International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 842806,33pages
doi:10.1155/2011/842806
Research Article
Shintani Functions on
SL
3
,
R
Keiju Sono
Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan
Correspondence should be addressed to Keiju Sono,[email protected] Received 25 May 2011; Revised 26 September 2011; Accepted 26 September 2011 Academic Editor: Andrei Volodin
Copyrightq2011 Keiju Sono. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate the Shintani functions attached to the spherical and nonspherical principal series representations ofSL3,R. We give the explicit formulas of the radial part of Shintani functions and evaluate the dimension of the space of Shintani functions.
1. Introduction
Shintani function is originally introduced by Shintani forp-adic linear groupGLn, k, where
k is a finite extension of thep-adic field Qp 1. He defined some “Whittaker function” on
GLn, kand obtained the explicit formulas of them. Moreover, he proved the uniqueness of
his function. Later, a more detail study of Shintani functions forGLnwas done by Murase
and Sugano2 see also3. They obtained new kinds of integral formulas for theL
-func-tions in terms of the global Shintani func-func-tions and proved the multiplicity one theorem of the local one at the finite primes.
On the other hand, the multiplicity and explicit formulas of the Archimedean Shintani functions were more recently investigated by some mathematicians. For example, Hirano
studied the Shintani functions onGL2,R 4andGL2,C 5, Tsuzuki onSU1,1 6and
Un,1 7, and Moriyama onSp2,R 8,9. They constructed the differential equations
satisfied by the radial part of the Shintani functions and obtained the explicit formulas by solving them. Most of them are expressed by some linear combinations of the Gaussian hypergeometric functions. Moreover, the dimensions of the spaces of Shintani functions are obtained, which are sometimes bigger than 1.
In this paper, we investigate the Shintani functions onG SL3,R, attached to the
onG. We take
H
H1 0
0 h2
∈G|H1, h2∈GL2,R×GL1,R
1.1
as a subgroup ofGand takeK SO3as a maximal compact subgroup ofG. Letπ be an
arbitrary irreducible unitary representation ofGandη, Vηan irreducible unitary
represen-tation ofH, and letC∞ηH\Gbe the space of smooth functionsF:G → VηsatisfyingFhg
ηhFgh, g∈H×G. We consider the intertwining spaceIη,πHomgC,Kπ, C∞ηH\G
and its restriction
Iη,π−→HomK
τ, C∞ηH\G∼Cη,τ∗L 1.2
to the minimal K-type τ, Vτof π, where τ∗, Vτ∗ is the contragredient representation of
τ, Vτ,L:H\G/KandC∞η,τ∗Lis the space of smooth functionsF:G → Vη⊗Vτ∗satisfying
Fhgk ηh⊗τ∗k−1Fgforh, g, k∈H×G×K. The function which belongs to the
image of above map is called the Shintani function. In this paper, we assume thatπ is the
irreducible unitary principal series representation ofGandη is the unitary character ofH.
The study of Shintani functions for the general unitary representationηofHis a further
pro-blem.
InSection 4, we investigate the Shintani functions attached to the spherical or class
oneprincipal series representations. These representations have uniqueK-fixed vector, and
hence the minimalK-type is one-dimensional. In this case, the explicit formulas of Shintani
functions are obtained by solving two Casimir equations which are characterized by the action of the center of universal enveloping algebra. We also obtain the necessary condition of the existence of nonzero Shintani functions and prove that the dimension of the space of
Shintani functions is equal to or less than 1Theorem 4.8.
On the other hand, inSection 5, we investigate the Shintani functions attached to the
nonspherical principal series representations, whose minimal K-type is three-dimensional
representation of K. In this case, we construct two kinds of differential equations. One is
the Casimir equation we used inSection 4, and the other is the gradient equation. The key
point is as follows. We have three different nonspherical principal series with the same
infinitesimal charactersZg → C. We cannot distinguish them only by the elements ofZg.
This is the reason we need the gradient operator which has distinct eigenvalues for diff
e-rent nonspherical principal series. By combining these equations, we obtain the explicit for-mulas of the Shintani functions, the necessary condition of the existence of nonzero Shintani functions, and prove that the dimension of the space of Shintani functions is equal to or less
than 1Theorem 5.7.
As an application of the results of this paper, our explicit formulas for Shintani functions will be useful to compute the local zeta integral in the theory of Murase and Sugano
2,3;see also10, in the case ofUn,1. Furthermore, the author thinks these results are
2. Preliminaries
2.1. Groups and Algebras
LetG be the real reductive Lie groupSL3,R andg sl3,Rits Lie algebra. The Cartan
involutionθ :G → Gis defined byθg tg−1g ∈ G, and its differentialdθ :g → g
is given bydθX −tX X ∈ g, where t means the transposition of matrices. Then the
fixed subgroup ofθinGis equal toK SO3, which is the maximal compact subgroup of
G. Next, we define another involutive automorphismσofGbyσg JgJ g ∈G, where
J diag−1,−1,1. Its differentialdσ :g → gis given bydσX JXJ X ∈g. The fixed
subgroupHofσinGis isomorphic toGL2,R, that is,
H
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
⎛ ⎜ ⎜ ⎝
a11 a12 0
a21 a22 0
0 0 a33
⎞ ⎟ ⎟ ⎠∈G
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
∼
GL2,R. 2.1
We define1 and−1 eigenspaces ofdθ,dσingby
k{X∈g|dθX X}, p{X ∈g|dθX −X},
h{X∈g|dσX X}, q{X ∈g|dσX −X}.
2.2
Then,k,hare the Lie algebras ofK, H, respectively. We havegk⊕ph⊕q. LetEij∈M3,R
be the matrix whosei, j-component is 1 and the other components are 01≤i, j ≤3. For
1≤i < j≤3, we putKijEij−Eji, XijEijEji, HijEii−Ejj. Then, we have
k
i<j
RKij, p
i<j
RXij⊕RH12⊕RH23. 2.3
Next, we take
Y1
⎛ ⎜ ⎜ ⎝
1 0 0
0 0 0
0 0 −1
⎞ ⎟ ⎟
⎠, Y2
⎛ ⎜ ⎜ ⎝
0 1 0
1 0 0
0 0 0
⎞ ⎟ ⎟
⎠, Y3
⎛ ⎜ ⎜ ⎝
0 −1 0
1 0 0
0 0 0
⎞ ⎟ ⎟
⎠, Y4
⎛ ⎜ ⎜ ⎝
0 0 0
0 1 0
0 0 −1
⎞ ⎟ ⎟ ⎠
2.4
as a basis ofh. We havep∩q RX13⊕RX23, and we takea RX13 as a maximal Abelian
subspace ofp∩q. We define a subgroupAofGby
Aexpa
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩at:
⎛ ⎜ ⎜ ⎝
cosht 0 sinht
0 1 0
sinht 0 cosht
⎞ ⎟ ⎟ ⎠|t∈R
⎫ ⎪ ⎪ ⎬ ⎪ ⎪
⎭. 2.5
Then,Ghas a decompositionGHAK. Throughout this paper, we putL:H\G/K.
2.2. The Principal Series Representations
As a representation ofG, we take the principal series representation defined as follows. Let
P0 be a minimal parabolic subgroup ofGgiven by the upper triangular matrices inGand
P0MAP0Nthe Langlands decomposition ofP0with
AP0
diaga1, a2, a3|ai>0, a1a2a31
,
MK∩diagonals in G,
N
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
⎛ ⎜ ⎜ ⎝
1 x1 x2
0 1 x3
0 0 1
⎞ ⎟ ⎟
⎠∈G|x1, x2, x3∈R
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭.
2.6
To define a principal series representation with respect to the minimal parabolic subgroupP0
ofG, we firstly fix a characterσofMand a linear formν∈a∗P
0⊗RCHomRaP0,C, where
aP0is the Lie algebra ofAP0. We write
νdiagt1, t2, t3
ν1t1ν2t2 2.7
for diagt1, t2, t3∈aP0. Then, we can define a representationσ⊗a
νofMA
P0 and extend this
toP0by the identificationP0/N MAP0, taking the trivial representation1N as the
repre-sentation ofN. Then, the induced representation
πσ,ν :C∞IndGP0σ⊗a νρ⊗1
N 2.8
is called the principal series representation ofG. Here,ρis the half sum of positive roots of
g,agiven byaρa2
1a2, foradiaga1, a2, a3∈AP0.
Concretely, the representation space is given by
C∞M,σK:f ∈C∞K|fmk σmfk, m∈M, k∈K, 2.9
and the action ofGis defined by
πσ,νxf
k akxνρfκkx x∈G, k∈K. 2.10
Here, forg ∈ G, g ngagκg ng ∈ N, ag ∈ AP0, κg ∈ Kis the Iwasawa
de-composition. Throughout this paper, we assume that the representationπσ,ν is irreducible.
Moreover, we assume thatν1, ν2are the elements of
√
−1R. Then, this representation becomes
unitary.
Next, we define charactersσj j0,1,2,3ofMas follows. The groupMconsisting of
four elements is a finite Abelian group of2,2-type, and its elements except for the unity are
given by
Table 1
m1 m2 m3
σ1 1 −1 −1
σ2 −1 1 −1
σ3 −1 −1 1
The setMconsists of 4 characters{σj |j 0,1,2,3}, whereσ0is the trivial character ofM
andσ1, σ2, σ3are defined byTable 1.
The following propositionsee11, Proposition 1.1 gives the correspondence
bet-ween the characterσofMand the minimalK-type of the principal series representationπσ,ν
ofG.
Proposition 2.1. 1Ifσis the trivial character ofM, the representationπσ,ν is spherical or class
one. That is, it has a uniqueK-invariant vector inHσ,ν.
2If σis not trivial, the minimal K-type of the restriction πσ,ν|K toK is a 3-dimensional
representation ofK, which is isomorphic to the unique standard oneτ2, V2. The multiplicity of this
minimalK-type is one:
dimCHomKτ2, Hσ,ν 1. 2.12
3. The Space of Shintani Functions
3.1. The Definition of Shintani Functions
As a representation ofH, we take the unitary characterη ηs,k :H → C× s∈
√
−1R, k ∈
{0,1}defined by
η
⎛ ⎜ ⎜ ⎝
⎛ ⎜ ⎜ ⎝
h11 h12 0
h21 h22 0
0 0 h1
⎞ ⎟ ⎟ ⎠
⎞ ⎟ ⎟
⎠detH1k|detH1|s−k, H1
h11 h12
h21 h22
∈GL2,R. 3.1
Letηηs,kbe the unitary character ofHdefined previously. We consider the induced
repre-sentationC∞IndGHηwith the representation space
Cη∞H\G F ∈C∞G|FhgηhFg, ∀h∈H, ∀g∈G. 3.2
Gacts on this space by right translation. Letπσ,νbe the principal series representation ofG.
We consider the intertwining space
Iη,πσ,ν :HomgC,K
πσ,ν, C∞ηH\G
We denote its image bySη,πσ,ν, that is,
Sη,πσ,ν :
T∈Iη,πσ,ν
ImageT. 3.4
We call the element ofSη,πσ,ν the Shintani function of typeη, πσ,ν. Letτ, Vτbe theK-type
of the principal series representationπσ,ν, and letι:τ → πσ,νbe theK-embedding ofτand
ι∗the pullback viaι. Then, the map
ι∗:Iη,πσ,ν −→HomK
τ, Cη∞H\G∼C∞η,τ∗L 3.5
gives the restriction ofT ∈Sη,πσ,νtoτ, whereτ∗is the contragredient representation ofτand
the spaceCη,τ∞∗Lis defined by
C∞η,τ∗L
F:G−→Vτ∗|F
hgk−1ηhτ∗kFg, ∀h, g, k∈H×G×K. 3.6
We denote the image ofIη,πσ,νinC∞η,τ∗LbyCη,τ∞∗Lπσ,ν, and the element of this space is called
the Shintani function of typeπσ,ν, η, τ.
3.2. The
A
-Radial Part
BecauseGhas the decompositionG HAK, the element ofC∞η,τLis characterized by its
restriction toA. We denote the centralizer and the normalizer ofAinK∩HbyZK∩HA,
NK∩HA, respectively. It is easy to verify thatK∩H,ZK∩HA, andNK∩HAare given as
follows.
Lemma 3.1. One has
K∩H
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎝
cosθ sinθ 0
−sinθ cosθ 0
0 0 1
⎞ ⎟ ⎟ ⎠|θ∈R
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎝
cosθ sinθ 0
sinθ −cosθ 0
0 0 −1
⎞ ⎟ ⎟ ⎠|θ∈R
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭,
ZK∩HA ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩I, ⎛ ⎜ ⎜ ⎝
−1 0 0 0 1 0
0 0 −1
⎞ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭,
NK∩HA ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩I, ⎛ ⎜ ⎜ ⎝
−1 0 0
0 −1 0
0 0 1
⎞ ⎟ ⎟ ⎠, ⎛ ⎜ ⎜ ⎝
−1 0 0 0 1 0
0 0 −1
⎞ ⎟ ⎟ ⎠, ⎛ ⎜ ⎜ ⎝
1 0 0
0 −1 0
0 0 −1
⎞ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭. 3.7
Letw0 : diag−1,−1,1. Then,w0ZK∩HAis the unique nontrivial element ofW
NK∩HA/ZK∩HA. Letη : H → C be the unitary character ofH and τ, Vτthe
finite-dimensional representation ofK. We denote byCW∞A;η, τthe space of smooth functions
F:A → Vτsatisfying the following conditions:
1ηmτmFa Fa ∀m∈ZK∩HA, ∀a∈A,
2ηw0τw0Fa F
a−1 ∀a∈A,
3ηlτlF1 F1 ∀l∈K∩H.
3.8
The following lemma is proved by Flensted and Jensensee12, Theorem 4.1.
Lemma 3.2. The restriction toAgives the following isomorphism:
C∞η,τL∼CW∞A;η, τ. 3.9
Through this isomorphism, we denote the image of Cη,τ∞Lπσ,ν in CW∞A;η, τ by
C∞WA;η, τπσ,ν. This is our target space in this paper. The following two lemmas are obvious.
Lemma 3.3. Fort /0, one has
gAda−1t h⊕a⊕k. 3.10
Lemma 3.4. LetF ∈C∞η,τL. ForX∈k,Y ∈h,Z∈a, one has
RAda−1t YZXFat ηYτ−X
Zfat. 3.11
4. Shintani Functions Attached to the Spherical Principal
Series Representations
Throughout this section, as a character ofM, we take the trivial characterσ σ0. Then, the
principal series representationπσ0,νis the spherical or class one principal series representation
whose minimal K-type is the one-dimensional trivial representation 1, V1 of K, which
occurs of multiplicity one inπσ0,ν|KProposition 2.1.
4.1. The Capelli Elements
LetZgbe the center of the universal enveloping algebraUgofg.Zghas two
indepen-dent generators, and they are obtained as the Capelli elements becausegsl3is of typeA2
see13. Fori1,2,3, we put
EiiEii−
1 3
3
k1
Ekk
The following proposition gives the explicit description of the independent generators of
Zg see11.
Proposition 4.1. The independent generators{Cp2, Cp3}ofZgare given as follows:
Cp2
E11−1E22E22 E331E11−1E331−E23E32−E13E31−E12E21,
Cp3
E11−1E22E331E12E23E31E13E21E32−
E11−1E23E32−E13E22E31
−E12E21
E331.
4.2
SinceCp2, Cp3are the elements ofZg, they act onπσ0,νas the scalar operators. And
since the space of Shintani functions is the image of thegC, K-homomorphism ofπσ0,ν, they
act on the space of Shintani functions as the same scalar operators, respectively.
4.2. Eigenvalues of
Cp
2,
Cp
3In order to construct the partial differential equations satisfied by spherical functions attached
to the spherical principal series, we have to compute the eigenvalues of the actions of the
Capelli elementsCp2, Cp3. For the spherical principal series representation, σ σ0 is the
trivial character ofM. Let f0 be the generator of the minimalK-type inHσ0,ν normalized
such thatf0 |K ≡1. The actions ofCp2, Cp3onf0are computed in11, and the result is as
follows.
Proposition 4.2. The Capelli elementsCp2, Cp3 act onf0by scalar multiples, and the eigenvalues
are given as follows:
Cp2f0− 1 3
ν2
1−ν1ν2ν22
f0,
Cp3f0− 1
272ν1−ν22ν2−ν1ν1ν2f0.
4.3
4.3. Construction of the Casimir Equations
Next, we compute the actions ofCp2,Cp3onFat ∈C∞η,1L|A. Here,η ηs,k is the unitary
character ofH.
Lemma 4.3. ForFat∈C∞η,τL|A, one has
RAda−1t Yi
Fat sFat i1,4,
RAda−1t Yi
Fat 0 i2,3,
RX13Fat
dF dtat.
Proof. By definition, we have
RAda−1t Yi
Fat
d duF
atexp
uAda−1t Yi
u0
d
duF
expuYiat
u0
d
duη
expuYi
u0Fat.
4.5
Since expuY1 diageu,1, e−u, we haveηexpuY1 eku·es−kuesu.
Therefore, we have
RAda−1t Y1
Fat sFat. 4.6
Next, since
expuY2
⎛ ⎜ ⎜ ⎝
coshu sinhu 0
sinhu coshu 0
0 0 1
⎞ ⎟ ⎟
⎠, 4.7
ηexpuY2 1k· |1|s−k1. Therefore, we have
RAda−1t Y2
Fat 0. 4.8
The computations of the actions of Ada−1t Yi i3,4are similar. Finally, since expuX13
au, we have
RX13Fat d
duFatu
u0
dF
dtat. 4.9
By simple computations of matrices, we have the following expressions of the
ele-ments inM3,R.
Lemma 4.4. One has
E11 cosh
2t1
3 cosh2tAd
a−1t Y1− 1
3Ad
a−1t Y4− 1
2tanh2tK13,
E22 1
3−H12H23,
E33 sinh
2t−1
3 cosh2tAd
a−1t Y1− 1
3Ad
a−1t Y4 1
2tanh2tK13,
E23 1
2 sinhtAd
a−1t Y3 1
2 tanhtK12
1
E13 1
2X13
1
2K13,
E12 1
2 coshtAd
a−1t Y2−1
2tanhtK23
1
2K12.
4.10
To make use ofLemma 3.4, we have to rewriteCp2,Cp3in the form of linear
combi-nations of the elements inAda−1t hak. To do this, we use the following formulas which can
be obtained by direct computation.
Lemma 4.5. One has
!
K13,Ad
a−1t Y1
"
−2 cosh2tX13,
!
K13,Ad
a−1t Y4
"
−cosh2tX13,
!
K12,Ad
a−1t Y3
"
sinhtX13,
!
K23,Ad
a−1t Y3
"
−2 sinh3t
cosh2tAd
a−1t Y12 sinhtAd
a−1t Y4
cosht
cosh2tK13,
K13, X13 2
cosh2tAd
a−1t Y1−2 tanh2tK13,
!
K23,Ad
a−1t Y2
"
−coshtX13,
!
K12,Ad
a−1t Y2
"
2 cosh3t
cosh2tAd
a−1t Y1−2 coshtAd
a−1t Y4−
sinht
cosh2tK13,
K12, X13 −
1
sinhtAd
a−1t Y3− 1
tanhtK12,
K23, X13 1
coshtAd
a−1t Y2−tanhtK23,
!
X13,Ad
a−1t Y2
"
−tanhtAda−1t Y2− 1
coshtK23,
!
K13,Ad
a−1t Y2
"
− 1
tanhtAd
a−1t Y3−cosh2t
sinht K12,
!
K13,Ad
a−1t Y3
"
tanhtAda−1t Y2−cosh2t
cosht K23,
!
X13,Ad
a−1t Y1
"
−2 tanh2tAda−1t Y1−
2
cosh2tK13,
!
X13,Ad
a−1t Y4
"
−tanh2tAda−1t Y1− 1
cosh2tK13,
!
K12,Ad
a−1t Y1
"
−cosh2t
cosht Ad
a−1t Y2−tanhtK23,
!
K12,Ad
a−1t Y4
"
1−sinh2t
cosht Ad
!
K23,Ad
a−1t Y1
"
−cosh2t
sinht Ad
a−1t Y3− 1
tanhtK12,
!
K23,Ad
a−1t Y4
"
−1cosh2t
sinht Ad
a−1t Y3− 2
tanhtK12.
4.11
Here,X, Y:XY−Y Xis the Lie bracket ong.
By usingLemma 4.5, we can rewriteCp2,Cp3as we wished. Now, since for allFat∈
C∞η,1L|A is annihilated by the action ofUgkand Ada−1t Y2UgAda−1t Y3Ug, and the
actions of Ada−1t Y1 and Adat−1Y4 onF are the same the multiplication bys, we may
regardCp2,Cp3as the elements inUgmodP, wherePis the subalgebra ofUgdefined
by
PAda−1t Y1−Ad
a−1t Y4
Ug Ada−1t Y2Ug Ad
a−1t Y3Ug Ugk. 4.12
Lemma 4.6. One has the congruences
Cp2≡ − 1 4X 2 13− 1 2 #
tanh2t 1
tanh2t
$ X13 # 1 4tanh
22t−1
3
$
Ada−1t Y1
2
−1modP,
Cp3≡ − 1 12
Ada−1t Y1
X2 13− 1 6 #
tanh2t 1
tanh2t
$
Ada−1t Y1
X13 # −1 27 1 12tanh
22t$Ada−1
t Y1 3 −1 3
Ada−1t Y1
modP.
4.13
By combiningProposition 4.2, and Lemmas4.3and4.6, we have the following two
dif-ferential equations.
Theorem 4.7. The Shintani functionFat∈C∞η,1Lπσ0,ν|Asatisfies the following equations:
1the differential equation obtained from the action ofCp2:
−1
4
d2F
dt2 at− 1 2
#
tanh2t 1
tanh2t
$
dF dtat
%#
−1
3
1
4tanh
22t$s2−1
&
Fat
λ2Fat;
4.14
2the differential equation obtained from the action ofCp3:
− 1
12s
d2F
dt2 at− 1 6
#
tanh2t 1
tanh2t
$
sdF dtat
%#
−2
27
1
12tanh
22t
$
s3−1
3s
&
Fat
λ3Fat.
Here,
λ2− 1 3
ν12−ν1ν2ν22
,
λ3−1
272ν1−ν22ν2−ν1ν1ν2
4.16
are the eigenvalues of the Capelli elements on principal series representations. Equations4.15−4.14×1/3sgive
%
1
27s
31
3λ2s−λ3
&
Fat 0. 4.17
Therefore, ifFatis not identically zero, we have
s3−3ν21−ν1ν2ν22
s 2ν1−ν22ν2−ν1ν1ν2 0. 4.18
By solving this equation, we have
s2ν1−ν2, 2ν2−ν1, −ν1−ν2. 4.19
Therefore, one of the necessary conditions of the existence of nontrivial Shintani functions is
that the parametersis one of the above three values. Now, we assume thatssatisfies this
con-dition. We putxtanh2t, F'x:Fatin4.14. Then, we have
−4x1−x2d
2F'
dx2x−41−x
2dF'
dxx
%#
−1
3
1
4x
$
s2−1−λ2
& '
Fx 0. 4.20
Next, we putF'x 1−xμG0x μ∈C. Then,G0satisfies
−4x1−xμ2d
2G
0
dx2 x
−48μ4x1−xμ1dG0
dx x
%
−4μμ−1x4μ−4μx
#
−1
3
1
4x
$
s2−1−λ2
&
1−xμG0x 0.
4.21
We want to divide the left-hand side of4.21by1−xμ1. To do this, we takeμ∈C so that
μsatisfies
−4μ24μ− 1
12s
2−1−λ
20. 4.22
The value ofμ∈C is as follows.
1Ifs2ν1−ν2, μ 2±ν2/4. So we takeμ 2ν2/4.
2Ifs2ν2−ν1, μ 2±ν1/4. So we takeμ 2ν1/4.
For thisμ, the left-hand side of4.21is divided by1−xμ1, and the equation becomes
xx−1d
2G
0
dx2 x
−112μxdG0
dx
#
μ2− 1
16s
2
$
G0x 0. 4.23
This is the Gaussian hypergeometric differential equation. Note that the Shintani function
FatonAis regular at the origin⇔x0. Therefore,G0x 1−x−μF'xis also regular
atx0. Equation4.23has just one solution which is regular aroundx0up to constant
multiples, and it is given by
G0x 2F1
α, β; 1;x, 4.24
where2F1is the Gaussian hypergeometric function andα, β∈C are defined by
1αβ12μ,
αβμ2− 1
16s
2. 4.25
Explicitly, by solving these equations,α, βare given as follows.
1In case ofs2ν1−ν2, α, β ν11/2,−ν1ν21/2.
2In case ofs2ν2−ν1, α, β ν21/2,−ν2ν11/2.
3In case ofs−ν1−ν2, α, β ν11/2,−ν21/2.
Finally, we consider the three conditions in3.8. Condition1is equivalent to−1k
Fat Fat. Condition2always holds. Condition3is equivalent to−1kF1 F1,
which holds if condition1is satisfied. Summing up, we have the following theorem.
Theorem 4.8. Letηηs,kbe the unitary character ofHdefined by3.1. Then, the necessary
condi-tion of the existence of the nontrivial elements inC∞η,1Lπσ
0,νis that
k0, s2ν1−ν2, 2ν2−ν1, −ν1−ν2. 4.26
Suppose that this condition is satisfied and nontrivial Shintani functions exist. If one puts x
tanh22t,Fat F'x∈Cη,∞1Lπσ0,ν|Ais given as follows (up to constant multiples).
1In case ofs2ν1−ν2, one has
'
Fx 1−x2ν2/4
2F1
#
ν11
2 ,
−ν1ν21
2 ; 1;x
$
. 4.27
2In case ofs2ν2−ν1, one has
'
Fx 1−x2ν1/4
2F1
#
ν21
2 ,
−ν2ν11
2 ; 1;x
$
3In case ofs−ν1−ν2, one has
'
Fx 1−x2ν1−ν2/4
2F1
#
ν11
2 ,
−ν21
2 ; 1;x
$
. 4.29
Especially, one has
dimC∞η,1Lπσ
0,ν≤1. 4.30
5. Shintani Functions Attached to the Nonspherical Principal
Series Representations
In this section, as a character ofM, we take a nontrivial characterσ σi i 1,2,3. Then,
the minimalK-type ofπσi,νis the three-dimensional representation ofKwhich is isomorphic
to the tautological representationτ2:KSO3,R→GL3,Rwhich occurs of multiplicity
one inπσi,ν|K. We takeτ2∗instead ofτ2as a minimalK-type ofπσi,ν. The representation space
ofτ2is denoted byVτ2 R3, and we takes1 t1,0,0,s2 t0,1,0,s3 t0,0,1as a
basis ofVτ2. LetΨ∈Cη,τ∞2Lπσi,ν be the Shintani function. Then,Ψis expressed by
ΨgF0
gs1G0
gs2H0
gs3. 5.1
Ψis characterized by its restriction toA. To investigateΨ|A, we construct two kinds of
dif-ferential equations. One is the Casimir equation of degree two and the other is the gradient
equationor the Dirac-Schmidt equation.
5.1. The Casimir Equation
Firstly, we construct the Casimir equation of degree two. Since the Capelli elementCp2acts
on the representation space of the principal series representationπσi,νas a scalar operatorλ2
-multipleand the space of Shintani functionsC∞η,τ2Lπ
σi,ν is the image ofgC, K
-homomor-phism of πσi,ν, Cp2 acts on this space as the same scalar operator. Since for all Ψat ∈
C∞η,τ2L|A is annihilated by the action of Ada−1t Y2Ug, Adat−1Y3Ugand the actions of
Ada−1t Y1and Ada−1t Y4onFare the samethe multiplication bys, we may regardCp2as
the element inUgmodP, wherePis a subalgebra ofUgdefined by
PAda−1
t
Y1−Ad
a−1t Y4
Ug Ada−1t Y2Ug Ad
a−1t Y3Ug. 5.2
By using Lemmas4.4and4.5, we can rewriteCp2inProposition 4.1as follows.
Lemma 5.1. One has
Cp2≡ −1
4X
2
13−
1 2
#
tanh2t 1
tanh2t
$
X13
#
1
4tanh
22t−1
3
$
Ada−1t Y1
2
−1
tanh2t
2 cosh2tAd
a−1t Y1K13
#
−1
4tanh
22t 1
4
$
#−1
4tanh
2t1
4
$
K212
#
1
4−
1
4tanh
2t$K2
23
·modAda−1t Y1−Ad
a−1t Y4
Ug Ada−1t Y2Ug Ad
a−1t Y3Ug
.
5.3
By using this lemma, the action ofCp2 onΨat F0ats1 G0ats2H0ats3 ∈
C∞η,τ2L|Acan be computed easily. We have
Cp2Ψat Fats1Gats2Hats3, 5.4
where
Fat −
1 4
d2F 0
dt2 at− 1 2
#
tanh2t 1
tanh2t
$
dF0
dt at
%#
1
4tanh
22t− 1
3
$
s21
4tanh
22t 1
4 tanh2t−
3 2
&
F0at− tanh2t
2 cosh2tsH0at,
Gat −1
4
d2G 0
dt2 at− 1 2
#
tanh2t 1
tanh2t
$
dG0
dt at
%#
1
4tanh
2t−1
3
$
s21
4tanh
2t 1
4 tanh2t −
3 2
&
G0at,
Hat −
1 4
d2H 0
dt2 at− 1 2
#
tanh2t 1
tanh2t
$
dH0
dt at
%#
1
4tanh
22t− 1
3
$
s21
4tanh
22t 1
4tanh
2t−3
2
&
H0at tanh2t
2 cosh2tsF0at.
5.5
SinceΨatsatisfiesCp2Ψat λ2Ψat, we have the following three differential equations.
Theorem 5.2. ForΨat F0ats1G0ats2H0ats3∈Cη,τ∞2Lπσi,ν|A, the functionsF0,G0,
H0satisfy the following equations:
−1
4
d2F 0
dt2 at− 1 2
#
tanh2t 1
tanh2t
$
dF0
dt at
%#
1
4tanh
22t−1
3
$
s21
4tanh
22t 1
4 tanh2t−
3 2
&
F0at
− tanh2t
2 cosh2tsH0at λ2F0at,
−1 4
d2G 0
dt2 at− 1 2
#
tanh2t 1
tanh2t
$
dG0
dt at
%#
1
4tanh
2t−1
3
$
s21
4tanh
2t 1
4 tanh2t −
3 2
&
G0at λ2G0at,
5.7
−1
4
d2H0
dt2 at− 1 2
#
tanh2t 1
tanh2t
$
dH0
dt at
%#
1
4tanh
22t−1
3
$
s21
4tanh
22t 1
4tanh
2t− 3
2
&
H0at
tanh2t
2 cosh2tsF0at λ2H0at.
5.8
5.2. The Gradient Equation
For the spherical functionΨg ∈ Cη,τ∞2L, we define the right gradient operator∇R as
fol-lows.
Definition 5.3. For the orthonormal basis{Xi}5i1ofp, the right gradient operator∇Ris defined by
∇RΨg:5
i1
RXiΨ⊗X∗i. 5.9
Here,Xi∗is the dual basis ofXiwith respect to the inner productX, Y∈p × p → TrXY∈
C.
The set {H12, H23, X12, X23, X13} becomes the orthonormal basis of p, and {1/3
2H12 H23,1/3H12 2H23,1/2X12,1/2X23,1/2X13}is its dual basis. Therefore,
the gradient operator∇Ris explicitly given by
∇RΨg 1
3RH12Ψ⊗2H12H23
1
3RH23Ψ⊗H122H23
1 2
i<j
RXij
Ψ⊗Xij.
5.10
We rewrite this by using the basis ofpC.
Claim 1. We define five elementswi 0≤i≤4inpCby
w0:−2
H23− √
−1X23
, w4:−2
H23 √
−1X23
,
w2: 2
32H12H23,
w1:X13 √
−1X12, w3:−X13
√
−1X12.
5.11
Table 2
w0 w1 w2 w3 w4
s1 0 −
1 4s3
√
−1s2 −
1 3s1
1 4s3−
√
−1s2 0
s2 1 2s2−
√
−1s3 −
√
−1
4 s1
1
6s2 −
√
−1
4 s1
1 2s2
√
−1s3
s3 − 1 2s3
√
−1s2 −
1 4s1
1 6s3
1 4s1
1 2−s3
√
−1s2
With this basis, the gradient operator∇Ris rewritten as
∇RΨ 1
16Rw4Ψ⊗w0
1
16Rw0Ψ⊗w4−
1
4Rw3Ψ⊗w1
− 1
4Rw1Ψ⊗w3
3
8Rw2Ψ⊗w2
1
4
%
1
4Rw4Ψ⊗w0
1
4Rw0Ψ⊗w4−Rw3Ψ⊗w1
−Rw1Ψ⊗w33
2Rw2Ψ⊗w2
&
.
5.12
The Lie algebrapCbecomes the representation space of the adjoint action ofK. We denote
this representation byτ4, W4. By the Clebsch-Gordan theorem,τ2⊗τ4 has the irreducible
decomposition
τ2⊗τ4∼τ2⊕τ4⊕τ6. 5.13
Here, eachτnis then1-dimensional irreducible representation ofK. In this decomposition,
the projector ofK-modules
pr2:τ2⊗τ4τ2, si⊗wj−→pr2
si⊗wj
, 5.14
is described as inTable 2.
∇RΨis aτ
2⊗τ2⊗pC-valued function. Then, by mappingsi⊗wktopr2si⊗wk, we
have aK-homomorphism
pr'2◦ ∇R:C∞η,τ2L−→C
∞
η,τ2L. 5.15
Since the minimal K-type τ2∗ occurs of multiplicity one, pr'2 ◦ ∇R is a map of constant
multiple. To compute the action of the gradient operatorpr'2◦ ∇Ron the space of the Shintani
functionsC∞η,τ2Lπ
σi,ν, we have to decompose wi i 0,1,2,3,4along the decomposition
Lemma 5.4. One has
w0
2 sinh2t
cosh2tAd
a−1t Y1−2Ad
a−1t Y4tanh2tK13
2√−1
sinhtAd
a−1t Y3
2√−1
tanhtK12,
w4 2 sinh
2t
cosh2tAd
a−1t Y1−2Ad
a−1t Y4tanh2tK13−2 √
−1
sinhtAd
a−1t Y3−2 √
−1
tanhtK12,
w2
2cosh2t1
3 cosh2t Ad
a−1t Y1− 2
3Ad
a−1t Y4−tanh2tK13,
w1X13 √
−1
coshtAd
a−1t Y2− √
−1 tanhtK23,
w3−X13 √
−1
coshtAd
a−1t Y2− √
−1 tanhtK23.
5.16
By using Lemmas3.4and4.3and the table of projections, we can compute the action
of the gradient operator. ForΨat F0ats1G0ats2H0ats3∈C∞η,τ2L|A, we have
pr'2◦∇RΨat Fats1Gats2Hats3, 5.17
where
Fat − #
1
2 cosh2t−
1 6
$
sF0at−
1 2
dH0
dt at−
1
2tanh2t tanhtH0at,
Gat −
1
3sG0at,
Hat #
1
2 cosh2t
1 6
$
sH0at−
1 2
dF0
dt at−
1 2
#
tanh2t 1
tanht
$
F0at.
5.18
On the other hand, the eigenvalue of the gradient operator on the spherical functions of the
principal series representationπσi,ν depends on the choice ofσi, denoted byλσi i 1,2,3.
These values are computed in11and they are as follows:
λσ1−
1
32ν1−ν2, λσ2−
1
32ν2−ν1, λσ3
1
3ν1ν2. 5.19
Therefore, sinceΨat F0ats1G0ats2H0ats3∈C∞η,τ2Lπσi,ν|Asatisfiespr'2◦∇
RΨa t
λσiΨat, we have the following three differential equations.
Theorem 5.5. ForΨat F0ats1G0ats2H0ats3∈Cη,τ∞2Lπσi,ν|A, one has
−
#
1
2 cosh2t−
1 6
$
sF0at− 1
2
dH0
dt at−
1
−1
3sG0at λσiG0at, 5.21
#
1
2 cosh2t
1 6
$
sH0at−
1 2
dF0
dt at−
1 2
#
tanh2t 1
tanht
$
F0at λσiH0at. 5.22
We consider the case ofσσ1. We haveλσ1 −1/32ν1−ν2. By5.21, ifs /−3λσ1
2ν1−ν2, we haveG0at≡0. Suppose thats2ν1−ν2. We putxtanh22t,G0at G1x
in5.21. Then, the equation becomes
−4x21−x2d2G1
dx2 x−4x1−x
2dG1
dx x
%#
1
4x
2−1
3x
$
s2−2λ
2x1
&
G1x 0.
5.23
We putG1x xα1−xβG'1x α, β∈C. Then, the equation becomes
−4x21−x2d
2G'
1
dx2 x 4x1−x
−2α−12α2β1xdG'1
dx x
%
−4αα−11−x2−4ββ−1x28αβx1−x−4α1−x24βx1−x
#1
4x
2−1
3x
$
s2−2λ2x1
& '
G1x 0.
5.24
We want to divide the left-hand side of5.24byx1−x. For this purpose,α, β ∈ C must
satisfy
−4α210,
−4ββ−1− 1
12s
2−λ
2−10.
5.25
The solutions areα ±1/2, β 2±ν2/4. We chooseα 1/2, β 2ν2/4. Then the
left-hand side of5.24can be divided byx1−x, and the equation becomes
xx−1d
2G'
1
dx2 x
−23ν2
2
xdG'1 dx x
1 4
−ν2
1ν1ν22ν24 'G1x 0. 5.26
This is a Gaussian hypergeometric differential equation, and its regular solution is given by
'
G1x 2F1
#
1
2ν11,
1
2ν2−
1
2ν11; 2;x
$
5.27
up to constant multiples.The other solutions are not regular aroundx0, since they contain
logx.Therefore, we have
G0at G0x Cx1/21−x2ν2/42F1
#
1
2ν11,
1
2ν2−
1
2ν11; 2;x
$
whereCis a constant number andxtanh22t. Next, we consider the equations satisfied by
F0andH0. From5.20and5.22, we have
dF0
dt at
#
1
cosh2t
1 3
$
sH0at− #
tanh2t 1
tanht
$
F0at−2λσ1H0at,
dH0
dt at −
#
1
cosh2t−
1 3
$
sF0at−tanh2t tanhtH0at−2λσ1F0at.
5.29
By differentiating both sides of5.29byt, we have
d2F0
dt2 at
3 tanh22t 2
tan h2t
2 tanh2t
tan ht −3−
4
3sλσ1
−
1
cosh22t−
1 9
s24λ2σ1
F0at
%
−
#
4 tanh2t
cosh2t
2
3tanh2t
2
3 tanh2t
2
sinh2t
$
s
4
#
tanh2t 1
tanh2t
$
λσ1 &
H0at,
d2H0
dt2 at
3 tanh22t 2 tanh2t2 tanhttanh2t−3
− 4
3sλσ1−
1
cosh22t−
1 9
s24λ2σ1
H0at
%#
4 tanh2t
cosh2t −
2
3tanh2t−
2
3 tanh2t
2
sinh2t
$
s
4
#
tan h2t 1
tanh2t
$
λσ1 &
F0at.
5.30
By inserting5.29and5.30into the Casimir equation5.6,5.8to eliminate the differential
terms, we have
#
1
3sλσ1−λ
2
σ1−
1
9s
2−λ
2
$
F0at 0, #
1
3sλσ1−λ
2
σ1−
1
9s
2−λ
2
$
H0at 0.
5.31
Therefore, if the parameter s ∈ C satisfies 1/3sλσ1 −λ2σ1 −1/9s
2 −λ
2/0, then F0at,
H0at ≡ 0. Suppose that1/3sλσ1 −λ
2
σ1−1/9s
2−λ
λ2−1/3ν12−ν1ν2ν22, we have
s2 2ν1−ν2s
ν12−ν1ν2−2ν22
0. 5.32
Therefore, we have
s−ν1−ν2, −ν12ν2. 5.33
That is,5.33is the necessary condition of the existence of nontrivialF0at, H0at. We put
F0at cosh−1/22tsinht−1F't,
H0at cosh−1/22tcosht−1H(t
5.34
and insert these into5.20,5.22. Then we have
−
#
1
2 cosh2t−
1 6
$
s 1
tanhtF't−
1 2
dH(
dt t λσ1
1
tanhtF't,
#
1
2 cosh2t
1 6
$
stanhtH(t−1
2
dF'
dtt λσ1tanhtH(t.
5.35
Next, we put tanh2tu, F't F'1u, H(t H(1u. Then, the above equations become
−
#
1−u
21u −
1 6
$
sF'1u−u1−u
dH(1
du u λσ1F'1u, 5.36
−1−udF'1
duu
#
1−u
21u
1 6
$
sH(1u λσ1H(1u. 5.37
For a while, we consider the case ofs0, that is,ηη0,kis a signature sgnkofH, where
sgnk
⎛ ⎜ ⎜ ⎝
⎛ ⎜ ⎜ ⎝
h11 h12 0
h21 h22 0
0 0 h1
⎞ ⎟ ⎟ ⎠
⎞ ⎟ ⎟
⎠detH1k|detH1|−k, H1
h11 h12
h21 h22
. 5.38
Then, by combining5.36,5.37, we have
u1−u2λσ1 d2F'
1
du2 u−u1−uλσ1 dF'1
duu−λ
3
σ1F'1u 0. 5.39
Suppose thatλσ1/0. Then, the equation becomes
u1−u2d
2F'
1
du2 u−u1−u
dF'1
duu−λ
2
We putF'1u 1−uλσ1F2u. Then,F2usatisfies
u1−ud
2F
2
du2 u−2λσ11u dF2
duu−λ
2
σ1F2u 0. 5.41
Equation 5.41 is the Gaussian hypergeometric differential equation. Now, since F2u
1−u−λσ1cosh1/22tsinhtF0atandF0atis regular atat1 ⇔t0⇔u0,F2umust
be regular atu0. The regular solution of5.41is given by
F2u u2F1λσ11, λσ11; 2;u 5.42
up to constant multiples. Therefore, we have
'
F1u C·u1−uλσ12F1λσ11, λσ11; 2;u C: constant. 5.43
Similarly, we have
(
H1u C·1−uλσ12F1λσ11, λσ1; 1;u
C: constant. 5.44
We want to find the relation betweenCandC. By expandingF'1uandH(1uaroundu0,
we have
'
F1u C·
u λσ 2 11
2 u
2u3P
1u
,
(
H1u C·
1λ2σ1uu2P2u
,
5.45
whereP1uandP2uare the analytic functions aroundu0. By inserting these into5.36,
we have
−3u1−u2Cλ2σ1uP3u
31uλσ1C
uu2P4u
, 5.46
whereP3uandP4uare also the analytic functions aroundu0. By comparing the
coef-ficients ofuof both sides, we have
C−λσ1C. 5.47
Summing up,F0atandH0atare given by
F0at −Cλσ1cosh
−1/22tsinht−1u1−uλσ12F1λσ
11, λσ11; 2;u,
H0at Ccosh−1/22tcosht−11−uλσ12F1λσ
11, λσ1; 1;u.
C: some constant,utanh2t. In our computation, we assumed thatλσ1/0, but the
result above holds without this assumption.
Next, we consider conditions1,2, and3in3.8. ForΨ tF0, G0, H0∈C∞sgnk,τ2
L, condition1is equivalent to
−1k· t−F
0at, G0at,−H0at tF0at, G0at, H0at. 5.49
This is equivalent to
k0⇒F0≡0, H0≡0,
k1⇒G0≡0.
5.50
Condition2is equivalent to
t−F
0at,−G0at, H0at tF0at, G0at, H0at. 5.51
The solutions we have always satisfy this condition. Condition3is equivalent to
tcosθF
01 sinθG01,−sinθF01 cosθG01, H01 tF01, G01, H01,
tcosθF
01 sinθG01,sinθF01−cosθG01,−H01 −1k· tF01, G01, H01.
5.52
for allθ∈R. SinceF01 G01 0,5.52are equivalent to
k0⇒H0≡0. 5.53
But this condition holds if condition5.50is satisfied. We have obtained a result about the
Shintani functions attached to the nonspherical principal series representationπσ1,ν. Note that
since the transformν1 →ν2, ν2 →ν1does not change the eigenvalue of Casimir operatorλ2
and changes the eigenvalue of gradient operatorλσ1toλσ2, this transform gives the result in
case ofσ σ2. Similarly, the transformν1 → −ν1, ν2 → −ν1ν2gives the result in case of
σσ3. Summing up these results, we have the following theorem.
Theorem 5.6. Letηsgnk k∈ {0,1}be a signature ofHdefined by5.38andτ2a
three-dimen-sional tautological representation ofK, and letΨ tF0, G0, H0 ∈ C∞sgnk,τ2Lπσ1,ν be a Shintani
function corresponding to the nonspherical principal series representationπσ1,νofG. Then, the
restric-tion ofΨtoAis given as follows.
1In case ofk0,
bif 2ν1−ν20, one has
⎛ ⎜ ⎜ ⎝
F0at
G0at
H0at ⎞ ⎟ ⎟ ⎠C·
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0
x1/21−x2ν2/4
2F1
#
1
2ν11,
1
2ν2−
1
2ν11,; 2;x
$
0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠.
xtanh22t, C: some constant.
5.54
2In case ofk1,
aif−ν1−ν2−ν12ν2/0,Ψ≡0;
bif−ν1−ν2−ν12ν2 0, one has
⎛ ⎜ ⎜ ⎝
F0at
G0at
H0at ⎞ ⎟ ⎟ ⎠C·
⎛ ⎜ ⎜ ⎝
−λσ1cosh
−1/22tsinht−1u1−uλσ12F
1λσ11, λσ11; 2;u
0
cosh−1/22tcosht−11−uλσ12F1λσ
11, λσ1; 1;u ⎞ ⎟ ⎟ ⎠.
utanh2t, C: some constant.
5.55
Especially, in any case, one has
dimC∞sgn
k,τ2Lπσ1,ν ≤1. 5.56
The transformν1 →ν2, ν2 →ν1gives the result in case ofσσ2and the transform
ν1→ −ν1, ν2 → −ν1ν2gives the result in case ofσσ3.
Next, we compute the Shintani functionsΨ∈Cη,τ∞2Lπσ
1,νfor general unitary character
ηηs,kunder the assumption that 1, ν1, ν2are linearly independent overQ. We have already
known that the necessary condition of the existence of non zeroΨis that the parametersis
one of 2ν1−ν2,2ν2−ν1, or−ν1−ν2, and we have already solved the differential equations in
case ofs2ν1−ν2. Hereafter, we suppose that the parametersis either 2ν2−ν1or−ν1−ν2.
We putw 1−u/1u,F'2w F'1u,H(2w H(1uin5.36,5.37. Then we have
−#w
2 −
1 6
$
sF'2w w1−wd
(
H2
dw w λσ1F'2w,
w1wdF'2
dww
#
w
2
1 6
$
sH(2w λσ1H(2w.
5.57
We put
'
F2w ∞
n0
anwnα, H(2w ∞
n0
