 0
k,j
In order to get an element Y ∈ A , we need to choose t−k elements
t
among the s− j elements of Y ∩ X and k elements among the j
0
s−j j
k,j
elements of Y −X . Therefore |A |= .
0 t (cid:18) t−k (cid:19)(cid:18) k (cid:19)
Sinceforall σ belongingtotheisotropygroup K , wehave d(σX,σX )=
t 0
k,j
d(X,X ) and |A | depends only on the number of elements of X and
0 t
6
Y, we obtain that Φ is invariant under the action of K . Then, for
t t
Φ to be a spherical function, we need still only to fulfill the normalization
t
condition Φ (X ) = 1 or equivalently:
t 0
iii) A0,0α0 = 1.
t t
So we obtain the function h by solving the following system of linear
t
equations arising from i), ii) and iii)
|Ck,k|αk +|Ck+1,k|αk+1 = 0, 0 ≤ k ≤ t−1
t t t t

 s −1
α = ,
0 (cid:18) t (cid:19)




where
k′,k ′
Ct = {Y ∈ Pt :Z ⊂ Y, ℓ(Y,X0)= k , ℓ(Z,X0) = k}, Z ∈ Pt−1.
We have Y = Z ∪ {y}. Then if y ∈ X we get k′ = k, and if
0
y ∈/ X then k′ = k + 1. A combinatorial computation gives us that
0
s−(t−1−k) n−(s+k)
k,k k+1,k
|C |= and |C |= .
t (cid:18) 1 (cid:19) t (cid:18) 1 (cid:19)
Solving this system we get
s −1 (−1)k(s−t+k)!(n−s−k)!
α0 = and αk = , 1 ≤ k ≤ t.
t (cid:18) t (cid:19) t s
(s−t)!(n−s)!
(cid:18) t (cid:19)
Then the spherical function Φ associated to Ht is given by
t
min(j,t)
s−j j
Φ (X) = (−1)k γk
t (cid:18) t−k (cid:19)(cid:18) k (cid:19) t
kX=k0
−1
where j = d(X ,X) and γk = s (s−t+k)!(n−s−k)!.
0 t (cid:18) t (cid:19) (s−t)!(n−s)!
We have to compute now the eigenvalues of M with respect to our
1
decomposition of L2(P ). As we know, the eigenvalues λ associated with
s t
Ht are given by λ = M (Φ )(X ). Then we have:
t 1 t 0
λ = (n−s)(s−t)−(s−t+1)t, 0≤ t ≤ s
t
7
We want to show that λ 6= −(n− s). Let us suppose that for some
t
t the eigenvalue λ = −(n−s). By replacing λ = −(n−s) in the last
t t
equation and dividing by s−t+1 6= 0, we obtain that t+s = n. But
since by hypothesis t+s < n, we conclude that λ 6= −(n−s); 0≤ t ≤ s
t
and R∗◦R is an automorphism. Then R is injective for all 0≤ s < m
s s s
and
L2(Pm) ∼= (ImR0)⊕(ImR0)⊥⊕···⊕(ImRm−1)⊥ (orthogonal sum)
Inthis way, for each 0≤ m ≤ n, wehave constructed all theirreducible
2
componentsforthenaturalrepresentation L2(P ) of S andtheassociated
m n
spherical functions.
(cid:3)
In order to prove that for 0≤ m ≤ n
2
L2(N) R→∗n−1 L2(Pn−1) → ··· R∗n→−m L2(Pn−m),
is a resolution for the Gel’fand space L2(Pn−m), we consider the following
preliminaries
Proposition 3 Let us define the S −invariant application C from P
n s s
to Pn−s, (0 ≤ s ≤ n2) by Cs(X) = N −X. The following combinatorial
properties of C hold:
s
Let Y ∈ Pt, X0,X ∈ Ps and Z ∈Pt−1
1. If ℓ(Y,X )= k then ℓ(C (X ),C (Y)) = k
0 s 0 s
2. Let Btk,j = {Y′ ∈ Pn−t : Y′ ⊃ X′, d(X′,X0′) = j, ℓ(X0′,Y′) = k}
where X′,X0′ ∈ Pn−s, then |Akt,j|= |Btk,j|.
3. Let Dtk′,k = {Y′ ∈ Pn−t : Y′ ⊂ Z′, ℓ(X0′,Z′) = k, ℓ(X0′,Y′) = k′}
where Z′ ∈ Pn−(t−1), X0′ ∈ Pn−s), then |Ctk′,k| = |Dtk′,k|, k′ =
k,k+1.
4. The map Cs is an Sn−invariant bijection and Cs−1 = Cn−s .
8
Proof: Since (N −X )−(N −Y) = Y −X we get the first statement.
0 0
In connection with the second statement, we note that, in order to find
Y′ ∈ Bk,j, we need to choose j − k elements of the j elements of
t
X′ −X′ and we need to select the (n−t)−(n−s+j −k) remaining
0
elements of the n−(n−s+j) elements of N −(X′ ∪(X′−X′)). Then
0 0
s−j j
k,j k,j
|B |= = |A |.
t (cid:18) s−j −(t−k) (cid:19)(cid:18) j −k (cid:19) t
Finally, if Y′ ∈ Dk′,k then Y′ = Z′ − {z} and (X′ − Y′) =
t 0
(X′ − Z′)∪(X′ ∩{z}). Now, if z ∈ X′ then |X′ −Y′| = k +1
0 0 0 0
and if z ∈/ X′ then |X′ −Y′| = |X′ −Z′| = k. Therefore |Dk,k| =
0 0 0 t
k+1,k
n−(t−1)−(n−s−j) = s−t+j +1 and |D |= (n−s−j).
t
(cid:3)
Using these properties and following the same procedure employed in
the proof of Theorem 1 we get the following Theorem:
Theorem 2 Let 0 ≤ s ≤ n
2
1. R∗ is an injective intertwining operator.
n−s
2. L2(Pn−s) ∼= ImRn∗−1⊕KerRn−1⊕···⊕KerRn−s. (orthogonal sum)
3. Thesphericalfunction ϕ associated toeachirreduciblesub-representation
t
Lt ∼= KerRn−t of (L2(Pn−s(N),Sn) isgivenby ϕt(X′)= Φt(Cn−s(X′), X′ ∈
Pn−s.
(cid:3)
Theorem 3 Let 0 ≤ s ≤ n2 L2(Ps) ∼= L2(Pn−s) and KerRn−s ∼=
KerR∗ .
s−1
Proof: We define Cs∗ : L2(Pn−s) → L2(Ps), 0 ≤ s ≤ n2 by Cs∗(f) =
f ◦Cs, (f ∈ L2(Pn−s). Let h ∈ KerRn−s and Z ∈ Ps−1 then
∗ ′
C (h)(Y) = h(N −Y)= h(Y ) =0.
s
ZX⊂Y ZX⊂Y Y′⊂XN−Z
Therefore Cs∗(KerRn−s) = KerRs∗−1. From this and from Theorems 1
and 2 we get this Theorem.
(cid:3)
9
3 The q-analogue Gel’fand space (GL (q),V ).
n m
Let V = Fn and V the set of all s−dimensional subspaces of V ,0 ≤ s≤
q s
n. We fix a basis {e ,e ,....,e } of V and let W ∈ V , be the subspace
1 2 n 0 s
of V generated by {e1,e2,....,es} and W0′ ∈ Vn−s the subspace generated
by {e ,....,e }.
s+1 n
We construct a resolution for the Gel’fand space L2(V ,GL (q)),
m n
throughthegeneralized Radontransformfollowing thesameprocedureused
for (L2(P ),S ),0 ≤ m ≤ n. The combinatorial calculations are obtained
m n 2
n
by taking the q-analogue of .
(cid:18) m (cid:19)
Definition 3 1. We define the q-analogue of n! by n! = n (n −
q q
1) ···1 where n = qn−1 and the q-analogue of n by
q q q q−1 (cid:18) m (cid:19)
n n!q
=
(cid:18) m (cid:19) (n−s)! s!
q q q
2. Let (0 ≤ s ≤ m). We define the q-analogue pseudo distances ℓ on
q
V ×V , by ℓ (U,W) = dimU−dim(U∩W), (U ∈ V ,W ∈ V ).
s m q s m
n
Remark 3 1. Thegaussianbinomialcoefficient givesthenum-
(cid:18) s (cid:19)
q
ber of s−dimensional subspaces of V.
2. If s = m then ℓ = d defines a distance on V × V which
q q m m
classifies the orbits of the action on V ×V of GL (q).
m m n
3. The natural unitary representation (L2(V ),τ) associated to the geo-
s
metric space (GL (q),V ) is a Gel’fand geometric space, i.e., the uni-
n s
tary representation (L2(V ),τ) , (where τ is defined by (τ f)(W) =
s g
f(g−1(W)) for W ∈ V ,f ∈ L2(V ),g ∈ GL (q) ), is multiplicity free.
s s n
The number of irreducible components of L2(V ) is s+1.
s
Let us consider the following Lemma.
Lemma 1 Let W ∈ V . The number of l−dimensional subspaces Z of
s
V such that the dimension of Z ∩W is t is given by:
n−s s
q(s−t)(l−t) .
(cid:18) l−t (cid:19) (cid:18) t (cid:19)
q q
10