Higher Weight on GL(3), II - The cusp forms

0.4 MB·

by  Jack Buttcane

HIGHER WEIGHT ON GL 3 , II:
p q
THE CUSP FORMS.
JACK BUTTCANE
7
1
0 Abstract. ThepurposeofthispaperistocollectandmakeexplicittheresultsofGel1fand,
2 Graev and Piatetski-Shapiro [10] and Miyazaki [23,24] for the GLp3q cusp forms which are
n non-trivial on SOp3,Rq. We give new descriptions of the spaces of cusp forms of mini-
a mal K-type and from the Fourier-Whittaker expansions of such forms give a complete and
J completely explicit spectral expansion for L2pSLp3,ZqzPSLp3,Rqq, accounting for multi-

6
plicities, in the style ofDuke, Friedlander andIwaniec’spaper [9]. We directly compute the
1
Jacquet integral for the Whittaker functions at the minimal K-type, improving Miyazaki’s

] computation. The primary tool will be the study of the differential operators coming from
T the Lie algebra on vector-valuedcusp forms.
N
.
h
t
a
m 1. Introduction
[ In the previous paper, we worked out the continuous andresidual spectra in the Langlands

1 decomposition in the case of L2 SL 3,Z PSL 3,R . We turn now specifically to the cusp-
v p p qz p qq
idal spectral decomposition. The initial decomposition into eigenspaces of the Casimir oper-
0
8 ators is originally due, in much greater generality, to Gel1fand, Graev and Piatetski-Shapiro
3 [10](basedonearlier workofGelfandandGraev), who described thedecompositioninterms
1
4
of integral operators (in the style of Selberg [26]) operating on representation spaces in the
0
. L2-space. In modern terminology, this is the study of irreducible unitary representations
1
0 and g,K modules, and in the particular case of PSL 3,R , the former was initiated by
7 Vahuptinskqi˘ı [28] and the latter by Howe [14] and Miyazakpi [23q].
1
From the representation-theoretic description, this paper will analyze the structure of the
:
v
“large”, i.e. cuspidal, part of the principal series representations as they decompose over
i
X
the entries of the Wigner D-matrices, the intertwining operators, and the Jacquet integral
r
as an operator from the principal series representation to its Whittaker model. From the
a
analytic perspective, we are decomposing the eigenspaces of the Casimir operators on the
L2-space via raising and lowering operators obtained from the Lie algebra (in the style of
Maass [20]); we show the equivalence of several descriptions of the minimal K-types, and
compute Mellin-Barnes integrals for the Whittaker functions attached to those K-types. We
will largely avoid the representation-theoretic description outside of these first two sections;
partial descriptions in that language maybe found in [22, Appendix A] and [7].
The goal of this paper is to describe, as completely, explicitly and uniformly as possible,
the spectral expansion of the cuspidal part of L2 SL 3,Z PSL 3,R , and provide all of
p p qz p qq
Date: 10 January 2017.
2010 Mathematics Subject Classification. Primary 11F72; Secondary 11F30.
During the time of this research,the author was supported by NSF grant DMS-1601919.
1
2 JACK BUTTCANE
the associated information necessary for a number theorist to apply analysis on this space.
We classify the spectral parameters of the cusp forms at the minimal K-types in Theorem
4, and then, taking as input the spectral parameters and Fourier-Whittaker coefficients of
such minimal cusp forms, generate the rest of the spectral expansion. The main theorem is
Theorem 5.
We give expressions for the Whittaker functions at the minimal K-types in Theorem 6;
these are due, in greatly different language, to Miyazaki [24] who obtained them by studying
their differential equations and then applying a certain induction argument, but we take this
a step farther by evaluating the base case directly from the Jacquet integral, thus fixing the
relationship between elements of the representation and the Whittaker model, as well as the
cusp forms, via the Fourier expansion. This is significant for the theory of special functions
on GL 3 because the Mellin-Barnes integral is simpler analytically, but the Jacquet integral
p q
has a certain algebraic structure that we rely on in constructions such as [5]. Conversely, we
must then rely on Miyazaki’s solution of the differential equations (or representation theory)
for the uniqueness property.
A primary tenet of this paper, in combination with the previous part [6], is to be self-
contained. WegenerallysucceedhereexceptforthreepieceswhicharecollectedintoTheorem
1: First, we do not prove the initial spectral decomposition; this is a standard proof using
the techniques of Selberg studying integral operators first carried out by [10] and can be
found in the first few pages of Harish-Chandra’s book [13], or Langland’s paper [19], or in
a number of other books, e.g. [25], see also [15, Appendix A]. Second, we quote the trivial
bound for the principal series representations; this is a bound on Langlands parameters for
the conjecturally non-existentent complementary series and can be found in [28]. Lastly, we
do not prove multiplicity one for Whittaker functions; from the analytic perspective, we are
avoiding the solution to a lengthy problem in partial differential equations, but it follows in
the representation-theoretic language from Shalika’s paper [27].
The proofs below will generally assume that the quotient is by Γ SL 3,Z , but of course
“ p q
only the Fourier expansion will see the particular discrete group in use.
2. The main theorems
We continue in the notation of part I [6].
Let A be the space of smooth scalar-valued cusp forms. Let Ad, d 0 be the space
ě
of smooth, vector-valued cusp forms with K-type Dd; that is f Ad satisfies f gk
P p q “
f g Dd k . (So in particular, f is a 2d 1 -dimensional row vector.) In Section 4.3, we
p q p q p ` q
describe the decomposition of A into the spaces Ad by the projection operators on K. As
discussed in the introduction, we will use three external results about these cusp forms.
Theorem 1.
(1) The space of scalar-valued cusp forms decomposes into a direct sum of simultaneous
eigenspaces Aµ of the Casimir differential operators ∆1 and ∆2 (see Section 4.2).
We parameterize the eigenspaces by the eigenvalues of the power function
∆ipρ µ λi µ pρ µ, λ1 µ 1 µ21`µ222`µ23, λ2 µ µ1µ2µ3.
` “ p q ` p q “ ´ p q “
HIGHER WEIGHT ON GLp3q, II: THE CUSP FORMS. 3
Again, the eigenspaces A further decompose into Ad. The space Ad is finite-
µ µ µ
dimensional.
(2) If Ad 0 with d 0,1 and µ of the form x it, x it, 2it , then x 1.
µ ‰ t u P t u p ` ´ ` ´ q | | ă 2
(3) The n-th Fourier coefficient of φ A lies in the span of the Jacquet integrals. That
µ
P
is,
φ ug ψ u du I w ug ψ u du,
n µ,f l n
p q p q “ p q p q
żU Z U R żU R
p qz p q p q
with I xyk p y f k for some smooth f k . The integral on the right con-
µ,f ρ µ
p q “ ` p q p q p q
verges absolutely on Re µ1 Re µ2 Re µ3 , and must be interpretted in the
p q ą p q ą p q
Riemann sense when some Re µ Re µ .
i j
p q “ p q
Proof. Part (1) is Theorem 3 and Lemma 18 in [13].
Part (2) is the unitary dual estimate noted in [22, Appendix A.1]; of course, stronger
estimates are known for d 0, e.g [17, Appendix 2], and the d 1 case should also follow by
“ “
Rankin-Selberg theory (as [16] does for the unramified case), but to the author’s knowledge,
no one has yet performed this computation.
For part (3), we note that φ generates a representation of G via right translation; it has
the Fr´echet topology generated from the Lie algebra. This representation is equivalent to a
subrepresentation [18, Theorem 8.37] of a principal series representation
f f xyvk p y χ v f k
ρ µ
t | p q “ ` p q p q p qu
for some µ and χ. The operator on the first representation which takes a cusp form to its n-
th Fourier coefficient has image in the Whittaker model, as does the Jacquet integral on the
second representation, and they respect the Fr´echet toplology. Shalika’s local multiplicity
one theorem [27, Theorem 3.1] states these operators (from the first representation to the
(cid:3)
Whittaker model) are identical.
In Section 5, we will show the Lie algebra gives rise to five differential-vector operators
Ya : Ad Ad a, a 2, 1,0,1,2 which are left-translation invariant. We describe the
`
Ñ “ ´ ´
cusp forms by their behavior under the action of these operators. In Sections 8.3 and 9.1,
we will show
Theorem 2. Define the differential operator
(1) Λx 27∆22 4 ∆1 x2 1 ∆1 4x2 1 2, x R.
“ ` ` ´ ` ´ P
Then for f Ad, the following a`re equivalent˘:` ˘
P
(1) f is orthogonal to Y1Ad 1 and Y2Ad 2.
´ ´
(2) f is a zero of Y 1, Y 2 and an eigenfunction of Y0.
´ ´
(3) d 0, d 1 or f is a zero of the operator Λ .
d´1
“ “ 2
We say that such an f is at its minimal K-type and we define Ad to be the space of such
˚
forms. Further define Adµ˚ be the subspace of simultaneous eigenfunctions of ∆1 and ∆2 with
eigenvalues matching p .
ρ µ
`
The value of a description as in part (3) is that it involves only the particular K-type we
are interested in, without reference to Ad 1 or Ad 2. This argument of Section 9.1 also gives
´ ´
an equivalent statement of the Strong Selberg Eigenvalue Conjecture:
4 JACK BUTTCANE
1
Conjecture 3. The null space of Λ in the cusp forms is trivial for x 0, .
x 2
P p q
In fact, we expect Λ to be a postive operator on the cusp forms.
x
For vectors in C2d 1, we again index from the central entry. Let vd be the 2d 1 -
` j
p ` q
dimensional row vector with entries
(2) vd δ ,
j,m1 m1 j
“ “
and set
1
(3) ud, vd 1 dvd .
j ˘ “ 2p j,m1 ˘p´ q ´j,m1q
In Section 8.3, we prove
Theorem 4. Supposeφ Ad . Thenits n-th Fouriercoefficientisa multiple offWd ,µ,ψ
P µ˚ p¨ nq
where f C2d 1 and µ can be taken as one of the following:
`
P
1
(1) For d 0, we use f 1 and Re µ 0 or µ x it, x it, 2it , x .
2
(2) For d “ 1, we use f “ u1, and pReqµ“ 0 or“µp `x ´it, `x ´it, 2qit|,| ăx 1.
0´ 2
“ “ p q “ “ p ` ´ ` ´ q | | ă
(3) For d 2, we use f ud, and µ d 1 it, d 1 it, 2it .
ě “ d` “ ´2 ` ´ ´2 ` ´
We note that the symmetric square of a h`olomorphic modular form˘of even weight k will
occur at minimal weight d 2k 1 in the form (3) with t 0, c.f. [7, Section 6.4].
“ ´ “
A by-product of the analysis of Section 8.3 is the explicit determination of all vectors
f C2d 1 such that fWd ,µ,ψ is identically zero, when µ is in the standard form (48).
` n
P p¨ q
This is the main obstruction in passing from the analysis of differential operators on the
power functions (that is, elements of the principal series representation) to analysis on the
Whittaker functions.
The vectors ud,ε , with δ 0,1 and ε 1, are the rows of the matrices Σd of Section
2j δ P t u “ ˘ χ
I.2.2.21 where 1`δ χ v and ε χ v or conversely χ χ . If Φ Aκ has
Fourier coefficipe´ntsqρΦ“pnqpuκ2´j,ε`δqWκp¨,µ,“ψnqpw`i´tqh the parameters “µ anp´d1quδ,κ2εj,ε δ “ fPgivµe˚n by
Theorem 4, then for all d, we`may construct two matrix-valued forms `
(4) Φd g ρ n ΣdWd γvg,µ,ψ ,
φ χ n
p q “ p q p q
γ U Z V SL 2,Z v V n N2
Pp p qÿqz p q ÿP ÿP
and Φd g defined similarly but using the spectral parameters µ instead. Note that this is
p q ´
not the dual form. We will insist on a slightly unusual normalization (70) of Φ.
InrSection 7, we look at the spaces of vector-valued forms generated by applying the Ya
operators to the minimal weight cusp forms. More precisely, we operate at the level of the
coefficient vectors. That is, if v lies in anappropriate subspace ofC2d 1 (therowspace of Σd),
` χ
then the entries of p y vDd k lie in the principal series representation and the entries of
ρ µ
vWd g,µ,ψ lie in t`hepWq hittpakqer model, and we will show the entries of vΦd g (or more
m
p q p q
precisely vΦd g ) are cusp forms. Since the Ya operators function identically on all three
p q
objects by left-translation invariance, we are free to study their behavior on objects of the
first type. r
1
Section and equation numbers beginning with an I reference [6].
HIGHER WEIGHT ON GLp3q, II: THE CUSP FORMS. 5
Now let S3κ be a basis of Aκ˚ for each κ, and take S3 to be the union of all the S3κ (in an
appropriately weird space of definition). As described above, for each Φ S3 and every d,
P
we may construct the two matrix-valued forms Φd and Φd. It is the case that the function
Φd will be identically zero when d is less than the minimal K-type of Φ, and in Section 9,
we pull everything together into the following very uniforrm expansion of cusp forms:
r
Theorem 5. For f A, we have
P
8 T
f g 2d 1 Tr Φd g f g Φd g dg .
1 1 1
p q “ p ` q p q p q p q
ΦÿPS3dÿ“0 ˆ żΓzG ˙
r
The equivalent statement for a vector-valued form f Ad is
P
T
(5) f g f g Φd g dg Φd g .
1 1 1
p q “ p q p q p q
ΦÿPS3żΓzG
This completes the spectral expansion. r
Each vector-valued cusp form of K-type Dd corresponds to 2d 1 scalar-valued forms,
`
and the multiplicities of a given Φ Sκ in the vector-valued forms are essentially the rank
3˚
P
of the matrices Σd, with the middle 2κ 1 rows removed if κ 2. By the types listed in
χ ´ ě
Theorem 4, these multiplicities are
d 1 d even,
(1) 2d`1 d odd,
´2
"
(2) d 1
`2
(3) d κ 1 d κ.
X ´2 \ ` ě
Finally, since many theorems on automorphic forms concern forms at the minimal K-
X \
types, in Section 6, we compute the Mellin-Barnes integrals of those Whittaker functions.
These have been computed by Miyazaki [24] for d 2, Manabe, Ishii and Oda [21] for d 1
ě “
and Bump [4] for d 0. In case d 1, the bases used are somewhat different than ours,
“ ě
and for d 2, the results here are somewhat stronger than Miyazaki’s because we have
ě
solidified the connection between the Jacquet-Whittaker function and the Mellin-Barnes
integral. (Theorem 5.9 of [24] contains the phrase “there is an element [of the Whittaker
model]”; we have very precisely answered the question of which vector in the principal series
representation gives rise to that element.)
For α 0,1 3, β,η Z3, ℓ Z2 and s C2 define
P t u P P P
Λdαpµq “ π´23`µ3´µ1Γ 1`α1`2µ1´µ2 Γ 1`α2`2µ1´µ3 Γ 1`α3`2µ2´µ3 ,
` 3 Γ˘ βi`s1 µi Γ ηi˘ s2`µi ˘
G d,β,η,s,µ i 1 `2´ `2` ,
“
p q “ ś Γ `s1`s2` ip2β˘i`ηi`q´2d ˘
ř
0 1 ´ ¯
G ℓ,s,µ G 0,0,0,s,µ , G ℓ,s,µ G 1, ℓ1,ℓ1,1 ℓ1 , ℓ2,ℓ2,1 ℓ2 ,s,µ ,
p q “ p q p q “ p p ´ q p ´ q q
and for d 2,
ě
(6) Gd ℓ,s,µ G d, d,0,ℓ1 , 0,d,ℓ2 ,s,µ .
p q “ p p q p q q
6 JACK BUTTCANE
Now for m d, write m εm with ε 1 and 0 m d, and set
1 1
| | ď “ “ ˘ ď ď
m
2d m
Gd s,µ εℓ Gd d m,ℓ ,s,µ ,
m1p q “d d m ℓ pp ´ q q
ˆ ` ˙ℓ 0 ˆ ˙
ÿ“
and take Gd s,µ to be the vector with coordinates Gd s,µ .
p q m1p q
Theorem 6. The Whittaker functions at the minimal K-types are
1 G0 s,µ ds
W0py,µ,ψ1,1q “4π2 pπy1q1´s1pπy2q1´s2Λ0 p µq 2πi 2,
żRepsq“s p0,0,0qp qp q
1 G1 s,µ ds
u10,´W1py,µ,ψ1,1q “4π2 pπy1q1´s1pπy2q1´s2Λ1 p µq 2πi 2,
żRepsq“s p0,1,1qp qp q
1 G1 s,µw3 ds
´?2u11,´W1py,µ,ψ1,1q “4π2 żRepsq“spπy1q1´s1pπy2q1´s2Λ1p1,0p,1qpµw3qqp2πiq2,
´?2u11,`W1py,µ,ψ1,1q “4π12 żRepsq“spπy1q1´s1pπy2q1´s2ΛG1p11,1p,s0q,pµµwwllqqp2dπsiq2,
and for µ d 1 it, d 1 it, 2it with d 2,
´2 ´2
“ p ` ´ ` ´ q ě
1 Gd s,µ ds
W´ddpy,µ,ψ1,1q “4π2 żRepsq“spπy1q1´s1pπy2q1´s2Λdp0,p0,1qpµqqp2πiq2, Wdd,m “ 0,
for any s R 2.
`
P p q
3. Acknowledgements
The author would like to thank Stephen D. Miller for his insight into the non-spherical
Maassforms; inparticular,Prof. Millerandtheauthorindependentlyandnear-simultaneously
arrived at forms of (9)-(11). The author would also like to thank James Cogdell and Joseph
Hundley for their comments and helpful discussions along the way.
4. Background
4.1. The Lie algebras. The complexified Lie algebra of G PSL 3,R is g sl 3 , the
space of complex matrices of trace zero, and the complexified“Lie algpebraqof K“ SpOq3,R
“ p q
is k, the space of anti-symmetric matrices. These matrices act as differential operators on
smooth functions of G by
d
(7) Xf g : f gexptX , X iY f Xf iYf,
p qp q “ dt p q p ` q “ `
ˇt 0
ˇ “
when the entries of X andY are real. Wewˇill not differentiate notationallybetween amatrix
ˇ
in the Lie algebra and the associated differential operator. Note that the commutator of the
HIGHER WEIGHT ON GLp3q, II: THE CUSP FORMS. 7
differentialoperatorsassociatedtotwosuchmatricesisthengivenbythedifferential operator
associated to the commutator of the matrices; that is,
X,Y : X Y Y X XY YX.
r s “ ˝ ´ ˝ “ ´
We will also need some knowledge of the adjoint operator: If f,h A, then
P
(8) Xf g h g dg f g Xh g dg,
p qp q p q “ p qp´ qp q
żΓ G żΓ G
z z
directly from the definitions (7) and the Haar measure dg.
We take as our basis of k the three matrices
0 0 1 0 1 0
¯
K 0 0 i , K 1 0 0 .
1 0
˘ “¨ 1 i ´0 ˛ “¨´0 0 0˛
˘
˝ ‚ ˝ ‚
In terms of the Z-Y-Zcoordinates of Section I.2.2, the associated differential operatorsacting
on a function of k α,β,γ , are
p q
K 1 e¯iγ icotβ γ β icscβ α , K0 γ.
˘ “ p B ¯B ´ B q “´B
On an entry of the Wigner-D matrix, the K 1 operators increase or decrease the column[3,
˘
Section 3.8],
K 1Dmd1,m d d 1 m m 1 Dmd1,m 1,
˘ “ p ` q´ p ˘ q ˘
and K0 does not, a
K0Dmd1,m “ imDmd1,m.
The Laplacian on K is given by
∆K K1K 1 K0K0 iK0,
“ ´ ´ `
and the Wigner-D matrix satisfies ∆ Dd d d 1 Dd.
k
“ p ` q
We extend to a basis for g by adding the five matrices
1 i 0 0 0 1 1 0 0
˘ ˘ ?6
X i 1 0 , X 0 0 i , X 0 1 0 .
2 1 0
˘ “¨˘0 ´0 0˛ ˘ “´¨ 1 i 0 ˛ “´ 3 ¨0 0 2˛
˘ ´
˝ ‚ ˝ ‚ ˝ ‚
Though we will not directly use it, these matrices are orthogonal to k and have identical
norm under the Killing form.
If we define the differential operators
Z˘2 “p2y2By2 ´y1By1q˘2iy2Bx2,
Z˘1 “´2ipy1x2Bx3 `y1Bx1q¯2y1y2Bx3,
Z0 “´?6y1By1,
and
Z Z i , Z Z , Z Z ,
2 2 α 1 1 0 0
˘ “ ˘ ˘ B ˘ “ ˘ “
r r r
8 JACK BUTTCANE
then as differential operators in Iwasawa UYK, i.e. g xyk, coordinates, our basis has the
“
following form
2
(9) X˘2 “j 2Dj2,˘2pkqZj ´ ?36D02,˘1pkqK˘1 ˘ 2i3?6D02,˘2pkqK0,
ÿ“´
2 r
(10) X˘1 “j 2Dj2,˘1pkqZj ´ 13p4D02,0pkq´1qK˘1 ˘ 2i3?6D02,˘1pkqK0 ˘i?2D01,˘1pkqBα,
ÿ“´
2 r
2 2 2
(11) X0 “j 2Dj,0pkqZj ´ D0,1pkqK´1 `D0,´1pkqK1 .
ÿ“´ ` ˘
These can be computed byrwriting
1
xykexptX xyx y yy k ,
˚ ´ ˚ ˚
“ p qp q
where kexptX x y k is computed from the Iwasawa decomposition of Section I.2.4. A
˚ ˚ ˚
somewhat less co“mputationally-intensive approach is to note that kexptX exp tkXk 1 k
´
and decompose kXk 1 over the more Iwasawa-friendly basis “ p q
´
(12) E1,2,E1,3,E2,3,E11 E22,E11 E33,E22 E33,K0,K 1,
´ ´ ´ ˘
where E δ indexing from the upper left. We have expressed, as we may, all of
ij k,l i k,j l
p q “ “ “
the trigonometric polynomials in α,β,γ in terms of the Wigner D-matrices in preparation
for application of the Clebsch-Gordan multiplication rules, see Section 4.5.
4.2. The Casimir Operators. The (normalized) Casimir operators are defined by
1 1
∆1 Ei,j Ej,i, ∆2 Ei,j Ej,k Ek,i ∆1.
“ ´2 ˝ “ 3 ˝ ˝ `
i,j i,j,k
ÿ ÿ
In the Lie algebra, these are sometimes called the Capelli elements, and their differential
operators may be computed through the Iwasawa-friendly basis (12) or through the expres-
sions already given for the X and K operators by expressing the E as linear combinations
j j ij
of either basis (and the identity matrix).
On functions of PSL 3,R SO 3,R these become [11, eqs (6.1.1)]
p q{ p q
2 2 2 2 2 2 2 2 2 2 2 2 2
∆˝1 “ ´y1By1 ´y2By2 `y1y2By1By2 ´y1px2 `y2qBx3 ´y1Bx1 ´y2Bx2 ´2y1x2Bx1Bx3,
2 2 2 2 3 2 2 2 2
∆˝2 “´y1y2By1By2 `y1y2By1By2 ´y1y2Bx3By1 `y1y2Bx2By1 ´2y1y2x2Bx1Bx3By2
2 2 2 2 2 2 2 2 2 2 2
`p´x2 `y2qy1y2Bx3By2 ´y1y2Bx1By2 `2y1y2Bx1Bx2Bx3 `2y1y2x2Bx2Bx3
2 2 2 2 2 2 2 2 2 2 2 2 2
`y1By1 ´y2By2 `2y1x2Bx1Bx3 `px2 `y2qy1Bx3 `y1Bx1 ´y2Bx2.
We will require expressions for the full operators.
The Lie algebra acts on smooth functions on K on the left by the operators [3, Section
3.8]
J1 i cosαcotβ α sinα β cosαcscβ γ ,
“ p B ` B ´ B q
J2 i sinαcotβ α cosα β sinαcscβ γ ,
“ p B ´ B ´ B q
HIGHER WEIGHT ON GLp3q, II: THE CUSP FORMS. 9
with
J1 iJ2 Dmd1,m d d 1 m1 m1 1 Dmd1 1,m.
p´ ˘ q “ p ` q´ p ˘ q ˘
We define the operators a
T1 “y1px2Bx3 `Bx1q, T4 “T3T2 `T1p1´y2By2q,
T2 “y1y2Bx3, T5 “T2p1`y2By2 ´y1By1q`T1T3,
T3 “y2Bx2, T6 “T1T2 ´T3 `y1By1T3,
then as functions of x1,x2,x3,y1,y2,α,β,γ, the full Casimir operators on SL 3 are given by
p q
(13) ∆1 ∆˝1 iJ1T1 iJ2T2 αT3,
“ ` ´ `B
(14) ∆2 ∆˝2 iJ1 T4 αT2 iJ2T5 αT6.
“ ` p `B q´ `B
4.3. The projection operators. Let C G,Dd be the space of smooth functions on G that
p q
lie in the span of the entries of Dd. That is f C G,Dd if and only if
P p q
r
f xyk f xy Dd k .
m1r,m m1,m
p q “ p q p q
m1,m d
| |ÿ| |ď
For f C G,Dd , taking the expansion into Wigner D-functions and evaluating at the
P p q
identity gives
r
f xyk 2d 1 f xykk Dd k dk Pdf xyk ,
p q “ p ` q p 1q m,mp 1q 1 “ m p q
m dżK m d
|ÿ|ď |ÿ|ď
r
where
Pdf g 2d 1 f gk Dd k dk
m m,m
p qp q “ p ` q p q p q
żK
is the projection onto the srpan of the m-th column of Dd.
The projection operators may also be written as
Pdf xyk 2d 1 f xyk Dd k dk Dd k .
m p q “ p ` q p 1q m1,mp 1q 1 m1,mp q
m1 dżK
| ÿ|ď
r
Also let
Pd Pd.
“ m
m d
|ÿ|ď
r r
Note that the projection operators Pmd commute with the differential operators ∆1 and ∆2,
but the projection onto the span of a single Wigner function Dd might not.
m1,m
Now if f C G,Dd , we may wrrite f in terms of row vector-valued functions which
P p q
transform by Dd,
r
fpxykq “ pflpxykqql, flpxykq :“ p2d`1q fpxykk1qDld k1´1 dk1 “ flpxyqDdpkq,
|lÿ|ďd żK ´ ¯
10 JACK BUTTCANE
so we may move to considering vector-valued functions, say C G,Dd is the space of smooth
p q
vector-valued functions f xyk f xy Dd k . Define the vector projection operators Pd :
p q “ p q p q m
C G,Dd C G,Dd by
p q Ñ p q
r pPmdfqpgq “ p2d`1q fpgk1qDmd k1´1 dk1.
żK ´ ¯
Note that we have an isomorphism C G,Dd C G,Dd 2d 1 as each scalar-valued func-
`
p q – p q
tiongivesrisetoonevector-valuedfunctionforeachrowofDd. Thesefunctionsareinherently
linear combinations of the rows of Dd kr,
p q
f xyk ,...,f xyk f xy Dd k ,
p ´dp q dp qq “ m1p q m1p q
m1 d
| ÿ|ď
and the entries may be written as
f xyk f xy Dd k .
m m1 m1,m
p q “ p q p q
m1 d
| ÿ|ď
The raising and lowering operators on K then give
K 1fm xyk d d 1 m m 1 fm 1 xyk ,
˘ p q “ p ` q´ p ˘ q ˘ p q
using fm “ 0 when |m| ą d. The opearators Pld and K˘1 have simple commutation relations,
PldK˘1 “r K˘1Pld¯1,
and this gives
r r
PldK˘1 “ dpd`1q´lpl ˘1qPld¯1.
4.4. The commutation relations. Wae record the commutation relations amongst the dif-
ferential operators. The commutators on the K are
j
K 1,K0 iK 1, K1,K 1 2iK0,
r ˘ s “ ¯ ˘ r ´ s “ ´
The commutators involving an X and a K are
j j
Xj,K0 ijXj, X 2,K 1 2X 1, X 1,K 1 ?6X0,
r s “ ´ r ˘ ¯ s “ ´ ˘ r ˘ ¯ s “ ´
X0,K 1 ?6X 1, X 1,K 1 2X 2, X 2,K 1 0.
r ˘ s “ ´ ˘ r ˘ ˘ s “ ´ ˘ r ˘ ˘ s “
The commutators amongst the X are
j
X 2,X0 0, X 1,X0 ?6K 1, X 2,X 1 0, X 2,X 1 2K 1,
r ˘ s “ r ˘ s “ ˘ r ˘ ˘ s “ r ˘ ¯ s “ ´ ˘
X2,X 2 4iK0, X1,X 1 2iK0.
r ´ s “ ´ r ´ s “
The relations with ∆ are
K
X0,∆K 6X0 ?6K 1X1 ?6K 1X 1,
r s “ ´ ´ ´ ` ´
X 1,∆K 6X 1 2iK0X 1 ?6K 1X0 2K 1X 2,
r ˘ s “ ˘ ˘ ˘ ´ ˘ ´ ¯ ˘
X 2,∆K 6X 2 4iK0X 2 2K 1X 1.
r ˘ s “ ˘ ˘ ˘ ´ ˘ ˘