Shintani Functions on SL 3 , R

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.


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 International Journal of Mathematics and Mathematical Sciences on G.We take 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 g C ,K π, C ∞ η H \G and its restriction 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 : 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.
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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 t g −1 g ∈ G , and its differential dθ : g → g is given by dθ X − t X 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 We define 1 and −1 eigenspaces of dθ, dσ in g by 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 Next, we take as a basis of h.We have p ∩ q RX 13 ⊕ RX 23 , and we take a RX 13 as a maximal Abelian subspace of p ∩ q.We define a subgroup A of G by 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 ⊗ R C.

The Principal Series Representations
As a representation of G, we take the principal series representation defined as follows.Let P 0 be a minimal parabolic subgroup of G given by the upper triangular matrices in G and P 0 MA P 0 N the Langlands decomposition of P 0 with

2.6
To define a principal series representation with respect to the minimal parabolic subgroup P 0 of G, we firstly fix a character σ of M and a linear form ν ∈ a * P 0 ⊗ R C Hom R a P 0 , C , where a P 0 is the Lie algebra of A P 0 .We write for diag t 1 , t 2 , t 3 ∈ a P 0 .Then, we can define a representation σ ⊗ a ν of MA P 0 and extend this to P 0 by the identification P 0 /N MA P 0 , taking the trivial representation 1 N as the representation of N.Then, the induced representation is called the principal series representation of G. Here, ρ is the half sum of positive roots of g, a given by a ρ a 2 1 a 2 , for a diag a 1 , a 2 , a 3 ∈ A P 0 .Concretely, the representation space is given by and the action of G is defined by Here, for g ∈ G, g n g a g κ g n g ∈ N, a g ∈ A P 0 , κ g ∈ K is the Iwasawa decomposition.Throughout this paper, we assume that the representation π σ,ν is irreducible.Moreover, we assume that ν 1 , ν 2 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 by The set M consists of 4 characters {σ j | j 0, 1, 2, 3}, where σ 0 is the trivial character of M and σ 1 , σ 2 , σ 3 are defined by 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 τ 2 , V 2 .The multiplicity of this minimal K-type is one:

The Definition of Shintani Functions
As a representation of H, we take the unitary character η η s,k : 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 G acts on this space by right translation.Let π σ,ν be the principal series representation of G.
We consider the intertwining space

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We denote its image by S η,π σ,ν , that is, S η,π σ,ν : Image T .

3.4
We call the element of S η,π σ,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 gives the restriction of T ∈ S η,π σ,ν to τ, where τ * is the contragredient representation of τ and the space C ∞ η,τ * L is defined by We denote the image of , and the element of this space is called the Shintani function of type π σ,ν , η, τ .

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.

3.7
Here, I is the unit element of GL 3, R .

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Let w 0 : diag −1, −1, 1 .Then, w 0 Z 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 finitedimensional representation of K. We denote by C ∞ W A; η, τ the space of smooth functions F : A → V τ satisfying the following conditions:

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: Through this isomorphism, we denote the image of . This is our target space in this paper.The following two lemmas are obvious.

Shintani Functions Attached to the Spherical Principal Series Representations
Throughout this section, as a character of M, we take the trivial character σ σ 0 .Then, the principal series representation π σ 0 ,ν is the spherical or class one principal series representation whose minimal

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 sl 3 is of type A 2 see 13 .For i 1, 2, 3, we put

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The following proposition gives the explicit description of the independent generators of Z g see 11 .
Proposition 4.1.The independent generators {Cp 2 , Cp 3 } of Z g are given as follows:

4.2
Since Cp 2 , Cp 3 are the elements of Z g , they act on π σ 0 ,ν as the scalar operators.And since the space of Shintani functions is the image of the g C , K -homomorphism of π σ 0 ,ν , they act on the space of Shintani functions as the same scalar operators, respectively.

Eigenvalues of Cp 2 , Cp 3
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 Cp 2 , Cp 3 .For the spherical principal series representation, σ σ 0 is the trivial character of M. Let f 0 be the generator of the minimal K-type in H σ 0 ,ν normalized such that f 0 | K ≡ 1.The actions of Cp 2 , Cp 3 on f 0 are computed in 11 , and the result is as follows.
Proposition 4.2.The Capelli elements Cp 2 , Cp 3 act on f 0 by scalar multiples, and the eigenvalues are given as follows:

Construction of the Casimir Equations
Next, we compute the actions of Cp 2 , Cp 3 on

4.4
International Journal of Mathematics and Mathematical Sciences 9 Proof.By definition, we have

4.5
Since exp uY 1 diag e u , 1, e −u , we have η exp uY 1 e ku • e s−k u e su .Therefore, we have The computations of the actions of Ad a −1 t Y i i 3, 4 are similar.Finally, since exp uX 13 a u , we have By simple computations of matrices, we have the following expressions of the elements in M 3, R .

Lemma 4.4. One has
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4.10
To make use of Lemma 3.4, we have to rewrite Cp 2 , Cp 3 in the form of linear combinations of the elements in Ad a −1 t h ak.To do this, we use the following formulas which can be obtained by direct computation.Lemma 4.5.One has International Journal of Mathematics and Mathematical Sciences 11

4.11
Here, X, Y : XY − Y X is the Lie bracket on g.
By using Lemma 4.5, we can rewrite Cp 2 , Cp 3 as we wished.Now, since for all Lemma 4.6.One has the congruences are the eigenvalues of the Capelli elements on principal series representations.Equations 4.15 − 4.14 × 1/3 s give Therefore, if F a t is not identically zero, we have By solving this equation, we have 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

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 The value of μ ∈ C is as follows.

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For this μ, the left-hand side of 4.21 is divided by 1 − x μ 1 , and the equation becomes 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, G 0 x 1 − x −μ F 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 where 2 F 1 is the Gaussian hypergeometric function and α, β ∈ C are defined by

4.25
Explicitly, by solving these equations, α, β are given as follows. 1 Finally, we consider the three conditions in 3.8 .Condition 1 is equivalent to −1 k F a t F a t .Condition 2 always holds.Condition 3 is equivalent to −1 k F 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

4.26
Suppose that this condition is satisfied and nontrivial Shintani functions exist.If one puts x

given as follows (up to constant multiples).
1 In case of s 2ν 1 − ν 2 , one has

4.28
International Journal of Mathematics and Mathematical Sciences 3 In case of s −ν 1 − ν 2 , one has

4.29
Especially, one has

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 σ i ,ν is the three-dimensional representation of K which is isomorphic to the tautological representation τ 2 : K SO 3, R → GL 3, R which occurs of multiplicity one in π σ i ,ν | K .We take τ * 2 instead of τ 2 as a minimal K-type of π σ i ,ν .The representation space of τ 2 is denoted by V τ 2 R 3 , and we take s 1 t 1, 0, 0 , s 2 t 0, 1, 0 , s 3 t 0, 0, 1 as a basis of V τ 2 .Let Ψ ∈ C ∞ η,τ 2 L π σ i ,ν be the Shintani function.Then, Ψ is expressed by 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 .

The Casimir Equation
Firstly, we construct the Casimir equation of degree two.Since the Capelli element Cp 2 acts on the representation space of the principal series representation π σ i ,ν as a scalar operator λ 2multiple and the space of Shintani functions C ∞ η,τ 2 L π σ i ,ν is the image of g C , K -homomorphism of π σ i ,ν , Cp 2 acts on this space as the same scalar operator.Since for all Ψ a t ∈ C ∞ η,τ 2 L | A is annihilated by the action of Ad a −1 t Y 2 U g , Ad a −1 t Y 3 U g and the actions of Ad a −1 t Y 1 and Ad a −1 t Y 4 on F are the same the multiplication by s , we may regard Cp 2 as the element in U g mod P , where P is a subalgebra of U g defined by By using Lemmas 4.4 and 4.5, we can rewrite Cp 2 in Proposition 4.1 as follows.

5.3
By using this lemma, the action of Cp 2 on Ψ a t F 0 a t s 1 G 0 a t s 2 H 0 a t s 3 ∈ C ∞ η,τ 2 L | A can be computed easily.We have where

The Gradient Equation
For the spherical function Ψ g ∈ C ∞ η,τ 2 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 Here, X * i is the dual basis of X i with respect to the inner product X, Y ∈ p × p → Tr XY ∈ C.

5.10
We rewrite this by using the basis of p C .
Claim 1.We define five elements

5.11
Then, {w i | 0 ≤ i ≤ 4} becomes the basis of p C .
With this basis, the gradient operator ∇ R is rewritten as

5.12
The Lie algebra p C becomes the representation space of the adjoint action of K. We denote this representation by τ 4 , W 4 .By the Clebsch-Gordan theorem, τ 2 ⊗ τ 4 has the irreducible decomposition

5.13
Here, each τ n is the n 1 -dimensional irreducible representation of K.In this decomposition, the projector of K-modules

5.15
Since the minimal K-type τ * 2 occurs of multiplicity one, p r 2 • ∇ R is a map of constant multiple.To compute the action of the gradient operator p r 2 • ∇ R on the space of the Shintani functions C ∞ η,τ 2 L π σ i ,ν , we have to decompose w i i 0, 1, 2, 3, 4 along the decomposition

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Lemma 5.4.One has

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 where

5.18
On the other hand, the eigenvalue of the gradient operator on the spherical functions of the principal series representation π σ i ,ν depends on the choice of σ i , denoted by λ σ i i 1, 2, 3 .These values are computed in 11 and they are as follows: we have the following three differential equations.

5.23
We put Then, the equation becomes

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

5.25
The solutions are α ±1/2, β 2 ± ν 2 /4.We choose α 1/2, β 2 ν 2 /4.Then the left-hand side of 5.24 can be divided by x 1 − x , and the equation becomes This is a Gaussian hypergeometric differential equation, and its regular solution is given by up to constant multiples.The other solutions are not regular around x 0, since they contain log x.Therefore, we have where C is a constant number and x tanh 2 2t .Next, we consider the equations satisfied by F 0 and H 0 .From 5.20 and 5.22 , we have

5.29
By differentiating both sides of 5.29 by t, we have

5.30
By inserting 5.29 and 5.30 into the Casimir equation 5.6 , 5.8 to eliminate the differential terms, we have

5.31
Therefore, if the parameter International Journal of Mathematics and Mathematical Sciences 21 Therefore, we have

5.33
That is, 5.33 is the necessary condition of the existence of nontrivial F 0 a t , H 0 a t .We put and insert these into 5.20 , 5.22 .Then we have

5.35
Next, we put tanh 2 t u, F t F 1 u , H t H 1 u .Then, the above equations become

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 Then, by combining 5.36 , 5.37 , we have

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We put Equation 5.41 is the Gaussian hypergeometric differential equation.Now, since F 2 u 1 − u −λ σ 1 cosh 1/2 2t sinh tF 0 a t and F 0 a t is regular at a t 1 ⇔ t 0 ⇔ u 0 , F 2 u must be regular at u 0. The regular solution of 5.41 is given by up to constant multiples.Therefore, we have Similarly, we have We want to find the relation between C and C .By expanding F 1 u and H 1 u around u 0, we have

5.45
where P 1 u and P 2 u are the analytic functions around u 0. By inserting these into 5.36 , we have where P 3 u and P 4 u are also the analytic functions around u 0. By comparing the coefficients of u of both sides, we have

5.47
Summing up, F 0 a t and H 0 a t are given by

5.48
International Journal of Mathematics and Mathematical Sciences 23 C : some constant, u tanh2 t .In our computation, we assumed that λ σ 1 / 0, but the result above holds without this assumption.

5.49
This is equivalent to

5.50
Condition 2 is equivalent to

5.51
The solutions we have always satisfy this condition.Condition 3 is equivalent to for all θ ∈ R .Since F 0 1 G 0 1 0, 5.52 are equivalent to k 0 ⇒ H 0 ≡ 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 σ 1 ,ν .Note that since the transform ν 1 → ν 2 , ν 2 → ν 1 does not change the eigenvalue of Casimir operator λ 2 and changes the eigenvalue of gradient operator λ σ 1 to λ σ 2 , this transform gives the result in case of σ σ 2 .Similarly, the transform ν 1 → −ν 1 , ν 2 → −ν 1 ν 2 gives the result in case of σ σ 3 .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 τ 2 a three-dimensional tautological representation of K, and let Ψ t F 0 , G 0 , H 0 ∈ C ∞ sgn k ,τ 2 L π σ 1 ,ν be a Shintani function corresponding to the nonspherical principal series representation π σ 1 ,ν of G.Then, the restriction of Ψ to A is given as follows.

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x tanh 2 2t , C : some constant .

5.55
Especially, in any case, one has
Next, we compute the Shintani functions Ψ ∈ C ∞ η,τ 2 L π σ 1 ,ν for general unitary character η η s,k under the assumption that 1, ν 1 , ν 2 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 2ν 1 − ν 2 , 2ν 2 − ν 1 , or −ν 1 − ν 2 , and we have already solved the differential equations in case of s 2ν 1 − ν 2 .Hereafter, we suppose that the parameter s is either 2ν .36 , 5.37 .Then we have

5.57
We put c n w n γ 5.58 α, γ ∈ C, a 0 , c 0 / 0 in 5.57 and compute the power series solutions.By inserting these series into 5.57 , we have

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, ν 1 , ν 2 are linearly independent over Q carefully .Therefore, from 5.59 , we have

5.61
By combining this and α γ, we have

5.62
Hereafter, we put A A ν 1 , ν 2 s/6 − λ σ 1 .Then, F 2 w resp., H 2 w are expressed by the linear combination of some power series ∞ n 0 a n w n A , ∞ n 0 a n w n−A resp., ∞ n 0 c n w n A , ∞ n 0 c n w n−A .That is, there exist common constants C , C − such that

5.63
By inserting International Journal of Mathematics and Mathematical Sciences into 5.57 and picking up the coefficients of w n A , we have the following recurrence relations: 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 1 a n for all n ≥ 0 by induction.Therefore, by inserting c n −1 n 1 a n , c n−1 −1 n a n−1 into 5.65 , we have

5.67
Thus, we have

5.69
Similarly, if the characteristic roots are α γ −A, by inserting c n w n−A 5.70 into 5.57 , we have

5.71
for all n ≥ 1 .From 5.68 , a 2n and a 2n−1 are expressed by

5.73
Thus, we have

5.74
And since c n −1 n 1 a n , we have

5.75
Similarly, by using

5.77
Therefore, F 2 w and H 2 w are expressed as follows:

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 0. Since all hypergeometric functions appearing in the right-hand sides of 5.78 are in the form of 2 F 1 a, b; a b; w 2 , 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 2

5.80
Here, γ is the Euler constant and ψ is defined by

5.81
We apply this formula to the right-hand sides of 5.78 .Firstly, the coefficient of log 1− w 2 of F 2 w equals H t cosh 1/2 2t cosh tH 0 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 F 0 a t , G 0 a t , and H 0 a t in any case.Summing up, we obtain the following theorem.

5.88
Let Ψ a t t F 0 a t , G 0 a t , H 0 a t ∈ C ∞ η,τ 2 L π σ 1 ,ν | A and suppose that the condition above is satisfied.Then, where C is some constant and F 2 w and H 2 w are the functions given by 5.85 .Especially, in any case, one has dim C ∞ η,τ 2 L π σ 1 ,ν ≤ 1.
Remark 5.8.By using the relation 1 − x w 2 and the formulas of the hypergeometric function

5.93
These computations are due to Professor T. Ishii.

Theorem 5 . 7 .
Assume that 1, ν 1 , ν 2 are linearly independent over Q.Let η η s,k be a unitary character of H defined by 3.1 and σ σ 1 the character of M.Then, the necessary condition of the existence of the nontrivial Shintani functions attached to the nonspherical principal series representation π σ 1 ,ν is that s −ν 1 − ν 2 or − ν 1 2ν 2 .
and the actions of Ad a −1 t Y 1 and Ad a −1 t Y 4 on F are the same the multiplication by s , we may regard Cp 2 , Cp 3 as the elements in U g mod P , where P is the subalgebra of U g defined by Since Ψ a t satisfies Cp 2 Ψ a t λ 2 Ψ a t , we have the following three differential equations.For Ψ a t F 0 a t s 1 G 0 a t s 2 H 0 a t s 3 ∈ C ∞ η,τ 2 L π σ i ,ν | A ,the functions F 0 , G 0 , H 0 satisfy the following equations: