Shintani Functions on 𝑆𝐿(3,𝐑)

Abstract

We investigate the Shintani functions attached to the spherical and nonspherical principal series representations of 𝑆𝐿(3,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 for 𝑝-adic linear group 𝐺𝐿(𝑛,𝑘), where 𝑘 is a finite extension of the 𝑝-adic field 𝐐_𝑝 [1]. He defined some “Whittaker function” on 𝐺𝐿(𝑛,𝑘) and obtained the explicit formulas of them. Moreover, he proved the uniqueness of his function. Later, a more detail study of Shintani functions for 𝐺𝐿(𝑛) was done by Murase and Sugano [2] (see also [3]). They obtained new kinds of integral formulas for the 𝐿-functions in terms of the global Shintani functions 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 on 𝐺𝐿(2,𝐑) [4] and 𝐺𝐿(2,𝐂) [5], Tsuzuki on 𝑆𝑈(1,1) [6] and 𝑈(𝑛,1) [7], and Moriyama on 𝑆𝑝(2,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 on 𝐺=𝑆𝐿(3,𝐑), attached to the principal series representations of 𝐺. Now we explain the definition of the Shintani functions on 𝐺. We take𝐻𝐻=¹00ℎ₂∈𝐺∣𝐻₁,ℎ₂∈𝐺𝐿(2,𝐑)×𝐺𝐿(1,𝐑) (1.1) as a subgroup of 𝐺 and take 𝐾=𝑆𝑂(3) as a maximal compact subgroup of 𝐺. Let 𝜋 be an arbitrary irreducible unitary representation of 𝐺 and (𝜂,𝑉_𝜂) an irreducible unitary representation of 𝐻, and let 𝐶^∞_𝜂(𝐻⧵𝐺) be the space of smooth functions 𝐹∶𝐺→𝑉_𝜂 satisfying 𝐹(ℎ𝑔)=𝜂(ℎ)𝐹(𝑔)((ℎ,𝑔)∈𝐻×𝐺). We consider the intertwining space 𝐼_𝜂,𝜋=Hom_{(𝔤𝐂,𝐾)}(𝜋,𝐶^∞_𝜂(𝐻⧵𝐺)) and its restriction𝐼_𝜂,𝜋⟶Hom_𝐾𝜏,𝐶^∞_𝜂(𝐻⧵𝐺)≅𝐶_𝜂,𝜏∗(𝐿) (1.2) to the minimal 𝐾-type (𝜏,𝑉_𝜏) of 𝜋, where (𝜏^∗,𝑉_𝜏∗) is the contragredient representation of (𝜏,𝑉_𝜏), 𝐿∶=𝐻⧵𝐺/𝐾 and 𝐶^∞_𝜂,𝜏∗(𝐿) is the space of smooth functions 𝐹∶𝐺→𝑉_𝜂⊗𝑉_𝜏∗ satisfying 𝐹(ℎ𝑔𝑘)=(𝜂(ℎ)⊗𝜏^∗(𝑘⁻¹))𝐹(𝑔) for (ℎ,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 that 𝜋 is the irreducible unitary principal series representation of 𝐺 and 𝜂 is the unitary character of 𝐻. The study of Shintani functions for the general unitary representation 𝜂 of 𝐻 is a further problem.

In Section 4, we investigate the Shintani functions attached to the spherical (or class one) principal series representations. These representations have unique 𝐾-fixed vector, and hence the minimal 𝐾-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 1 (Theorem 4.8).

On the other hand, in Section 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 in Section 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 characters 𝑍(𝔤)→𝐂. We cannot distinguish them only by the elements of 𝑍(𝔤). This is the reason we need the gradient operator which has distinct eigenvalues for different nonspherical principal series. By combining these equations, we obtain the explicit formulas 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 1 (Theorem 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 also [10], in the case of 𝑈(𝑛,1)). Furthermore, the author thinks these results are interesting themselves in view of the harmonic analysis on homogeneous spaces.

2. Preliminaries

2.1. Groups and Algebras

Let 𝐺 be the real reductive Lie group 𝑆𝐿(3,𝐑) and 𝔤=𝔰𝔩(3,𝐑) its Lie algebra. The Cartan involution 𝜃∶𝐺→𝐺 is defined by 𝜃(𝑔)=(^𝑡𝑔)⁻¹(𝑔∈𝐺), and its differential 𝑑𝜃∶𝔤→𝔤 is given by 𝑑𝜃(𝑋)=−^𝑡𝑋(𝑋∈𝔤), where 𝑡 means the transposition of matrices. Then the fixed subgroup of 𝜃 in 𝐺 is equal to 𝐾=𝑆𝑂(3), which is the maximal compact subgroup of 𝐺. Next, we define another involutive automorphism 𝜎 of 𝐺 by 𝜎(𝑔)=𝐽𝑔𝐽(𝑔∈𝐺), where 𝐽=diag(−1,−1,1). Its differential 𝑑𝜎∶𝔤→𝔤 is given by 𝑑𝜎(𝑋)=𝐽𝑋𝐽(𝑋∈𝔤). The fixed subgroup 𝐻 of 𝜎 in 𝐺 is isomorphic to 𝐺𝐿(2,𝐑), that is, 𝑎𝐻=¹¹𝑎₁₂0𝑎₂₁𝑎₂₂000𝑎₃₃∈𝐺≅𝐺𝐿(2,𝐑). (2.1) We define +1 and −1 eigenspaces of 𝑑𝜃, 𝑑𝜎 in 𝔤 by {(𝑋)}{(𝑋)},{(𝑋)}{(𝑋)}.𝔨=𝑋∈𝔤∣𝑑𝜃=𝑋,𝔭=𝑋∈𝔤∣𝑑𝜃=−𝑋𝔥=𝑋∈𝔤∣𝑑𝜎=𝑋,𝔮=𝑋∈𝔤∣𝑑𝜎=−𝑋 (2.2) Then, 𝔨,𝔥 are the Lie algebras of 𝐾,𝐻, respectively. We have 𝔤=𝔨⊕𝔭=𝔥⊕𝔮. Let 𝐸_𝑖𝑗∈𝑀(3,𝐑) be the matrix whose (𝑖,𝑗)-component is 1 and the other components are 0 (1≤𝑖,𝑗≤3). For 1≤𝑖<𝑗≤3, we put 𝐾_𝑖𝑗=𝐸_𝑖𝑗−𝐸_𝑗𝑖,𝑋_𝑖𝑗=𝐸_𝑖𝑗+𝐸_𝑗𝑖,𝐻_𝑖𝑗=𝐸_𝑖𝑖−𝐸_𝑗𝑗. Then, we have𝔨=_𝑖<𝑗𝐑𝐾_𝑖𝑗,𝔭=_𝑖<𝑗𝐑𝑋_𝑖𝑗⊕𝐑𝐻₁₂⊕𝐑𝐻₂₃. (2.3) Next, we take𝑌₁=10000000−1,𝑌₂=010100000,𝑌₃=0−10100000,𝑌₄=00001000−1 (2.4) as a basis of 𝔥. We have 𝔭∩𝔮=𝐑𝑋₁₃⊕𝐑𝑋₂₃, and we take 𝔞=𝐑𝑋₁₃ as a maximal Abelian subspace of 𝔭∩𝔮. We define a subgroup 𝐴 of 𝐺 by𝐴=exp(𝔞)=𝑎_𝑡∶=.cosh𝑡0sinh𝑡010sinh𝑡0cosh𝑡∣𝑡∈𝐑 (2.5) Then, 𝐺 has a decomposition 𝐺=𝐻𝐴𝐾. Throughout this paper, we put 𝐿∶=𝐻⧵𝐺/𝐾.

For a Lie algebra 𝔩, we denote its complexification by 𝔩_𝐂, that is, 𝔩_𝐂=𝔩⊗_𝐑𝐂.

2.2. The Principal Series Representations

As a representation of 𝐺, we take the principal series representation defined as follows. Let 𝑃₀ be a minimal parabolic subgroup of 𝐺 given by the upper triangular matrices in 𝐺 and 𝑃₀=𝑀𝐴_𝑃0𝑁 the Langlands decomposition of 𝑃₀ with𝐴_𝑃0=diag𝑎₁,𝑎₂,𝑎₃∣𝑎_𝑖>0,𝑎₁𝑎₂𝑎₃=1,𝑀=𝐾∩diagonalsin𝐺,𝑁=1𝑥₁𝑥₂01𝑥₃001∈𝐺∣𝑥₁,𝑥₂,𝑥₃∈𝐑. (2.6) To define a principal series representation with respect to the minimal parabolic subgroup 𝑃₀ of 𝐺, we firstly fix a character 𝜎 of 𝑀 and a linear form 𝜈∈𝔞^∗_𝑃0⊗_𝐑𝐂=Hom_𝐑(𝔞_𝑃0,𝐂), where 𝔞_𝑃0 is the Lie algebra of 𝐴_𝑃0. We write𝜈diag𝑡₁,𝑡₂,𝑡₃=𝜈₁𝑡₁+𝜈₂𝑡₂ (2.7) for diag(𝑡₁,𝑡₂,𝑡₃)∈𝔞_𝑃0. Then, we can define a representation 𝜎⊗𝑎^𝜈 of 𝑀𝐴_𝑃0 and extend this to 𝑃₀ by the identification 𝑃₀/𝑁≃𝑀𝐴_𝑃0, taking the trivial representation 𝟏_𝑁 as the representation of 𝑁. Then, the induced representation𝜋_𝜎,𝜈∶=𝐶^∞Ind^𝐺_𝑃0𝜎⊗𝑎^𝜈+𝜌⊗1_𝑁 (2.8) is called the principal series representation of 𝐺. Here, 𝜌 is the half sum of positive roots of (𝔤,𝔞) given by 𝑎^𝜌=𝑎²₁𝑎₂, for 𝑎=diag(𝑎₁,𝑎₂,𝑎₃)∈𝐴_𝑃0.

Concretely, the representation space is given by𝐶^∞_()𝑀,𝜎(𝐾)∶=𝑓∈𝐶^∞(𝐾)∣𝑓(𝑚𝑘)=𝜎(𝑚)𝑓(𝑘),𝑚∈𝑀,𝑘∈𝐾, (2.9) and the action of 𝐺 is defined by𝜋_𝜎,𝜈(𝑥)𝑓(𝑘)=𝑎(𝑘𝑥)^𝜈+𝜌𝑓(𝜅(𝑘𝑥))(𝑥∈𝐺,𝑘∈𝐾). (2.10) Here, for 𝑔∈𝐺,𝑔=𝑛(𝑔)𝑎(𝑔)𝜅(𝑔)(𝑛(𝑔)∈𝑁,𝑎(𝑔)∈𝐴_𝑃0,𝜅(𝑔)∈𝐾) is the Iwasawa decomposition. Throughout this paper, we assume that the representation 𝜋_𝜎,𝜈 is irreducible. Moreover, we assume that 𝜈₁,𝜈₂ are the elements of √−1𝐑. Then, this representation becomes unitary.

Next, we define characters 𝜎_𝑗(𝑗=0,1,2,3) of 𝑀 as follows. The group 𝑀 consisting of four elements is a finite Abelian group of (2,2)-type, and its elements except for the unity are given by𝑚₁=diag(1,−1,−1),𝑚₂=diag(−1,1,−1),𝑚₃=diag(−1,−1,1). (2.11) The set 𝑀 consists of 4 characters {𝜎_𝑗∣𝑗=0,1,2,3}, where 𝜎₀ is the trivial character of 𝑀 and 𝜎₁,𝜎₂,𝜎₃ are defined by Table 1.

Table 1 

The following proposition (see [11, Proposition 1.1]) gives the correspondence between the character 𝜎 of 𝑀 and the minimal 𝐾-type of the principal series representation 𝜋_𝜎,𝜈 of 𝐺.

Proposition 2.1. (1) If 𝜎 is the trivial character of 𝑀, the representation 𝜋_𝜎,𝜈 is spherical or class one. That is, it has a unique 𝐾-invariant vector in 𝐻_𝜎,𝜈.
(2) If 𝜎 is not trivial, the minimal 𝐾-type of the restriction 𝜋_𝜎,𝜈|_𝐾 to 𝐾 is a 3-dimensional representation of 𝐾, which is isomorphic to the unique standard one (𝜏₂,𝑉₂). The multiplicity of this minimal 𝐾-type is one: dim_𝐂Hom_𝐾𝜏₂,𝐻_𝜎,𝜈=1. (2.12)

3. The Space of Shintani Functions

3.1. The Definition of Shintani Functions

As a representation of 𝐻, we take the unitary character 𝜂=𝜂_𝑠,𝑘∶𝐻→𝐂^×(𝑠∈√−1𝐑,𝑘∈{0,1}) defined by 𝜂ℎ₁₁ℎ₁₂0ℎ₂₁ℎ₂₂000ℎ₁=det𝐻₁^𝑘det𝐻₁^𝑠−𝑘,𝐻₁=ℎ₁₁ℎ₁₂ℎ₂₁ℎ₂₂∈𝐺𝐿(2,𝐑). (3.1) Let 𝜂=𝜂_𝑠,𝑘 be the unitary character of 𝐻 defined previously. We consider the induced representation 𝐶^∞Ind^𝐺_𝐻(𝜂) with the representation space 𝐶^∞_𝜂(𝐻⧵𝐺)=𝐹∈𝐶^∞(𝐺)∣𝐹ℎ𝑔=𝜂(ℎ)𝐹𝑔,∀ℎ∈𝐻,∀𝑔∈𝐺. (3.2)  𝐺 acts on this space by right translation. Let 𝜋_𝜎,𝜈 be the principal series representation of 𝐺. We consider the intertwining space 𝐼_{𝜂,𝜋𝜎,𝜈}∶=Hom_{(𝔤𝐂,𝐾)}𝜋_𝜎,𝜈,𝐶^∞_𝜂(𝐻⧵𝐺). (3.3) We denote its image by 𝑆_{𝜂,𝜋𝜎,𝜈}, that is, 𝑆_{𝜂,𝜋𝜎,𝜈}∶=_{𝑇∈𝐼𝜂,𝜋𝜎,𝜈}Image(𝑇). (3.4) We call the element of 𝑆_{𝜂,𝜋𝜎,𝜈} the Shintani function of type (𝜂,𝜋_𝜎,𝜈). Let (𝜏,𝑉_𝜏) be the 𝐾-type of the principal series representation 𝜋_𝜎,𝜈, and let 𝜄∶𝜏→𝜋_𝜎,𝜈 be the 𝐾-embedding of 𝜏 and 𝜄^∗ the pullback via 𝜄. Then, the map 𝜄^∗∶𝐼_{𝜂,𝜋𝜎,𝜈}⟶Hom_𝐾𝜏,𝐶^∞_𝜂(𝐻⧵𝐺)≅𝐶^∞_𝜂,𝜏∗(𝐿) (3.5) gives the restriction of 𝑇∈𝑆_{𝜂,𝜋𝜎,𝜈} to 𝜏, where 𝜏^∗ is the contragredient representation of 𝜏 and the space 𝐶^∞_𝜂,𝜏∗(𝐿) is defined by 𝐶^∞_𝜂,𝜏∗(𝐿)=𝐹∶𝐺⟶𝑉_𝜏∗∣𝐹ℎ𝑔𝑘⁻¹=𝜂(ℎ)𝜏^∗(𝑘)𝐹𝑔,∀ℎ,𝑔,𝑘∈𝐻×𝐺×𝐾. (3.6) We denote the image of 𝐼_{𝜂,𝜋𝜎,𝜈} in 𝐶^∞_𝜂,𝜏∗(𝐿) by 𝐶^∞_𝜂,𝜏∗(𝐿)_{𝜋𝜎,𝜈}, and the element of this space is called the Shintani function of type (𝜋_𝜎,𝜈,𝜂,𝜏).

3.2. The 𝐴-Radial Part

Because 𝐺 has the decomposition 𝐺=𝐻𝐴𝐾, the element of 𝐶^∞_𝜂,𝜏(𝐿) is characterized by its restriction to 𝐴. We denote the centralizer and the normalizer of 𝐴 in 𝐾∩𝐻 by 𝑍_𝐾∩𝐻(𝐴), 𝑁_𝐾∩𝐻(𝐴), respectively. It is easy to verify that 𝐾∩𝐻, 𝑍_𝐾∩𝐻(𝐴), and 𝑁_𝐾∩𝐻(𝐴) are given as follows.

Lemma 3.1. One has 𝐾∩𝐻=cos𝜃sin𝜃0−sin𝜃cos𝜃0001∣𝜃∈𝐑cos𝜃sin𝜃0sin𝜃−cos𝜃000,𝑍−1∣𝜃∈𝐑_𝐾∩𝐻(𝐴)=,𝑁𝐼,−10001000−1_𝐾∩𝐻(𝐴)=𝐼,,,.−1000−10001−10001000−11000−1000−1 (3.7) Here, 𝐼 is the unit element of 𝐺𝐿(3,𝐑).

Let 𝑤₀∶=diag(−1,−1,1). Then, 𝑤₀𝑍_𝐾∩𝐻(𝐴) is the unique nontrivial element of 𝑊=𝑁_𝐾∩𝐻(𝐴)/𝑍_𝐾∩𝐻(𝐴). Let 𝜂∶𝐻→𝐂 be the unitary character of 𝐻 and (𝜏,𝑉_𝜏) the finite-dimensional representation of 𝐾. We denote by 𝐶^∞_𝑊(𝐴;𝜂,𝜏) the space of smooth functions 𝐹∶𝐴→𝑉_𝜏 satisfying the following conditions: (1)𝜂(𝑚)𝜏(𝑚)𝐹(𝑎)=𝐹(𝑎)∀𝑚∈𝑍_𝐾∩𝐻(𝐴),(2)𝜂𝑤,∀𝑎∈𝐴₀𝜏𝑤₀𝐹(𝑎)=𝐹𝑎⁻¹(),(3)𝜂(𝑙)𝜏(𝑙)𝐹(1)(1).∀𝑎∈𝐴=𝐹)(∀𝑙∈𝐾∩𝐻 (3.8) The following lemma is proved by Flensted and Jensen (see [12, Theorem 4.1]).

Lemma 3.2. The restriction to 𝐴 gives the following isomorphism: 𝐶^∞_𝜂,𝜏(𝐿)≅𝐶^∞_𝑊𝐴;𝜂,𝜏. (3.9)

Through this isomorphism, we denote the image of 𝐶^∞_𝜂,𝜏(𝐿)_{𝜋𝜎,𝜈} in 𝐶^∞_𝑊(𝐴;𝜂,𝜏) by 𝐶^∞_𝑊(𝐴;𝜂,𝜏)_{𝜋𝜎,𝜈}. This is our target space in this paper. The following two lemmas are obvious.

Lemma 3.3. For 𝑡≠0, one has 𝔤=Ad𝑎_𝑡⁻¹𝔥⊕𝔞⊕𝔨. (3.10)

Lemma 3.4. Let 𝐹∈𝐶^∞_𝜂,𝜏(𝐿). For 𝑋∈𝔨, 𝑌∈𝔥, 𝑍∈𝔞, one has 𝑅Ad𝑎_𝑡⁻¹𝑌𝑍𝑋𝐹𝑎_𝑡=𝜂(𝑌)𝜏(−𝑋)𝑍𝑓𝑎_𝑡. (3.11)

4. Shintani Functions Attached to the Spherical Principal Series Representations

Throughout this section, as a character of 𝑀, we take the trivial character 𝜎=𝜎₀. Then, the principal series representation 𝜋_𝜎0,𝜈 is the spherical or class one principal series representation whose minimal 𝐾-type is the one-dimensional trivial representation (𝟏,𝑉_𝟏) of 𝐾, which occurs of multiplicity one in 𝜋_𝜎0,𝜈|_𝐾 (Proposition 2.1).

4.1. The Capelli Elements

Let 𝑍(𝔤) be the center of the universal enveloping algebra 𝑈(𝔤) of 𝔤. 𝑍(𝔤) has two independent generators, and they are obtained as the Capelli elements because 𝔤=𝔰𝔩₃ is of type 𝐴₂ (see [13]). For 𝑖=1,2,3, we put 𝐸^′_𝑖𝑖=𝐸_𝑖𝑖−13³_𝑘=1𝐸_𝑘𝑘. (4.1) The following proposition gives the explicit description of the independent generators of 𝑍(𝔤) (see [11]).

Proposition 4.1. The independent generators {𝐶𝑝₂,𝐶𝑝₃} of 𝑍(𝔤) are given as follows: 𝐶𝑝₂=𝐸^′₁₁−1𝐸^′₂₂+𝐸^′₂₂𝐸^′₃₃+1+𝐸^′₁₁−1𝐸^′₃₃+1−𝐸₂₃𝐸₃₂−𝐸₁₃𝐸₃₁−𝐸₁₂𝐸₂₁,𝐶𝑝₃=𝐸^′₁₁−1𝐸^′₂₂𝐸^′₃₃+1+𝐸₁₂𝐸₂₃𝐸₃₁+𝐸₁₃𝐸₂₁𝐸₃₂−𝐸^′₁₁−1𝐸₂₃𝐸₃₂−𝐸₁₃𝐸^′₂₂𝐸₃₁−𝐸₁₂𝐸₂₁𝐸^′₃₃+1. (4.2)

Since 𝐶𝑝₂,𝐶𝑝₃ are the elements of 𝑍(𝔤), they act on 𝜋_𝜎0,𝜈 as the scalar operators. And since the space of Shintani functions is the image of the (𝔤_𝐂,𝐾)-homomorphism of 𝜋_𝜎0,𝜈, they act on the space of Shintani functions as the same scalar operators, respectively.

4.2. Eigenvalues of 𝐶𝑝₂, 𝐶𝑝₃

In 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 elements 𝐶𝑝₂, 𝐶𝑝₃. For the spherical principal series representation, 𝜎=𝜎₀ is the trivial character of 𝑀. Let 𝑓₀ be the generator of the minimal 𝐾-type in 𝐻_𝜎0,𝜈 normalized such that 𝑓₀∣𝐾≡1. The actions of 𝐶𝑝₂,𝐶𝑝₃ on 𝑓₀ are computed in [11], and the result is as follows.

Proposition 4.2. The Capelli elements 𝐶𝑝₂,𝐶𝑝₃ act on 𝑓₀ by scalar multiples, and the eigenvalues are given as follows: 𝐶𝑝₂𝑓₀1=−3𝜈²₁−𝜈₁𝜈₂+𝜈²₂𝑓₀,𝐶𝑝₃𝑓₀1=−272𝜈₁−𝜈₂2𝜈₂−𝜈₁𝜈₁+𝜈₂𝑓₀. (4.3)

4.3. Construction of the Casimir Equations

Next, we compute the actions of 𝐶𝑝₂, 𝐶𝑝₃ on 𝐹(𝑎_𝑡)∈𝐶^∞_𝜂,𝟏(𝐿)|_𝐴. Here, 𝜂=𝜂_𝑠,𝑘 is the unitary character of 𝐻.

Lemma 4.3. For 𝐹(𝑎_𝑡)∈𝐶^∞_𝜂,𝜏(𝐿)|_𝐴, one has 𝑅Ad𝑎_𝑡⁻¹𝑌_𝑖𝐹𝑎_𝑡=𝑠𝐹𝑎_𝑡(),𝑅𝑖=1,4Ad𝑎_𝑡⁻¹𝑌_𝑖𝐹𝑎_𝑡(),𝑅𝑋=0𝑖=2,3₁₃𝐹𝑎_𝑡=𝑑𝐹𝑎𝑑𝑡^𝑡. (4.4)

Proof. By definition, we have 𝑅Ad𝑎_𝑡⁻¹𝑌_𝑖𝐹𝑎_𝑡=𝑑𝐹𝑎𝑑𝑢^𝑡exp𝑢Ad𝑎_𝑡⁻¹𝑌_𝑖𝑢=0=𝑑𝐹𝑑𝑢exp𝑢𝑌_𝑖𝑎_𝑡𝑢=0=𝑑𝜂𝑑𝑢exp𝑢𝑌_𝑖𝑢=0𝐹𝑎_𝑡. (4.5) Since exp(𝑢𝑌₁)=diag(𝑒^𝑢,1,𝑒^−𝑢), we have 𝜂((exp(𝑢𝑌₁))=𝑒^𝑘𝑢⋅𝑒^{(𝑠−𝑘)𝑢}=𝑒^𝑠𝑢.
Therefore, we have 𝑅Ad𝑎_𝑡⁻¹𝑌₁𝐹𝑎_𝑡=𝑠𝐹𝑎_𝑡. (4.6) Next, since exp𝑢𝑌₂=,cosh𝑢sinh𝑢0sinh𝑢cosh𝑢0001 (4.7)   𝜂(exp(𝑢𝑌₂))=1^𝑘⋅|1|^𝑠−𝑘=1. Therefore, we have 𝑅Ad𝑎_𝑡⁻¹𝑌₂𝐹𝑎_𝑡=0. (4.8) The computations of the actions of Ad(𝑎_𝑡⁻¹)(𝑌_𝑖)(𝑖=3,4) are similar. Finally, since exp(𝑢𝑋₁₃)=𝑎_𝑢, we have 𝑅𝑋₁₃𝐹𝑎_𝑡=𝑑𝐹𝑎𝑑𝑢^𝑡+𝑢_𝑢=0=𝑑𝐹𝑎𝑑𝑡^𝑡. (4.9)

By simple computations of matrices, we have the following expressions of the elements in 𝑀(3,𝐑).

Lemma 4.4. One has 𝐸^′₁₁=cosh²𝑡+13cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−13Ad𝑎_𝑡⁻¹𝑌₄−12tanh(2𝑡)𝐾₁₃,𝐸^′₂₂=13−𝐻₁₂+𝐻₂₃,𝐸^′₃₃=sinh²𝑡−13cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−13Ad𝑎_𝑡⁻¹𝑌₄+12tanh(2𝑡)𝐾₁₃,𝐸₂₃=12sinh𝑡Ad𝑎_𝑡⁻¹𝑌₃+1𝐾2tanh𝑡¹²+12𝐾₂₃,𝐸₁₃=12𝑋₁₃+12𝐾₁₃,𝐸₁₂=12cosh𝑡Ad𝑎_𝑡⁻¹𝑌₂−12tanh𝑡𝐾₂₃+12𝐾₁₂. (4.10)

To make use of Lemma 3.4, we have to rewrite 𝐶𝑝₂, 𝐶𝑝₃ in the form of linear combinations of the elements in (Ad(𝑎_𝑡⁻¹)𝔥)𝔞𝔨. To do this, we use the following formulas which can be obtained by direct computation.

Lemma 4.5. One has 𝐾₁₃,Ad𝑎_𝑡⁻¹𝑌₁=−2cosh(2𝑡)𝑋₁₃,𝐾₁₃,Ad𝑎_𝑡⁻¹𝑌₄=−cosh(2𝑡)𝑋₁₃,𝐾₁₂,Ad𝑎_𝑡⁻¹𝑌₃=sinh𝑡𝑋₁₃,𝐾₂₃,Ad𝑎_𝑡⁻¹𝑌₃=−2sinh³𝑡cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁+2sinh𝑡Ad𝑎_𝑡⁻¹𝑌₄+cosh𝑡()𝐾cosh2𝑡¹³,𝐾₁₃,𝑋₁₃=2cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−2tanh(2𝑡)𝐾₁₃,𝐾₂₃,Ad𝑎_𝑡⁻¹𝑌₂=−cosh𝑡𝑋₁₃,𝐾₁₂,Ad𝑎_𝑡⁻¹𝑌₂=2cosh³𝑡cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−2cosh𝑡Ad𝑎_𝑡⁻¹𝑌₄−sinh𝑡()𝐾cosh2𝑡¹³,𝐾₁₂,𝑋₁₃1=−sinh𝑡Ad𝑎_𝑡⁻¹𝑌₃−1𝐾tanh𝑡¹²,𝐾₂₃,𝑋₁₃=1cosh𝑡Ad𝑎_𝑡⁻¹𝑌₂−tanh𝑡𝐾₂₃,𝑋₁₃,Ad𝑎_𝑡⁻¹𝑌₂=−tanh𝑡Ad𝑎_𝑡⁻¹𝑌₂−1𝐾cosh𝑡²³,𝐾₁₃,Ad𝑎_𝑡⁻¹𝑌₂1=−tanh𝑡Ad𝑎_𝑡⁻¹𝑌₃−cosh(2𝑡)𝐾sinh𝑡¹²,𝐾₁₃,Ad𝑎_𝑡⁻¹𝑌₃=tanh𝑡Ad𝑎_𝑡⁻¹𝑌₂−cosh(2𝑡)𝐾cosh𝑡²³,𝑋₁₃,Ad𝑎_𝑡⁻¹𝑌₁=−2tanh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−2()𝐾cosh2𝑡¹³,𝑋₁₃,Ad𝑎_𝑡⁻¹𝑌₄=−tanh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−1()𝐾cosh2𝑡¹³,𝐾₁₂,Ad𝑎_𝑡⁻¹𝑌₁()=−cosh2𝑡cosh𝑡Ad𝑎_𝑡⁻¹𝑌₂−tanh𝑡𝐾₂₃,𝐾₁₂,Ad𝑎_𝑡⁻¹𝑌₄=1−sinh²𝑡cosh𝑡Ad𝑎_𝑡⁻¹𝑌₂−2tanh𝑡𝐾₂₃,𝐾₂₃,Ad𝑎_𝑡⁻¹𝑌₁()=−cosh2𝑡sinh𝑡Ad𝑎_𝑡⁻¹𝑌₃−1𝐾tanh𝑡¹²,𝐾₂₃,Ad𝑎_𝑡⁻¹𝑌₄=−1+cosh²𝑡sinh𝑡Ad𝑎_𝑡⁻¹𝑌₃−2𝐾tanh𝑡¹². (4.11) Here, [𝑋,𝑌]∶=𝑋𝑌−𝑌𝑋 is the Lie bracket on 𝔤.

By using Lemma 4.5, we can rewrite 𝐶𝑝₂, 𝐶𝑝₃ as we wished. Now, since for all 𝐹(𝑎_𝑡)∈𝐶^∞_𝜂,𝟏(𝐿)|_𝐴 is annihilated by the action of 𝑈(𝔤)𝔨 and Ad(𝑎_𝑡⁻¹)𝑌₂𝑈(𝔤)Ad(𝑎_𝑡⁻¹)𝑌₃𝑈(𝔤), and the actions of Ad(𝑎_𝑡⁻¹)𝑌₁ and Ad(𝑎_𝑡⁻¹)𝑌₄ on 𝐹 are the same (the multiplication by 𝑠), we may regard 𝐶𝑝₂, 𝐶𝑝₃ as the elements in 𝑈(𝔤)(mod𝔓), where 𝔓 is the subalgebra of 𝑈(𝔤) defined by 𝔓=Ad𝑎_𝑡⁻¹𝑌₁−Ad𝑎_𝑡⁻¹𝑌₄𝑈(𝔤)+Ad𝑎_𝑡⁻¹𝑌₂𝑈(𝔤)+Ad𝑎_𝑡⁻¹𝑌₃𝑈(𝔤)+𝑈(𝔤)𝔨. (4.12)

Lemma 4.6. One has the congruences 𝐶𝑝₂1≡−4𝑋²₁₃−12()+1tanh2𝑡()𝑋tanh2𝑡¹³+14tanh²()−12𝑡3Ad𝑎_𝑡⁻¹𝑌₁²−1(mod𝔓),𝐶𝑝₃1≡−12Ad𝑎_𝑡⁻¹𝑌₁𝑋²₁₃−16()+1tanh2𝑡tanh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁𝑋₁₃+−1+12712tanh²(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁³−13Ad𝑎_𝑡⁻¹𝑌₁(mod𝔓). (4.13)

By combining Proposition 4.2, and Lemmas 4.3 and 4.6, we have the following two differential equations.

Theorem 4.7. The Shintani function 𝐹(𝑎_𝑡)∈𝐶^∞_𝜂,𝟏(𝐿)_{𝜋𝜎0,𝜈}|_𝐴 satisfies the following equations:
(1) the differential equation obtained from the action of 𝐶𝑝₂: −14𝑑²𝐹𝑑𝑡²𝑎_𝑡−12()+1tanh2𝑡tanh(2𝑡)𝑑𝐹𝑎𝑑𝑡^𝑡+−13+14tanh²(2𝑡)𝑠²−1𝐹𝑎_𝑡=𝜆₂𝐹𝑎_𝑡; (4.14)
(2) the differential equation obtained from the action of 𝐶𝑝₃: −1𝑠𝑑12²𝐹𝑑𝑡²𝑎_𝑡−16()+1tanh2𝑡()𝑠tanh2𝑡𝑑𝐹𝑎𝑑𝑡^𝑡+−2+12712tanh²(2𝑡)𝑠³−13𝑠𝐹𝑎_𝑡=𝜆₃𝐹𝑎_𝑡. (4.15) Here, 𝜆₂1=−3𝜈²₁−𝜈₁𝜈₂+𝜈²₂,𝜆₃1=−272𝜈₁−𝜈₂2𝜈₂−𝜈₁𝜈₁+𝜈₂ (4.16) are the eigenvalues of the Capelli elements on principal series representations.
Equations (4.15)−(4.14) ×(1/3)𝑠 give 1𝑠27³+13𝜆₂𝑠−𝜆₃𝐹𝑎_𝑡=0. (4.17)

Therefore, if 𝐹(𝑎_𝑡) is not identically zero, we have 𝑠³−3𝜈²₁−𝜈₁𝜈₂+𝜈²₂𝑠+2𝜈₁−𝜈₂2𝜈₂−𝜈₁𝜈₁+𝜈₂=0. (4.18) By solving this equation, we have 𝑠=2𝜈₁−𝜈₂,2𝜈₂−𝜈₁,−𝜈₁−𝜈₂. (4.19) Therefore, one of the necessary conditions of the existence of nontrivial Shintani functions is that the parameter 𝑠 is one of the above three values. Now, we assume that 𝑠 satisfies this condition. We put 𝑥=tanh(2𝑡),𝐹(𝑥)∶=𝐹(𝑎_𝑡) in (4.14). Then, we have −4𝑥(1−𝑥)²𝑑²𝐹𝑑𝑥²(𝑥)−4(1−𝑥)²𝑑𝐹(𝑥)+−1𝑑𝑥3+14𝑥𝑠²−1−𝜆₂𝐹(𝑥)=0. (4.20) Next, we put 𝐹(𝑥)=(1−𝑥)^𝜇𝐺₀(𝑥)(𝜇∈𝐂). Then, 𝐺₀ satisfies −4𝑥(1−𝑥)^𝜇+2𝑑²𝐺₀𝑑𝑥²(𝑥)+−4+8𝜇+4𝑥(1−𝑥)^𝜇+1𝑑𝐺₀(𝑥)+−1𝑑𝑥−4𝜇𝜇−1𝑥+4𝜇−4𝜇𝑥+3+14𝑥𝑠²−1−𝜆₂(1−𝑥)^𝜇𝐺₀(𝑥)=0. (4.21) We want to divide the left-hand side of (4.21) by (1−𝑥)^𝜇+1. To do this, we take 𝜇∈𝐂 so that 𝜇 satisfies −4𝜇²1+4𝜇−𝑠12²−1−𝜆₂=0. (4.22) The value of 𝜇∈𝐂 is as follows. (1)If 𝑠=2𝜈₁−𝜈₂,𝜇=(2±𝜈₂)/4. So we take 𝜇=(2+𝜈₂)/4. (2)If 𝑠=2𝜈₂−𝜈₁,𝜇=(2±𝜈₁)/4. So we take 𝜇=(2+𝜈₁)/4. (3)If 𝑠=−𝜈₁−𝜈₂,𝜇={2±(𝜈₁−𝜈₂)}/4. So we take 𝜇=(2+𝜈₁−𝜈₂)/4.

For this 𝜇, the left-hand side of (4.21) is divided by (1−𝑥)^𝜇+1, and the equation becomes 𝑥()𝑑𝑥−1²𝐺₀𝑑𝑥²(𝑥)+−1+1+2𝜇𝑥𝑑𝐺₀+𝜇𝑑𝑥²−1𝑠16²𝐺₀(𝑥)=0. (4.23) This is the Gaussian hypergeometric differential equation. Note that the Shintani function 𝐹(𝑎_𝑡) on 𝐴 is regular at the origin (⇔𝑥=0). Therefore, 𝐺₀(𝑥)=(1−𝑥)^−𝜇𝐹(𝑥) is also regular at 𝑥=0. Equation (4.23) has just one solution which is regular around 𝑥=0 (up to constant multiples), and it is given by 𝐺₀(𝑥)=₂𝐹₁𝛼,𝛽;1;𝑥, (4.24) where ₂𝐹₁ is the Gaussian hypergeometric function and 𝛼,𝛽∈𝐂 are defined by 1+𝛼+𝛽=1+2𝜇,𝛼𝛽=𝜇²−1𝑠16². (4.25) Explicitly, by solving these equations, 𝛼,𝛽 are given as follows. (1)In case of 𝑠=2𝜈₁−𝜈₂,(𝛼,𝛽)=((𝜈₁+1)/2,(−𝜈₁+𝜈₂+1)/2). (2)In case of 𝑠=2𝜈₂−𝜈₁,(𝛼,𝛽)=((𝜈₂+1)/2,(−𝜈₂+𝜈₁+1)/2). (3)In case of 𝑠=−𝜈₁−𝜈₂,(𝛼,𝛽)=((𝜈₁+1)/2,(−𝜈₂+1)/2).

Finally, we consider the three conditions in (3.8). Condition (1) is equivalent to (−1)^𝑘𝐹(𝑎_𝑡)=𝐹(𝑎_𝑡). Condition (2) always holds. Condition (3) is equivalent to (−1)^𝑘𝐹(1)=𝐹(1), which holds if condition (1) is satisfied. Summing up, we have the following theorem.

Theorem 4.8. Let 𝜂=𝜂_𝑠,𝑘 be the unitary character of 𝐻 defined by (3.1). Then, the necessary condition of the existence of the nontrivial elements in 𝐶^∞_𝜂,𝟏(𝐿)_{𝜋𝜎0,𝜈} is that 𝑘=0,𝑠=2𝜈₁−𝜈₂,2𝜈₂−𝜈₁,−𝜈₁−𝜈₂. (4.26) Suppose that this condition is satisfied and nontrivial Shintani functions exist. If one puts 𝑥=tanh²(2𝑡), 𝐹(𝑎_𝑡)=𝐹(𝑥)∈𝐶^∞_𝜂,𝟏(𝐿)_{𝜋𝜎0,𝜈}|_𝐴 is given as follows (up to constant multiples). (1)In case of 𝑠=2𝜈₁−𝜈₂, one has 𝐹(𝑥)=(1−𝑥)^(2+𝜈2₂^)/4𝐹₁𝜈₁+12,−𝜈₁+𝜈₂+12;1;𝑥. (4.27)(2)In case of 𝑠=2𝜈₂−𝜈₁, one has 𝐹(𝑥)=(1−𝑥)^(2+𝜈1₂^)/4𝐹₁𝜈₂+12,−𝜈₂+𝜈₁+12;1;𝑥. (4.28)(3)In case of 𝑠=−𝜈₁−𝜈₂, one has 𝐹(𝑥)=(1−𝑥)^{(2+𝜈1−𝜈2}₂^)/4𝐹₁𝜈₁+12,−𝜈₂+12;1;𝑥. (4.29) Especially, one hasdim𝐶^∞_𝜂,𝟏(𝐿)_{𝜋𝜎0,𝜈}≤1. (4.30)

5. Shintani Functions Attached to the Nonspherical Principal Series Representations

In this section, as a character of 𝑀, we take a nontrivial character 𝜎=𝜎_𝑖(𝑖=1,2,3). Then, the minimal 𝐾-type of 𝜋_{𝜎𝑖,𝜈} is the three-dimensional representation of 𝐾 which is isomorphic to the tautological representation 𝜏₂∶𝐾=𝑆𝑂(3,𝐑)↪𝐺𝐿(3,𝐑) which occurs of multiplicity one in 𝜋_{𝜎𝑖,𝜈}|_𝐾. We take 𝜏^∗₂ instead of 𝜏₂ as a minimal 𝐾-type of 𝜋_{𝜎𝑖,𝜈}. The representation space of 𝜏₂ is denoted by 𝑉_𝜏2(=𝐑³), and we take 𝑠₁=^𝑡(1,0,0), 𝑠₂=^𝑡(0,1,0), 𝑠₃=^𝑡(0,0,1) as a basis of 𝑉_𝜏2. Let Ψ∈𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎𝑖,𝜈} be the Shintani function. Then, Ψ is expressed by Ψ𝑔=𝐹₀𝑔𝑠₁+𝐺₀𝑔𝑠₂+𝐻₀𝑔𝑠₃. (5.1)  Ψ is characterized by its restriction to 𝐴. To investigate Ψ|_𝐴, we construct two kinds of differential equations. One is the Casimir equation of degree two and the other is the gradient equation (or the Dirac-Schmidt equation).

5.1. The Casimir Equation

Firstly, we construct the Casimir equation of degree two. Since the Capelli element 𝐶𝑝₂ acts on the representation space of the principal series representation 𝜋_{𝜎𝑖,𝜈} as a scalar operator (𝜆₂-multiple) and the space of Shintani functions 𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎𝑖,𝜈} is the image of (𝔤_𝐂,𝐾)-homomorphism of 𝜋_{𝜎𝑖,𝜈}, 𝐶𝑝₂ acts on this space as the same scalar operator. Since for all Ψ(𝑎_𝑡)∈𝐶^∞_𝜂,𝜏2(𝐿)|_𝐴 is annihilated by the action of Ad(𝑎_𝑡⁻¹)𝑌₂𝑈(𝔤),Ad(𝑎_𝑡⁻¹)𝑌₃𝑈(𝔤) and the actions of Ad(𝑎_𝑡⁻¹)𝑌₁ and Ad(𝑎_𝑡⁻¹)𝑌₄ on 𝐹 are the same (the multiplication by 𝑠), we may regard 𝐶𝑝₂ as the element in 𝑈(𝔤)(mod𝔓^′), where 𝔓^′ is a subalgebra of 𝑈(𝔤) defined by 𝔓^′=Ad𝑎_𝑡⁻¹𝑌₁−Ad𝑎_𝑡⁻¹𝑌₄𝑈(𝔤)+Ad𝑎_𝑡⁻¹𝑌₂𝑈(𝔤)+Ad𝑎_𝑡⁻¹𝑌₃𝑈(𝔤). (5.2) By using Lemmas 4.4 and 4.5, we can rewrite 𝐶𝑝₂ in Proposition 4.1 as follows.

Lemma 5.1. One has 𝐶𝑝₂1≡−4𝑋²₁₃−12()+1tanh2𝑡()𝑋tanh2𝑡¹³+14tanh²()−12𝑡3Ad𝑎_𝑡⁻¹𝑌₁²+()−1tanh2𝑡2cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁𝐾₁₃+−14tanh²()+12𝑡4𝐾²₁₃+−14tanh²1𝑡+4𝐾²₁₂+14−14tanh²𝑡𝐾²₂₃⋅modAd𝑎_𝑡⁻¹𝑌₁−Ad𝑎_𝑡⁻¹𝑌₄𝑈(𝔤)+Ad𝑎_𝑡⁻¹𝑌₂𝑈(𝔤)+Ad𝑎_𝑡⁻¹𝑌₃𝑈(𝔤). (5.3)

By using this lemma, the action of 𝐶𝑝₂ on Ψ(𝑎_𝑡)=𝐹₀(𝑎_𝑡)𝑠₁+𝐺₀(𝑎_𝑡)𝑠₂+𝐻₀(𝑎_𝑡)𝑠₃∈𝐶^∞_𝜂,𝜏2(𝐿)|_𝐴 can be computed easily. We have 𝐶𝑝₂Ψ𝑎_𝑡=𝐹^′𝑎_𝑡𝑠₁+𝐺^′𝑎_𝑡𝑠₂+𝐻^′𝑎_𝑡𝑠₃, (5.4) where 𝐹^′𝑎_𝑡1=−4𝑑²𝐹₀𝑑𝑡²𝑎_𝑡−12()+1tanh2𝑡tanh(2𝑡)𝑑𝐹₀𝑎𝑑𝑡^𝑡+14tanh²()−12𝑡3𝑠²+14tanh²()+12𝑡4tanh²𝑡−32𝐹₀𝑎_𝑡−tanh(2𝑡)2cosh(2𝑡)𝑠𝐻₀𝑎_𝑡,𝐺^′𝑎_𝑡1=−4𝑑²𝐺₀𝑑𝑡²𝑎_𝑡−12()+1tanh2𝑡tanh(2𝑡)𝑑𝐺₀𝑎𝑑𝑡^𝑡+14tanh²1𝑡−3𝑠²+14tanh²1𝑡+4tanh²𝑡−32𝐺₀𝑎_𝑡,𝐻^′𝑎_𝑡1=−4𝑑²𝐻₀𝑑𝑡²𝑎_𝑡−12()+1tanh2𝑡tanh(2𝑡)𝑑𝐻₀𝑎𝑑𝑡^𝑡+14tanh²()−12𝑡3𝑠²+14tanh²()+12𝑡4tanh²3𝑡−2𝐻₀𝑎_𝑡+tanh(2𝑡)2cosh(2𝑡)𝑠𝐹₀𝑎_𝑡. (5.5) Since Ψ(𝑎_𝑡) satisfies 𝐶𝑝₂Ψ(𝑎_𝑡)=𝜆₂Ψ(𝑎_𝑡), we have the following three differential equations.

Theorem 5.2. For Ψ(𝑎_𝑡)=𝐹₀(𝑎_𝑡)𝑠₁+𝐺₀(𝑎_𝑡)𝑠₂+𝐻₀(𝑎_𝑡)𝑠₃∈𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎𝑖,𝜈}|_𝐴, the functions 𝐹₀, 𝐺₀, 𝐻₀ satisfy the following equations: −14𝑑²𝐹₀𝑑𝑡²𝑎_𝑡−12()+1tanh2𝑡tanh(2𝑡)𝑑𝐹₀𝑎𝑑𝑡^𝑡+14tanh²()−12𝑡3𝑠²+14tanh²()+12𝑡4tanh²𝑡−32𝐹₀𝑎_𝑡−tanh(2𝑡)2cosh(2𝑡)𝑠𝐻₀𝑎_𝑡=𝜆₂𝐹₀𝑎_𝑡, (5.6)−14𝑑²𝐺₀𝑑𝑡²𝑎_𝑡−12()+1tanh2𝑡tanh(2𝑡)𝑑𝐺₀𝑎𝑑𝑡^𝑡+14tanh²1𝑡−3𝑠²+14tanh²1𝑡+4tanh²𝑡−32𝐺₀𝑎_𝑡=𝜆₂𝐺₀𝑎_𝑡, (5.7)−14𝑑²𝐻₀𝑑𝑡²𝑎_𝑡−12()+1tanh2𝑡tanh(2𝑡)𝑑𝐻₀𝑎𝑑𝑡^𝑡+14tanh²()−12𝑡3𝑠²+14tanh²()+12𝑡4tanh²3𝑡−2𝐻₀𝑎_𝑡+tanh(2𝑡)2cosh(2𝑡)𝑠𝐹₀𝑎_𝑡=𝜆₂𝐻₀𝑎_𝑡. (5.8)

5.2. The Gradient Equation

For the spherical function Ψ(𝑔)∈𝐶^∞_𝜂,𝜏2(𝐿), we define the right gradient operator ∇^𝑅 as follows.

Definition 5.3. For the orthonormal basis {𝑋_𝑖}⁵_𝑖=1 of 𝔭, the right gradient operator ∇^𝑅 is defined by ∇^𝑅Ψ𝑔∶=⁵_𝑖=1𝑅𝑋_𝑖Ψ⊗𝑋^∗_𝑖. (5.9) Here, 𝑋^∗_𝑖 is the dual basis of 𝑋_𝑖 with respect to the inner product (𝑋,𝑌)∈𝔭×𝔭→Tr(𝑋𝑌)∈𝐂.
The set {𝐻₁₂,𝐻₂₃,𝑋₁₂,𝑋₂₃,𝑋₁₃} becomes the orthonormal basis of 𝔭, and {(1/3)(2𝐻₁₂+𝐻₂₃),(1/3)(𝐻₁₂+2𝐻₂₃),(1/2)𝑋₁₂,(1/2)𝑋₂₃,(1/2)𝑋₁₃} is its dual basis. Therefore, the gradient operator ∇^𝑅 is explicitly given by ∇^𝑅Ψ𝑔=13𝑅𝐻₁₂Ψ⊗2𝐻₁₂+𝐻₂₃+13𝑅𝐻₂₃Ψ⊗𝐻₁₂+2𝐻₂₃+12_𝑖<𝑗𝑅𝑋_𝑖𝑗Ψ⊗𝑋_𝑖𝑗. (5.10) We rewrite this by using the basis of 𝔭_𝐂.

Claim 1. We define five elements 𝑤_𝑖(0≤𝑖≤4) in 𝔭_𝐂 by𝑤₀∶=−2𝐻₂₃−√−1𝑋₂₃,𝑤₄∶=−2𝐻₂₃+√−1𝑋₂₃,𝑤₂2∶=32𝐻₁₂+𝐻₂₃,𝑤₁∶=𝑋₁₃+√−1𝑋₁₂,𝑤₃∶=−𝑋₁₃+√−1𝑋₁₂. (5.11) Then, {𝑤_𝑖∣0≤𝑖≤4} becomes the basis of 𝔭_𝐂.

With this basis, the gradient operator ∇^𝑅 is rewritten as ∇^𝑅1Ψ=𝑅𝑤16⁴Ψ⊗𝑤₀+1𝑅𝑤16⁰Ψ⊗𝑤₄−14𝑅𝑤₃Ψ⊗𝑤₁−14𝑅𝑤₁Ψ⊗𝑤₃+38𝑅𝑤₂Ψ⊗𝑤₂=1414𝑅𝑤₄Ψ⊗𝑤₀+14𝑅𝑤₀Ψ⊗𝑤₄−𝑅𝑤₃Ψ⊗𝑤₁−𝑅𝑤₁Ψ⊗𝑤₃+32𝑅𝑤₂Ψ⊗𝑤₂. (5.12) The Lie algebra 𝔭_𝐂 becomes the representation space of the adjoint action of 𝐾. We denote this representation by (𝜏₄,𝑊₄). By the Clebsch-Gordan theorem, 𝜏₂⊗𝜏₄ has the irreducible decomposition 𝜏₂⊗𝜏₄≅𝜏₂⊕𝜏₄⊕𝜏₆. (5.13) Here, each 𝜏_𝑛 is the (𝑛+1)-dimensional irreducible representation of 𝐾. In this decomposition, the projector of 𝐾-modules 𝑝𝑟₂∶𝜏₂⊗𝜏₄↠𝜏₂,𝑠_𝑖⊗𝑤_𝑗⟼𝑝𝑟₂𝑠_𝑖⊗𝑤_𝑗, (5.14) is described as in Table 2.

Table 2 

∇^𝑅Ψ is a 𝜏₂⊗(𝜏₂⊗𝔭_𝐂)-valued function. Then, by mapping 𝑠_𝑖⊗𝑤_𝑘 to 𝑝𝑟₂(𝑠_𝑖⊗𝑤_𝑘), we have a 𝐾-homomorphism 𝑝𝑟₂∘∇^𝑅∶𝐶^∞_𝜂,𝜏2(𝐿)⟶𝐶^∞_𝜂,𝜏2(𝐿). (5.15) Since the minimal 𝐾-type 𝜏^∗₂ occurs of multiplicity one, 𝑝𝑟₂∘∇^𝑅 is a map of constant multiple. To compute the action of the gradient operator 𝑝𝑟₂∘∇^𝑅 on the space of the Shintani functions 𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎𝑖,𝜈}, we have to decompose 𝑤_𝑖(𝑖=0,1,2,3,4) along the decomposition 𝔤_𝐂=Ad(𝑎_𝑡⁻¹)𝔥_𝐂⊕𝔞_𝐂⊕𝔨_𝐂.

Lemma 5.4. One has 𝑤₀=2sinh²𝑡cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−2Ad𝑎_𝑡⁻¹𝑌₄+tanh(2𝑡)𝐾₁₃+2√−1sinh𝑡Ad𝑎_𝑡⁻¹𝑌₃+2√−1𝐾tanh𝑡¹²,𝑤₄=2sinh²𝑡cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−2Ad𝑎_𝑡⁻¹𝑌₄+tanh(2𝑡)𝐾₁₃−2√−1sinh𝑡Ad𝑎_𝑡⁻¹𝑌₃−2√−1𝐾tanh𝑡¹²,𝑤₂=2cosh²𝑡+13cosh(2𝑡)Ad𝑎_𝑡⁻¹𝑌₁−23Ad𝑎_𝑡⁻¹𝑌₄−tanh(2𝑡)𝐾₁₃,𝑤₁=𝑋₁₃+√−1cosh𝑡Ad𝑎_𝑡⁻¹𝑌₂−√−1tanh𝑡𝐾₂₃,𝑤₃=−𝑋₁₃+√−1cosh𝑡Ad𝑎_𝑡⁻¹𝑌₂−√−1tanh𝑡𝐾₂₃. (5.16)

By using Lemmas 3.4 and 4.3 and the table of projections, we can compute the action of the gradient operator. For Ψ(𝑎_𝑡)=𝐹₀(𝑎_𝑡)𝑠₁+𝐺₀(𝑎_𝑡)𝑠₂+𝐻₀(𝑎_𝑡)𝑠₃∈𝐶^∞_𝜂,𝜏2(𝐿)|_𝐴, we have 𝑝𝑟₂∘∇^𝑅Ψ𝑎_𝑡=𝐹^′′𝑎_𝑡𝑠₁+𝐺^′′𝑎_𝑡𝑠₂+𝐻^′′𝑎_𝑡𝑠₃, (5.17) where 𝐹^′′𝑎_𝑡1=−()−12cosh2𝑡6𝑠𝐹₀𝑎_𝑡−12𝑑𝐻₀𝑎𝑑𝑡^𝑡−12(tanh(2𝑡)+tanh𝑡)𝐻₀𝑎_𝑡,𝐺^′′𝑎_𝑡1=−3𝑠𝐺₀𝑎_𝑡,𝐻^′′𝑎_𝑡=1()+12cosh2𝑡6𝑠𝐻₀𝑎_𝑡−12𝑑𝐹₀𝑎𝑑𝑡^𝑡−12()+1tanh2𝑡𝐹tanh𝑡⁰𝑎_𝑡. (5.18) On the other hand, the eigenvalue of the gradient operator on the spherical functions of the principal series representation 𝜋_{𝜎𝑖,𝜈} depends on the choice of 𝜎_𝑖, denoted by 𝜆_𝜎𝑖(𝑖=1,2,3). These values are computed in [11] and they are as follows: 𝜆_𝜎11=−32𝜈₁−𝜈₂,𝜆_𝜎21=−32𝜈₂−𝜈₁,𝜆_𝜎3=13𝜈₁+𝜈₂. (5.19) Therefore, since Ψ(𝑎_𝑡)=𝐹₀(𝑎_𝑡)𝑠₁+𝐺₀(𝑎_𝑡)𝑠₂+𝐻₀(𝑎_𝑡)𝑠₃∈𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎𝑖,𝜈}|_𝐴 satisfies 𝑝𝑟₂∘∇^𝑅Ψ(𝑎_𝑡)=𝜆_𝜎𝑖Ψ(𝑎_𝑡), we have the following three differential equations.

Theorem 5.5. For Ψ(𝑎_𝑡)=𝐹₀(𝑎_𝑡)𝑠₁+𝐺₀(𝑎_𝑡)𝑠₂+𝐻₀(𝑎_𝑡)𝑠₃∈𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎𝑖,𝜈}|_𝐴, one has −1()−12cosh2𝑡6𝑠𝐹₀𝑎_𝑡−12𝑑𝐻₀𝑎𝑑𝑡^𝑡−12(tanh(2𝑡)+tanh𝑡)𝐻₀𝑎_𝑡=𝜆_𝜎𝑖𝐹₀𝑎_𝑡,−13𝑠𝐺₀𝑎_𝑡=𝜆_𝜎𝑖𝐺₀𝑎_𝑡,1()+12cosh2𝑡6𝑠𝐻₀𝑎_𝑡−12𝑑𝐹₀𝑎𝑑𝑡^𝑡−12()+1tanh2𝑡𝐹tanh𝑡⁰𝑎_𝑡=𝜆_𝜎𝑖𝐻₀𝑎_𝑡. (5.20)(5.21)(5.22)

We consider the case of 𝜎=𝜎₁. We have 𝜆_𝜎1=−(1/3)(2𝜈₁−𝜈₂). By (5.21), if 𝑠≠−3𝜆_𝜎1=2𝜈₁−𝜈₂, we have 𝐺₀(𝑎_𝑡)≡0. Suppose that 𝑠=2𝜈₁−𝜈₂. We put 𝑥=tanh²(2𝑡), 𝐺₀(𝑎_𝑡)=𝐺₁(𝑥) in (5.21). Then, the equation becomes −4𝑥²(1−𝑥)²𝑑²𝐺₁𝑑𝑥²(𝑥)−4𝑥(1−𝑥)²𝑑𝐺₁d𝑥(𝑥)+14𝑥²−13𝑥𝑠²−2+𝜆₂𝑥+1𝐺₁(𝑥)=0. (5.23) We put 𝐺₁(𝑥)=𝑥^𝛼(1−𝑥)^𝛽𝐺₁(𝑥)(𝛼,𝛽∈𝐂). Then, the equation becomes −4𝑥²(1−𝑥)²𝑑²𝐺₁𝑑𝑥²(𝑥)(𝑥𝑑𝐺+4𝑥1−𝑥)−2𝛼−1+2𝛼+2𝛽+1¹(𝑥)+()𝑑𝑥−4𝛼𝛼−1)(1−𝑥²−4𝛽𝛽−1𝑥²+8𝛼𝛽𝑥(1−𝑥)−4𝛼(1−𝑥)²()+1+4𝛽𝑥1−𝑥4𝑥²−13𝑥𝑠²−2+𝜆₂𝑥+1𝐺₁(𝑥)=0. (5.24) We want to divide the left-hand side of (5.24) by 𝑥(1−𝑥). For this purpose, 𝛼,𝛽∈𝐂 must satisfy −4𝛼²−1+1=0,−4𝛽𝛽−1𝑠12²−𝜆₂−1=0. (5.25) The solutions are 𝛼=±1/2,𝛽=(2±𝜈₂)/4. We choose 𝛼=1/2,𝛽=(2+𝜈₂)/4. Then the left-hand side of (5.24) can be divided by 𝑥(1−𝑥), and the equation becomes 𝑥()𝑑𝑥−1²𝐺₁𝑑𝑥²(𝑥)+𝜈−2+3+²2𝑥𝑑𝐺₁(𝑥)+1𝑑𝑥4−𝜈²₁+𝜈₁𝜈₂+2𝜈₂+4𝐺₁(𝑥)=0. (5.26) This is a Gaussian hypergeometric differential equation, and its regular solution is given by 𝐺₁(𝑥)=₂𝐹₁12𝜈₁1+1,2𝜈₂−12𝜈₁+1;2;𝑥 (5.27) up to constant multiples. (The other solutions are not regular around 𝑥=0, since they contain log𝑥.) Therefore, we have 𝐺₀𝑎_𝑡=𝐺₀(𝑥)=𝐶𝑥^1/2(1−𝑥)^(2+𝜈2₂^)/4𝐹₁12𝜈₁1+1,2𝜈₂−12𝜈₁+1;2;𝑥, (5.28) where 𝐶 is a constant number and 𝑥=tanh²(2𝑡). Next, we consider the equations satisfied by 𝐹₀ and 𝐻₀. From (5.20) and (5.22), we have 𝑑𝐹₀𝑎𝑑𝑡^𝑡=1()+1cosh2𝑡3𝑠𝐻₀𝑎_𝑡−()+1tanh2𝑡𝐹tanh𝑡⁰𝑎_𝑡−2𝜆_𝜎1𝐻₀𝑎_𝑡,𝑑𝐻₀𝑎𝑑𝑡^𝑡1=−()−1cosh2𝑡3𝑠𝐹₀𝑎_𝑡−(tanh(2𝑡)+tanh𝑡)𝐻₀𝑎_𝑡−2𝜆_𝜎1𝐹₀𝑎_𝑡. (5.29) By differentiating both sides of (5.29) by 𝑡, we have 𝑑²𝐹₀𝑑𝑡²𝑎_𝑡=3tanh²()+22𝑡tanh²𝑡+2tanh(2𝑡)tanh𝑡4−3−3𝑠𝜆_𝜎1−1cosh²()−12𝑡9𝑠²+4𝜆²_𝜎1𝐹₀𝑎_𝑡+−4tanh(2𝑡)()+2cosh2𝑡3()+2tanh2𝑡()+23tanh2𝑡()𝑠()+1sinh2𝑡+4tanh2𝑡()𝜆tanh2𝑡^𝜎1𝐻₀𝑎_𝑡,𝑑²𝐻₀𝑑𝑡²𝑎_𝑡=3tanh²(2𝑡)+2tanh²()−4𝑡+2tanh𝑡tanh2𝑡−33𝑠𝜆_𝜎1−1cosh²()−12𝑡9𝑠²+4𝜆²_𝜎1𝐻₀𝑎_𝑡+4tanh(2𝑡)()−2cosh2𝑡3()−2tanh2𝑡()+23tanh2𝑡()𝑠sinh2𝑡+4tanh()+12𝑡()𝜆tanh2𝑡^𝜎1𝐹₀𝑎_𝑡. (5.30) By inserting (5.29) and (5.30) into the Casimir equation (5.6), (5.8) to eliminate the differential terms, we have 13𝑠𝜆_𝜎1−𝜆²_𝜎1−19𝑠²−𝜆₂𝐹₀𝑎_𝑡1=0,3𝑠𝜆_𝜎1−𝜆²_𝜎1−19𝑠²−𝜆₂𝐻₀𝑎_𝑡=0. (5.31) Therefore, if the parameter 𝑠∈𝐂 satisfies (1/3)𝑠𝜆_𝜎1−𝜆²_𝜎1−(1/9)𝑠²−𝜆₂≠0, then 𝐹₀(𝑎_𝑡),𝐻₀(𝑎_𝑡)≡0. Suppose that (1/3)𝑠𝜆_𝜎1−𝜆²_𝜎1−(1/9)𝑠²−𝜆₂=0. Since 𝜆_𝜎1=−(1/3)(2𝜈₁−𝜈₂),𝜆₂=−(1/3)(𝜈²₁−𝜈₁𝜈₂+𝜈²₂), we have 𝑠²+2𝜈₁−𝜈₂𝑠+𝜈²₁−𝜈₁𝜈₂−2𝜈²₂=0. (5.32) Therefore, we have 𝑠=−𝜈₁−𝜈₂,−𝜈₁+2𝜈₂. (5.33) That is, (5.33) is the necessary condition of the existence of nontrivial 𝐹₀(𝑎_𝑡),𝐻₀(𝑎_𝑡). We put 𝐹₀𝑎_𝑡=cosh^−1/2(2𝑡)(sinh𝑡)⁻¹𝐹(𝑡),𝐻₀𝑎_𝑡=cosh^−1/2(2𝑡)(cosh𝑡)⁻¹𝐻(𝑡) (5.34) and insert these into (5.20), (5.22). Then we have −1()−12cosh2𝑡6𝑠1𝐹(𝑡)−1tanh𝑡2𝑑𝐻(𝑡)𝑑𝑡=𝜆_𝜎11𝐹(𝑡),1tanh𝑡()+12cosh2𝑡6𝐻(𝑡)−1𝑠tanh𝑡2𝑑𝐹(𝑡)𝑑𝑡=𝜆_𝜎1tanh𝑡𝐻(𝑡). (5.35) Next, we put tanh²𝑡=𝑢,𝐹(𝑡)=𝐹₁(𝑢),𝐻(𝑡)=𝐻₁(𝑢). Then, the above equations become −1−𝑢2()−11+𝑢6𝑠𝐹₁(𝑢)()𝑑𝐻−𝑢1−𝑢¹(𝑢)𝑑𝑢=𝜆_𝜎1𝐹₁(𝑢),−()𝑑𝐹1−𝑢¹(𝑢)+𝑑𝑢1−𝑢2()+11+𝑢6𝑠𝐻₁(𝑢)=𝜆_𝜎1𝐻₁(𝑢). (5.36)(5.37) For a while, we consider the case of 𝑠=0, that is, 𝜂=𝜂_0,𝑘 is a signature sgn_(𝑘) of 𝐻, where sgn_(𝑘)ℎ₁₁ℎ₁₂0ℎ₂₁ℎ₂₂000ℎ₁=det𝐻₁^𝑘det𝐻₁^−𝑘,𝐻₁=ℎ₁₁ℎ₁₂ℎ₂₁ℎ₂₂. (5.38) Then, by combining (5.36), (5.37), we have 𝑢(1−𝑢)²𝜆_𝜎1𝑑²𝐹₁𝑑𝑢²(𝑢)−𝑢(1−𝑢)𝜆_𝜎1𝑑𝐹₁(𝑢)𝑑𝑢−𝜆³_𝜎1𝐹₁(𝑢)=0. (5.39) Suppose that 𝜆_𝜎1≠0. Then, the equation becomes 𝑢(1−𝑢)²𝑑²𝐹₁𝑑𝑢²(𝑢)()𝑑𝐹−𝑢1−𝑢¹(𝑢)𝑑𝑢−𝜆²_𝜎1𝐹₁(𝑢)=0. (5.40) We put 𝐹₁(𝑢)=(1−𝑢)^𝜆𝜎1𝐹₂(𝑢). Then, 𝐹₂(𝑢) satisfies 𝑢()𝑑1−𝑢²𝐹₂𝑑𝑢²(𝑢)−2𝜆_𝜎1+1𝑢𝑑𝐹₂(𝑢)𝑑𝑢−𝜆²_𝜎1𝐹₂(𝑢)=0. (5.41) Equation (5.41) is the Gaussian hypergeometric differential equation. Now, since 𝐹₂(𝑢)=(1−𝑢)^−𝜆𝜎1cosh^1/2(2𝑡)sinh𝑡𝐹₀(𝑎_𝑡) and 𝐹₀(𝑎_𝑡) is regular at 𝑎_𝑡=1(⇔𝑡=0⇔𝑢=0), 𝐹₂(𝑢) must be regular at 𝑢=0. The regular solution of (5.41) is given by 𝐹₂(𝑢)=𝑢₂𝐹₁𝜆_𝜎1+1,𝜆_𝜎1+1;2;𝑢 (5.42) up to constant multiples. Therefore, we have 𝐹₁(𝑢)=𝐶⋅𝑢(1−𝑢)^𝜆𝜎1₂𝐹₁𝜆_𝜎1+1,𝜆_𝜎1+1;2;𝑢(𝐶∶constant). (5.43) Similarly, we have 𝐻₁(𝑢)=𝐶^′⋅(1−𝑢)^𝜆𝜎1₂𝐹₁𝜆_𝜎1+1,𝜆_𝜎1;1;𝑢𝐶^′∶constant. (5.44) We want to find the relation between 𝐶 and 𝐶^′. By expanding 𝐹₁(𝑢) and 𝐻₁(𝑢) around 𝑢=0, we have 𝐹₁(𝑢)𝜆=𝐶⋅𝑢+^𝜎21+12𝑢²+𝑢³𝑃₁(𝑢),𝐻₁(𝑢)=𝐶^′⋅1+𝜆²_𝜎1𝑢+𝑢²𝑃₂(𝑢), (5.45) where 𝑃₁(𝑢) and 𝑃₂(𝑢) are the analytic functions around 𝑢=0. By inserting these into (5.36), we have −3𝑢1−𝑢²𝐶^′𝜆²_𝜎1+𝑢𝑃₃(𝑢)=3(1+𝑢)𝜆_𝜎1𝐶𝑢+𝑢²𝑃₄(𝑢), (5.46) where 𝑃₃(𝑢) and 𝑃₄(𝑢) are also the analytic functions around 𝑢=0. By comparing the coefficients of 𝑢 of both sides, we have 𝐶=−𝜆_𝜎1𝐶^′. (5.47) Summing up, 𝐹₀(𝑎_𝑡) and 𝐻₀(𝑎_𝑡) are given by𝐹₀𝑎_𝑡=−𝐶^′𝜆_𝜎1cosh^−1/2(2𝑡)(sinh𝑡)⁻¹𝑢(1−𝑢)^𝜆𝜎1₂𝐹₁𝜆_𝜎1+1,𝜆_𝜎1,𝐻+1;2;𝑢₀𝑎_𝑡=𝐶^′cosh^−1/2(2𝑡)(cosh𝑡)⁻¹(1−𝑢)^𝜆𝜎1₂𝐹₁𝜆_𝜎1+1,𝜆_𝜎1;1;𝑢. (5.48)

(𝐶^′: some constant, 𝑢=tanh²𝑡). In our computation, we assumed that 𝜆_𝜎1≠0, but the result above holds without this assumption.

Next, we consider conditions (1), (2), and (3) in (3.8). For Ψ=^𝑡(𝐹₀,𝐺₀,𝐻₀)∈𝐶^∞_{sgn(𝑘),𝜏2}(𝐿), condition (1) is equivalent to (−1)^𝑘⋅^𝑡−𝐹₀𝑎_𝑡,𝐺₀𝑎_𝑡,−𝐻₀𝑎_𝑡=^𝑡𝐹₀𝑎_𝑡,𝐺₀𝑎_𝑡,𝐻₀𝑎_𝑡. (5.49) This is equivalent to 𝑘=0⟹𝐹₀≡0,𝐻₀≡0,𝑘=1⟹𝐺₀≡0. (5.50) Condition (2) is equivalent to^𝑡−𝐹₀𝑎_𝑡,−𝐺₀𝑎_𝑡,𝐻₀𝑎_𝑡=^𝑡𝐹₀𝑎_𝑡,𝐺₀𝑎_𝑡,𝐻₀𝑎_𝑡. (5.51) The solutions we have always satisfy this condition. Condition (3) is equivalent to ^𝑡cos𝜃𝐹₀(1)+sin𝜃𝐺₀(1),−sin𝜃𝐹₀(1)+cos𝜃𝐺₀(1),𝐻₀(1)=^𝑡𝐹₀(1),𝐺₀(1),𝐻₀(1),_𝑡cos𝜃𝐹₀(1)+sin𝜃𝐺₀(1),sin𝜃𝐹₀(1)−cos𝜃𝐺₀(1),−𝐻₀(1)=(−1)^𝑘⋅^𝑡𝐹₀(1),𝐺₀(1),𝐻₀(1). (5.52) (for all 𝜃∈𝐑). Since 𝐹₀(1)=𝐺₀(1)=0, (5.52) are equivalent to 𝑘=0⟹𝐻₀≡0. (5.53) But this condition holds if condition (5.50) is satisfied. We have obtained a result about the Shintani functions attached to the nonspherical principal series representation 𝜋_𝜎1,𝜈. Note that since the transform 𝜈₁↦𝜈₂,𝜈₂↦𝜈₁ does not change the eigenvalue of Casimir operator 𝜆₂ and changes the eigenvalue of gradient operator 𝜆_𝜎1 to 𝜆_𝜎2, this transform gives the result in case of 𝜎=𝜎₂. Similarly, the transform 𝜈₁↦−𝜈₁,𝜈₂↦−𝜈₁+𝜈₂ gives the result in case of 𝜎=𝜎₃. Summing up these results, we have the following theorem.

Theorem 5.6. Let 𝜂=sgn_(𝑘)(𝑘∈{0,1}) be a signature of 𝐻 defined by (5.38) and 𝜏₂ a three-dimensional tautological representation of 𝐾, and let Ψ=^𝑡(𝐹₀,𝐺₀,𝐻₀)∈𝐶^∞_{sgn(𝑘),𝜏2}(𝐿)_{𝜋𝜎1,𝜈} be a Shintani function corresponding to the nonspherical principal series representation 𝜋_𝜎1,𝜈 of 𝐺. Then, the restriction of Ψ to 𝐴 is given as follows.
(1)In case of 𝑘=0,(a) if 2𝜈₁−𝜈₂≠0, Ψ≡0;(b) if 2𝜈₁−𝜈₂=0, one has 𝐹₀𝑎_𝑡𝐺₀𝑎_𝑡𝐻₀𝑎_𝑡0𝑥=𝐶⋅^1/2(1−𝑥)^(2+𝜈2₂^)/4𝐹₁12𝜈₁1+1,2𝜈₂−12𝜈₁0.+1,;2;𝑥𝑥=tanh²(2𝑡),𝐶∶someconstant. (5.54)(2)In case of 𝑘=1, (a) if (−𝜈₁−𝜈₂)(−𝜈₁+2𝜈₂)≠0, Ψ≡0;(b) if (−𝜈₁−𝜈₂)(−𝜈₁+2𝜈₂)=0, one has 𝐹₀𝑎_𝑡𝐺₀𝑎_𝑡𝐻₀𝑎_𝑡=𝐶⋅−𝜆_𝜎1cosh^−1/2(2𝑡)(sinh𝑡)⁻¹𝑢(1−𝑢)^𝜆𝜎1₂𝐹₁𝜆_𝜎1+1,𝜆_𝜎10+1;2;𝑢cosh^−1/2(2𝑡)(cosh𝑡)⁻¹(1−𝑢)^𝜆𝜎1₂𝐹₁𝜆_𝜎1+1,𝜆_𝜎1.;1;𝑢𝑢=tanh²𝑡,𝐶∶someconstant. (5.55) Especially, in any case, one has dim𝐶^∞_{sgn(𝑘),𝜏2}(𝐿)_{𝜋𝜎1,𝜈}≤1. (5.56) The transform 𝜈₁↦𝜈₂,𝜈₂↦𝜈₁ gives the result in case of 𝜎=𝜎₂ and the transform 𝜈₁↦−𝜈₁,𝜈₂↦−𝜈₁+𝜈₂ gives the result in case of 𝜎=𝜎₃.

Next, we compute the Shintani functions Ψ∈𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎1,𝜈} for general unitary character 𝜂=𝜂_𝑠,𝑘 under the assumption that 1,𝜈₁,𝜈₂ are linearly independent over 𝐐. We have already known that the necessary condition of the existence of non zero Ψ is that the parameter 𝑠 is one of 2𝜈₁−𝜈₂,2𝜈₂−𝜈₁, or −𝜈₁−𝜈₂, and we have already solved the differential equations in case of 𝑠=2𝜈₁−𝜈₂. Hereafter, we suppose that the parameter 𝑠 is either 2𝜈₂−𝜈₁ or −𝜈₁−𝜈₂. We put 𝑤=(1−𝑢)/(1+𝑢), 𝐹₂(𝑤)=𝐹₁(𝑢), 𝐻₂(𝑤)=𝐻₁(𝑢) in (5.36), (5.37). Then we have −𝑤2−16𝑠𝐹₂(𝑤)()𝑑𝐻+𝑤1−𝑤²(𝑤)𝑑𝑤=𝜆_𝜎1𝐹₂(𝑤),𝑤()𝑑𝐹1+𝑤²(𝑤)+𝑤𝑑𝑤2+16𝑠𝐻₂(𝑤)=𝜆_𝜎1𝐻₂(𝑤). (5.57) We put 𝐹₂(𝑤)=^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝛼,𝐻₂(𝑤)=^∞_𝑛=0𝑐_𝑛𝑤^𝑛+𝛾 (5.58)  (𝛼,𝛾∈𝐂,𝑎₀,𝑐₀≠0) in (5.57) and compute the power series solutions. By inserting these series into (5.57), we have 𝑠6𝑎₀𝑤^𝛼+𝑤^𝛼+1𝑄₁(𝑤)+𝛾𝑐₀𝑤^𝛾+𝑤^𝛾+1𝑄₂(𝑤)=𝜆_𝜎1𝑎₀𝑤^𝛼+𝑤^𝛼+1𝑄₃(𝑤),𝛼𝑎₀𝑤^𝛼+𝑤^𝛼+1𝑄₄(𝑤)+𝑠6𝑐₀𝑤^𝛾+𝑤^𝛾+1𝑄₅(𝑤)=𝜆_𝜎1𝑐₀𝑤^𝛾+𝑤^𝛾+1𝑄₆(𝑤), (5.59) where 𝑄_𝑖(𝑤)(𝑖=1,…,6) are the analytic functions around 𝑤=0. By comparing the lowest terms in power series, easily we have 𝛼=𝛾 (in this argument, we use the fact that 1,𝜈₁,𝜈₂ are linearly independent over 𝐐 carefully). Therefore, from (5.59), we have 𝑠6−𝜆_𝜎1+𝛾𝑐₀=0,𝛼𝑎₀+𝑠6−𝜆_𝜎1𝑐₀=0. (5.60) Since (𝑎₀,𝑐₀)≠(0,0), we have 𝑠6−𝜆_𝜎1²−𝛼𝛾=0. (5.61) By combining this and 𝛼=𝛾, we have 𝑠𝛼=𝛾=±6−𝜆_𝜎1. (5.62) Hereafter, we put 𝐴=𝐴(𝜈₁,𝜈₂)=(𝑠/6)−𝜆_𝜎1. Then, 𝐹₂(𝑤) (resp., 𝐻₂(𝑤)) are expressed by the linear combination of some power series ∑^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝐴, ∑^∞_𝑛=0𝑎^′_𝑛𝑤^𝑛−𝐴 (resp., ∑^∞_𝑛=0𝑐_𝑛𝑤^𝑛+𝐴, ∑^∞_𝑛=0𝑐^′_𝑛𝑤^𝑛−𝐴). That is, there exist common constants 𝐶₊, 𝐶₋ such that 𝐹₂(𝑤)=𝐶₊^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝐴+𝐶₋^∞_𝑛=0𝑎^′_𝑛𝑤^𝑛−𝐴,𝐻₂(𝑤)=𝐶₊^∞_𝑛=0𝑐_𝑛𝑤^𝑛+𝐴+𝐶₋^∞_𝑛=0𝑐^′_𝑛𝑤^𝑛−𝐴. (5.63) By inserting 𝐹₂(𝑤)=^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝐴,𝐻₂(𝑤)=^∞_𝑛=0𝑐_𝑛𝑤^𝑛+𝐴 (5.64) into (5.57) and picking up the coefficients of 𝑤^𝑛+𝐴, we have the following recurrence relations: −𝑠2𝑎_𝑛−1+𝐴𝑎_𝑛−(𝑛+𝐴−1)𝑐_𝑛−1+(𝑛+𝐴)𝑐_𝑛=0,(𝑛+𝐴−1)𝑎_𝑛−1+(𝑛+𝐴)𝑎_𝑛+𝑠2𝑐_𝑛−1+𝐴𝑐_𝑛=0, (5.65)(5.66) for all 𝑛≥0. Here, we assume that 𝑎_𝑙=𝑐_𝑙=0 if 𝑙<0. From (5.65), (5.66), easily we have 𝑐_𝑛=(−1)^𝑛+1𝑎_𝑛 for all 𝑛≥0 by induction. Therefore, by inserting 𝑐_𝑛=(−1)^𝑛+1𝑎_𝑛,𝑐_𝑛−1=(−1)^𝑛𝑎_𝑛−1 into (5.65), we have 𝐴+(𝑛+𝐴)(−1)^𝑛+1𝑎_𝑛=𝑠2+(𝑛+𝐴−1)(−1)^𝑛𝑎_𝑛−1. (5.67) Thus, we have 𝑎_𝑛=^𝑛_𝑘=1𝑠/2+(𝑛+𝐴−1)(−1)^𝑘𝐴+(𝑘+𝐴)(−1)^𝑘+1𝑎₀,𝑐_𝑛=(−1)^𝑛+1𝑛_𝑘=1𝑠/2+(𝑛+𝐴−1)(−1)^𝑘𝐴+(𝑘+𝐴)(−1)^𝑘+1𝑎₀. (5.68)(5.69) Similarly, if the characteristic roots are 𝛼=𝛾=−𝐴, by inserting 𝐹₂(𝑤)=^∞_𝑛=0𝑎^′_𝑛𝑤^𝑛−𝐴,𝐻₂(𝑤)=^∞_𝑛=0𝑐^′_𝑛𝑤^𝑛−𝐴 (5.70) into (5.57), we have 𝑎^′_𝑛=^𝑛_𝑘=1𝑠/2+(𝑛−𝐴−1)(−1)^𝑘𝐴+(𝑘−𝐴)(−1)^𝑘+1𝑎₀,𝑐^′_𝑛=(−1)^𝑛𝑛_𝑘=1𝑠/2+(𝑛−𝐴−1)(−1)^𝑘𝐴+(𝑘−𝐴)(−1)^𝑘+1𝑎₀. (5.71) (for all 𝑛≥1). From (5.68), 𝑎_2𝑛 and 𝑎_2𝑛−1 are expressed by𝑎_2𝑛=(𝑠/4+𝐴/2+1/2)_𝑛(−𝑠/4+𝐴/2)_𝑛𝑛!(𝐴+1/2)_𝑛𝑎₀,𝑎_2𝑛−1()=−𝑠/4+𝐴/2+1/2_𝑛−1(−𝑠/4+𝐴/2)_𝑛(𝑛−1)!(𝐴+1/2)_𝑛𝑎₀. (5.72) Here, for 𝛼∈𝐂,𝑛∈𝐙_>0, we define (𝛼)_𝑛=𝛼(𝛼+1)(𝛼+2)⋯(𝛼+𝑛−1) and (𝛼)₀∶=1. Therefore, if we normalize 𝑎₀=1, we have ^∞_𝑛=0𝑎_2𝑛𝑤^2𝑛=^∞_𝑛=0(𝑠/4+𝐴/2+1/2)_𝑛(−𝑠/4+𝐴/2)_𝑛𝑛!(𝐴+1/2)_𝑛𝑤^2𝑛=₂𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴21;𝐴+2;𝑤²,_∞𝑛=1𝑎_2𝑛−1𝑤^2𝑛−1=−−𝑠/4+𝐴/2𝐴+1/2^∞_𝑛=1(𝑠/4+𝐴/2+1/2)_𝑛−1(−𝑠/4+𝐴/2+1)_𝑛−1(𝑛−1)!(𝐴+3/2)_𝑛−1𝑤^2𝑛−2=⋅𝑤𝑠/4−𝐴/2𝑤𝐴+1/2^∞_𝑛=0(𝑠/4+𝐴/2+1/2)_𝑛(−𝑠/4+𝐴/2+1)_𝑛𝑛!(𝐴+3/2)_𝑛𝑤^2𝑛=𝑠/4−𝐴/2𝑤𝐴+1/2²𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴23+1;𝐴+2;𝑤². (5.73) Thus, we have ^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝐴=𝑤^𝐴₂𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴21;𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤𝐴+1/2²𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴23+1;𝐴+2;𝑤². (5.74) And since 𝑐_𝑛=(−1)^𝑛+1𝑎_𝑛, we have ^∞_𝑛=0𝑐_𝑛𝑤^𝑛+𝐴=𝑤^𝐴−₂𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴21;𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤𝐴+1/2²𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴23+1;𝐴+2;𝑤². (5.75) Similarly, by using 𝑎^′_2𝑛=(−𝑠/4−𝐴/2+1/2)_𝑛(𝑠/4−𝐴/2)_𝑛𝑛!(−𝐴+1/2)_𝑛𝑎^′₀,𝑎^′_2𝑛−1()=−−𝑠/4−𝐴/2+1/2_𝑛−1(𝑠/4−𝐴/2)_𝑛(𝑛−1)!(−𝐴+1/2)_𝑛𝑎^′₀, (5.76) and 𝑐^′_𝑛=(−1)^𝑛𝑎^′_𝑛, if we normalize 𝑎^′₀=1, we have ^∞_𝑛=0𝑎^′_𝑛𝑤^𝑛−𝐴=𝑤^−𝐴₂𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴21;−𝐴+2;𝑤²−𝑠/4−𝐴/2𝑤−𝐴+1/2²𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴23+1;−𝐴+2;𝑤²,_∞𝑛=0𝑐^′_𝑛𝑤^𝑛−𝐴=𝑤^−𝐴₂𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴21;−𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤−𝐴+1/2²𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴23+1;−𝐴+2;𝑤². (5.77) Therefore, 𝐹₂(𝑤) and 𝐻₂(𝑤) are expressed as follows: 𝐹₂(𝑤)=𝐶₊𝑤^𝐴₂𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴21;𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤𝐴+1/2²𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴23+1;𝐴+2;𝑤²+𝐶₋𝑤^−𝐴₂𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴21;−𝐴+2;𝑤²−𝑠/4−𝐴/2𝑤−𝐴+1/2²𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴23+1;−𝐴+2;𝑤²,𝐻₂(𝑤)=𝐶₊𝑤^𝐴−₂𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴21;𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤𝐴+1/2²𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴23+1;𝐴+2;𝑤²+𝐶₋𝑤^−𝐴₂𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴21;−𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤−𝐴+1/2²𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴23+1;−𝐴+2;𝑤². (5.78) We want the relation between 𝐶₊ and 𝐶₋. We can find the relation by using the regularity of the Shintani function and the asymptotic formula of Gaussian hypergeometric function ₂𝐹₁. Now, since 𝐹₀(𝑎_𝑡) and 𝐻₀(𝑎_𝑡) are regular around 𝑡=0(⇔𝑤=1) and since 𝐹(𝑡)=cosh^1/2(2𝑡)sinh𝑡𝐹₀𝑎_𝑡,𝐻(𝑡)=cosh^1/2(2𝑡)cosh𝑡𝐻₀𝑎_𝑡, (5.79)𝐹₂(𝑤)=𝐹(𝑡) and 𝐻₂(𝑤)=𝐻(𝑡) must be regular around 𝑤=1 and 𝐹₂(𝑤) must satisfy 𝐹₂(1)=0. Since all hypergeometric functions appearing in the right-hand sides of (5.78) are in the form of ₂𝐹₁(𝑎,𝑏;𝑎+𝑏;𝑤²), to investigate the behavior of the right-hand sides of (5.78) around 𝑤=1, we use the following asymptotic formula.

Formula 1
We have ₂𝐹₁()Γ()()(𝑎)(𝑏)𝑎,𝑏;𝑎+𝑏;𝑧=−𝑎+𝑏log1−𝑧+𝜓+𝜓+2𝛾Γ(𝑎)Γ(𝑏)(1+𝑂(𝑧−1))(𝑧⟶1). (5.80) Here, 𝛾 is the Euler constant and 𝜓 is defined by 𝜓(𝑧)=^∞_𝑘=11𝑘−1𝑘+𝑧−1−𝛾. (5.81)

We apply this formula to the right-hand sides of (5.78). Firstly, the coefficient of log(1−𝑤²) of 𝐹₂(𝑤) equals 𝐶₊−Γ(𝐴+1/2)Γ()Γ()−𝑠/4+𝐴/2+1/2−𝑠/4+𝐴/2𝑠/4−𝐴/2Γ()𝐴+1/2𝐴+3/2Γ(𝑠/4+𝐴/2+1/2)Γ(−𝑠/4+𝐴/2+1)+𝐶₋−Γ(−𝐴+1/2)Γ()Γ()+−𝑠/4−𝐴/2+1/2𝑠/4−𝐴/2𝑠/4−𝐴/2Γ()−𝐴+1/2−𝐴+3/2Γ(−𝑠/4−𝐴/2+1/2)Γ(𝑠/4−𝐴/2+1)=𝐶₊⋅0+𝐶₋⋅0=0. (5.82) Therefore, 𝐹₂(𝑤) is regular around 𝑤=1 regardless of the values of 𝐶₊, 𝐶₋. Next, the coefficient of log(1−𝑤²) of 𝐻₂(𝑤) equals 𝐶₊Γ(𝐴+1/2)Γ()Γ()−𝑠/4+𝐴/2+1/2−𝑠/4+𝐴/2𝑠/4−𝐴/2Γ()𝐴+1/2𝐴+3/2Γ(𝑠/4+𝐴/2+1/2)Γ(−𝑠/4+𝐴/2+1)+𝐶₋−Γ(−𝐴+1/2)Γ()Γ()−−𝑠/4−𝐴/2+1/2𝑠/4−𝐴/2𝑠/4−𝐴/2Γ()−𝐴+1/2−𝐴+3/2Γ()Γ()=()−𝑠/4−𝐴/2+1/2𝑠/4−𝐴/2+12Γ𝐴+1/2Γ()Γ()𝐶𝑠/4+𝐴/2+1/2−𝑠/4+𝐴/2⁺−2Γ(−𝐴+1/2)Γ()Γ()𝐶−𝑠/4−𝐴/2+1/2𝑠/4−𝐴/2⁻. (5.83) Since this coefficient must be 0, we have 𝐶₊=Γ(−𝐴+1/2)Γ()Γ(),𝐶−𝑠/4−𝐴/2+1/2𝑠/4−𝐴/2₋=Γ(𝐴+1/2)Γ(𝑠/4+𝐴/2+1/2)Γ(−𝑠/4+𝐴/2) (5.84) up to (the same) constant multiples. Note that we can easily verify that 𝐹₂(1)=0 for these 𝐶₊, 𝐶₋ by using formulas Γ(𝑧+1)=𝑧Γ(𝑧),𝜓(𝑧+1)=𝜓(𝑧)+1/𝑧. Now, we have completely determined 𝐹₂(𝑤),𝐻₂(𝑤). We have 𝐹₂(𝑤)=Γ(−𝐴+1/2)Γ()Γ()𝑤−𝑠/4−𝐴/2+1/2𝑠/4−𝐴/2^𝐴×₂𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴21;𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤𝐴+1/2²𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴23+1;𝐴+2;𝑤²+Γ(𝐴+1/2)Γ()Γ()𝑤𝑠/4+𝐴/2+1/2−𝑠/4+𝐴/2^−𝐴×₂𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴21;−𝐴+2;𝑤²−𝑠/4−𝐴/2𝑤−𝐴+1/2²𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴23+1;−𝐴+2;𝑤²,𝐻₂(𝑤)=Γ(−𝐴+1/2)Γ()Γ()𝑤−𝑠/4−𝐴/2+1/2𝑠/4−𝐴/2^𝐴×−₂𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴21;𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤𝐴+1/2²𝐹₁𝑠4+𝐴2+12𝑠,−4+𝐴23+1;𝐴+2;𝑤²+Γ(𝐴+1/2)Γ()Γ()𝑤𝑠/4+𝐴/2+1/2−𝑠/4+𝐴/2^−𝐴×₂𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴21;−𝐴+2;𝑤²+𝑠/4−𝐴/2𝑤−𝐴+1/2²𝐹₁−𝑠4−𝐴2+12,𝑠4−𝐴23+1;−𝐴+2;𝑤² (5.85) up to (the same) constant multiples. Here, 𝐴=𝐴(𝜈₁,𝜈₂)=𝑠/6−𝜆_𝜎1, 𝜆_𝜎1=−(1/3)(2𝜈₁−𝜈₂), 𝑤=(1−𝑢)/(1+𝑢)=(1−tanh²𝑡)/(1+tanh²𝑡), 𝐹₂(𝑤)=𝐹(𝑡)=cosh^1/2(2𝑡)sinh𝑡𝐹₀(𝑎_𝑡), 𝐻₂(𝑤)=𝐻(𝑡)=cosh^1/2(2𝑡)cosh𝑡𝐻₀(𝑎_𝑡). By the same argument we have done when 𝜂 is the signature, we can easily verify that conditions (1), (2), and (3) in (3.8) are equivalent to condition (5.50). We have already computed 𝐹₀(𝑎_𝑡), 𝐺₀(𝑎_𝑡), and 𝐻₀(𝑎_𝑡) in any case. Summing up, we obtain the following theorem.

Theorem 5.7. Assume that 1,𝜈₁,𝜈₂ are linearly independent over 𝐐. Let 𝜂=𝜂_s,𝑘 be a unitary character of 𝐻 defined by (3.1) and 𝜎=𝜎₁ the character of 𝑀. Then, the necessary condition of the existence of the nontrivial Shintani functions attached to the nonspherical principal series representation 𝜋_𝜎1,𝜈 is that 𝑘=0,𝑠=2𝜈₁−𝜈₂ (5.86) or 𝑘=1,𝑠=−𝜈₁−𝜈₂or−𝜈₁+2𝜈₂. (5.87) That is, if the condition above is not satisfied, one has 𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎1,𝜈}={0}. (5.88) Let Ψ(𝑎_𝑡)=^𝑡(𝐹₀(𝑎_𝑡),𝐺₀(𝑎_𝑡),𝐻₀(𝑎_𝑡))∈𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎1,𝜈}|_𝐴 and suppose that the condition above is satisfied. Then,(1)if 𝑘=0 and 𝑠=2𝜈₁−𝜈₂, one has 𝐹₀𝑎_𝑡𝐺₀𝑎_𝑡𝐻₀𝑎_𝑡0𝑥=𝐶⋅^1/2(1−𝑥)^(2+𝜈2₂^)/4𝐹₁12𝜈₁1+1,2𝜈₂−12𝜈₁0.+1,;2;𝑥𝑥=tanh²(2𝑡),𝐶∶someconstant; (5.89)(2)if 𝑘=1 and 𝑠=−𝜈₁−𝜈₂ or −𝜈₁+2𝜈₂, one has 𝐹₀𝑎_𝑡𝐺₀𝑎_𝑡𝐻₀𝑎_𝑡=𝐶⋅cosh^−1/2(2𝑡)sinh⁻¹𝑡𝐹₂(𝑤)0cosh^−1/2(2𝑡)cosh⁻¹𝑡𝐻₂(𝑤), (5.90)where 𝐶 is some constant and 𝐹₂(𝑤) and 𝐻₂(𝑤) are the functions given by (5.85). Especially, in any case, one hasdim𝐶^∞_𝜂,𝜏2(𝐿)_{𝜋𝜎1,𝜈}≤1. (5.91) The transform 𝜈₁↦𝜈₂,𝜈₂↦𝜈₁ gives the result in case of 𝜎=𝜎₂, and the transform 𝜈₁↦−𝜈₁,𝜈₂↦−𝜈₁+𝜈₂ gives the result in case of 𝜎=𝜎₃.

Remark 5.8. By using the relation 1−𝑥=𝑤² and the formulas of the hypergeometric function Γ(𝛼)Γ𝛽Γ𝛾²𝐹₁=Γ(𝛼)Γ𝛼,𝛽;𝛾;𝑧𝛽−𝛼Γ()𝛾−𝛼1−𝑧₂^−𝛼𝐹₁1𝛼,𝛾−𝛽;𝛼−𝛽+1;+Γ𝛽Γ1−𝑧𝛼−𝛽Γ()𝛾−𝛽1−𝑧₂^−𝛽𝐹₁1𝛽,𝛾−𝛼;𝛽−𝛼+1;,1−𝑧₂𝐹₁𝛼,𝛽,𝛾;𝑧=(1−𝑧)₂^−𝛼𝐹₁𝑧𝛼,𝛾−𝛽;𝛾;=()𝑧−11−𝑧₂^{𝛾−𝛼−𝛽}𝐹₁𝛾−𝛼,𝛾−𝛽;𝛾;𝑧, (5.92) we can rewrite 𝐹₂(𝑤), 𝐻₂(𝑤) in Theorem 5.7 as functions in 𝑥=tanh²(2𝑡). We put 𝐹₂(𝑤)=𝐹₃(𝑥), 𝐻₂(𝑤)=𝐻₃(𝑥). Then, we have 𝐹₃(𝑥)=−𝑠4+𝐴2−(1−𝑥)₂^𝐴/2𝐹₁𝑠4+𝐴+12𝑠,−4+𝐴2+();1;𝑥1−𝑥₂^(𝐴+1)/2𝐹₁𝑠4+𝐴+12𝑠,−4+𝐴2,𝐻+1;1;𝑥₃(𝑥)=−𝑠4+𝐴2(1−𝑥)₂^𝐴/2𝐹₁𝑠4+𝐴+12𝑠,−4+𝐴2+();1;𝑥1−𝑥₂^(𝐴+1)/2𝐹₁𝑠4+𝐴+12𝑠,−4+𝐴2+1;1;𝑥. (5.93) These computations are due to Professor T. Ishii.

Acknowledgments

The author would like to express his gratitude to Professor T. Oda for his constant encouragement and many advices. He also thanks Professor M. Hirano, T. Ishii, T. Moriyama, M. Tsuzuki, and T. Miyazaki for helpful discussion.