International Journal of Mathematics and Mathematical SciencesVolume 2011 (2011), Article ID 842806, 33 pagesdoi:10.1155/2011/842806

Research Article

Shintani Functions on

Keiju Sono

Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan

Received 25 May 2011; Revised 26 September 2011; Accepted 26 September 2011

Academic Editor: Andrei Volodin

Copyright © 2011 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.

Abstract.

We investigate the Shintani functions attached to the spherical and nonspherical principal series representations of . 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 -adic linear group , where is a finite extension of the -adic field [1]. He defined some “Whittaker function” on and obtained the explicit formulas of them. Moreover, he proved the uniqueness of his function. Later, a more detail study of Shintani functions for was done by Murase and Sugano [2] (see also [3]). They obtained new kinds of integral formulas for the -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 [4] and [5], Tsuzuki on [6] and [7], and Moriyama on [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 , attached to the principal series representations of . Now we explain the definition of the Shintani functions on . We take as a subgroup of and take as a maximal compact subgroup of . Let be an arbitrary irreducible unitary representation of and an irreducible unitary representation of , and let be the space of smooth functions satisfying . We consider the intertwining space and its restriction to the minimal -type of , where is the contragredient representation of , and is the space of smooth functions satisfying for . The function which belongs to the image of above map is called the Shintani function. In this paper, we assume that is the irreducible unitary principal series representation of and is the unitary character of . The study of Shintani functions for the general unitary representation of 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 -fixed vector, and hence the minimal -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 -type is three-dimensional representation of . 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 . We cannot distinguish them only by the elements of . 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 ). 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 be the real reductive Lie group and its Lie algebra. The Cartan involution is defined by , and its differential is given by , where means the transposition of matrices. Then the fixed subgroup of in is equal to , which is the maximal compact subgroup of . Next, we define another involutive automorphism of by , where . Its differential is given by . The fixed subgroup of in is isomorphic to , that is, We define +1 and −1 eigenspaces of , in by Then, are the Lie algebras of , respectively. We have . Let be the matrix whose -component is 1 and the other components are 0 . For , we put . Then, we have Next, we take as a basis of . We have , and we take as a maximal Abelian subspace of . We define a subgroup of by Then, has a decomposition . Throughout this paper, we put .

For a Lie algebra , we denote its complexification by , that is, .

2.2. The Principal Series Representations

As a representation of , we take the principal series representation defined as follows. Let be a minimal parabolic subgroup of given by the upper triangular matrices in and the Langlands decomposition of with To define a principal series representation with respect to the minimal parabolic subgroup of , we firstly fix a character of and a linear form , where is the Lie algebra of . We write for . Then, we can define a representation of and extend this to by the identification , taking the trivial representation as the representation of . Then, the induced representation is called the principal series representation of . Here, is the half sum of positive roots of given by , for .

Concretely, the representation space is given by and the action of is defined by Here, for is the Iwasawa decomposition. Throughout this paper, we assume that the representation is irreducible. Moreover, we assume that are the elements of . Then, this representation becomes unitary.

Next, we define characters of as follows. The group consisting of four elements is a finite Abelian group of -type, and its elements except for the unity are given by The set consists of 4 characters , where is the trivial character of and are defined by Table 1.

Table 1

1
−1
−1

−1
1
−1

−1
−1
1

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

Proposition 2.1. (1) If is the trivial character of , the representation is spherical or class one. That is, it has a unique -invariant vector in . (2) If is not trivial, the minimal -type of the restriction to is a 3-dimensional representation of , which is isomorphic to the unique standard one . The multiplicity of this minimal -type is one:

3. The Space of Shintani Functions

3.1. The Definition of Shintani Functions

As a representation of , we take the unitary character defined by Let be the unitary character of defined previously. We consider the induced representation with the representation space   acts on this space by right translation. Let be the principal series representation of . We consider the intertwining space We denote its image by , that is, We call the element of the Shintani function of type . Let be the -type of the principal series representation , and let be the -embedding of and the pullback via . Then, the map gives the restriction of to , where is the contragredient representation of and the space is defined by We denote the image of in by , and the element of this space is called the Shintani function of type .

3.2. The -Radial Part

Because has the decomposition , the element of is characterized by its restriction to . We denote the centralizer and the normalizer of in by , , respectively. It is easy to verify that , , and are given as follows.

Lemma 3.1. One has Here, is the unit element of .

Let . Then, is the unique nontrivial element of . Let be the unitary character of and the finite-dimensional representation of . We denote by the space of smooth functions satisfying the following conditions: The following lemma is proved by Flensted and Jensen (see [12, Theorem 4.1]).

Lemma 3.2. The restriction to gives the following isomorphism:

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

Lemma 3.3. For , one has

Lemma 3.4. Let . For , , , one has

4. Shintani Functions Attached to the Spherical Principal Series Representations

Throughout this section, as a character of , we take the trivial character . Then, the principal series representation is the spherical or class one principal series representation whose minimal -type is the one-dimensional trivial representation of , which occurs of multiplicity one in (Proposition 2.1).

4.1. The Capelli Elements

Let be the center of the universal enveloping algebra of . has two independent generators, and they are obtained as the Capelli elements because is of type (see [13]). For , we put The following proposition gives the explicit description of the independent generators of (see [11]).

Proposition 4.1. The independent generators of are given as follows:

Since are the elements of , they act on as the scalar operators. And since the space of Shintani functions is the image of the -homomorphism of , they act on the space of Shintani functions as the same scalar operators, respectively.

4.2. Eigenvalues of ,

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 , . For the spherical principal series representation, is the trivial character of . Let be the generator of the minimal -type in normalized such that . The actions of on are computed in [11], and the result is as follows.

Proposition 4.2. The Capelli elements act on by scalar multiples, and the eigenvalues are given as follows:

4.3. Construction of the Casimir Equations

Next, we compute the actions of , on . Here, is the unitary character of .

Lemma 4.3. For , one has

Proof. By definition, we have Since , we have .Therefore, we have Next, since   . Therefore, we have The computations of the actions of are similar. Finally, since , we have

By simple computations of matrices, we have the following expressions of the elements in .

Lemma 4.4. One has

To make use of Lemma 3.4, we have to rewrite , in the form of linear combinations of the elements in . To do this, we use the following formulas which can be obtained by direct computation.

Lemma 4.5. One has Here, is the Lie bracket on .

By using Lemma 4.5, we can rewrite , as we wished. Now, since for all is annihilated by the action of and , and the actions of and on are the same (the multiplication by ), we may regard , as the elements in , where is the subalgebra of defined by

Lemma 4.6. One has the congruences

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 satisfies the following equations: (1) the differential equation obtained from the action of : (2) the differential equation obtained from the action of : Here, are the eigenvalues of the Capelli elements on principal series representations. Equations (4.15)−(4.14) give

Therefore, if 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 is one of the above three values. Now, we assume that satisfies this condition. We put in (4.14). Then, we have Next, we put . Then, satisfies We want to divide the left-hand side of (4.21) by . To do this, we take so that satisfies The value of is as follows. (1)If . So we take . (2)If . So we take . (3)If . So we take .

For this , the left-hand side of (4.21) is divided by , and the equation becomes This is the Gaussian hypergeometric differential equation. Note that the Shintani function on is regular at the origin (). Therefore, is also regular at . Equation (4.23) has just one solution which is regular around (up to constant multiples), and it is given by where is the Gaussian hypergeometric function and are defined by Explicitly, by solving these equations, are given as follows. (1)In case of . (2)In case of . (3)In case of .

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

Theorem 4.8. Let be the unitary character of defined by (3.1). Then, the necessary condition of the existence of the nontrivial elements in is that Suppose that this condition is satisfied and nontrivial Shintani functions exist. If one puts , is given as follows (up to constant multiples). (1)In case of , one has (2)In case of , one has (3)In case of , one has Especially, one has

5. Shintani Functions Attached to the Nonspherical Principal Series Representations

In this section, as a character of , we take a nontrivial character . Then, the minimal -type of is the three-dimensional representation of which is isomorphic to the tautological representation which occurs of multiplicity one in . We take instead of as a minimal -type of . The representation space of is denoted by ), and we take , , as a basis of . Let be the Shintani function. Then, is expressed by   is characterized by its restriction to . To investigate , 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 acts on the representation space of the principal series representation as a scalar operator (-multiple) and the space of Shintani functions is the image of -homomorphism of , acts on this space as the same scalar operator. Since for all is annihilated by the action of and the actions of and on are the same (the multiplication by ), we may regard as the element in , where is a subalgebra of defined by By using Lemmas 4.4 and 4.5, we can rewrite in Proposition 4.1 as follows.

Lemma 5.1. One has

By using this lemma, the action of on can be computed easily. We have where Since satisfies , we have the following three differential equations.

Theorem 5.2. For , the functions , , satisfy the following equations:

5.2. The Gradient Equation

For the spherical function , we define the right gradient operator as follows.

Definition 5.3. For the orthonormal basis of , the right gradient operator is defined by Here, is the dual basis of with respect to the inner product .The set becomes the orthonormal basis of , and , is its dual basis. Therefore, the gradient operator is explicitly given by We rewrite this by using the basis of .

Claim 1. We define five elements in by Then, becomes the basis of .

With this basis, the gradient operator is rewritten as The Lie algebra becomes the representation space of the adjoint action of . We denote this representation by . By the Clebsch-Gordan theorem, has the irreducible decomposition Here, each is the -dimensional irreducible representation of . In this decomposition, the projector of -modules is described as in Table 2.

Table 2

0

0

is a -valued function. Then, by mapping to , we have a -homomorphism Since the minimal -type occurs of multiplicity one, is a map of constant multiple. To compute the action of the gradient operator on the space of the Shintani functions , we have to decompose along the decomposition .

Lemma 5.4. One has

By using Lemmas 3.4 and 4.3 and the table of projections, we can compute the action of the gradient operator. For , we have where On the other hand, the eigenvalue of the gradient operator on the spherical functions of the principal series representation depends on the choice of , denoted by . These values are computed in [11] and they are as follows: Therefore, since satisfies , we have the following three differential equations.

Theorem 5.5. For , one has

We consider the case of . We have . By (5.21), if , we have . Suppose that . We put , in (5.21). Then, the equation becomes We put . Then, the equation becomes We want to divide the left-hand side of (5.24) by . For this purpose, must satisfy The solutions are . We choose . Then the left-hand side of (5.24) can be divided by , 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 , since they contain .) Therefore, we have where is a constant number and . Next, we consider the equations satisfied by and . From (5.20) and (5.22), we have By differentiating both sides of (5.29) by , we have By inserting (5.29) and (5.30) into the Casimir equation (5.6), (5.8) to eliminate the differential terms, we have Therefore, if the parameter satisfies , then . Suppose that . Since , we have Therefore, we have That is, (5.33) is the necessary condition of the existence of nontrivial . We put and insert these into (5.20), (5.22). Then we have Next, we put . Then, the above equations become For a while, we consider the case of , that is, is a signature of , where Then, by combining (5.36), (5.37), we have Suppose that . Then, the equation becomes We put . Then, satisfies Equation (5.41) is the Gaussian hypergeometric differential equation. Now, since and is regular at , must be regular at . 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 and . By expanding and around , we have where and are the analytic functions around . By inserting these into (5.36), we have where and are also the analytic functions around . By comparing the coefficients of of both sides, we have Summing up, and are given by

(: some constant, ). In our computation, we assumed that , but the result above holds without this assumption.

Next, we consider conditions (1), (2), and (3) in (3.8). For , condition (1) is equivalent to This is equivalent to Condition (2) is equivalent to The solutions we have always satisfy this condition. Condition (3) is equivalent to (for all ). Since , (5.52) are equivalent to 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 . Note that since the transform does not change the eigenvalue of Casimir operator and changes the eigenvalue of gradient operator to , this transform gives the result in case of . Similarly, the transform gives the result in case of . Summing up these results, we have the following theorem.

Theorem 5.6. Let be a signature of defined by (5.38) and a three-dimensional tautological representation of , and let be a Shintani function corresponding to the nonspherical principal series representation of . Then, the restriction of to is given as follows. (1)In case of ,(a) if , ;(b) if , one has (2)In case of , (a) if , ;(b) if , one has Especially, in any case, one has The transform gives the result in case of and the transform gives the result in case of .

Next, we compute the Shintani functions for general unitary character under the assumption that are linearly independent over . We have already known that the necessary condition of the existence of non zero is that the parameter is one of , or , and we have already solved the differential equations in case of . Hereafter, we suppose that the parameter is either or . We put , , in (5.36), (5.37). Then we have We put  () in (5.57) and compute the power series solutions. By inserting these series into (5.57), we have where are the analytic functions around . By comparing the lowest terms in power series, easily we have (in this argument, we use the fact that are linearly independent over carefully). Therefore, from (5.59), we have Since , we have By combining this and , we have Hereafter, we put . Then, (resp., ) are expressed by the linear combination of some power series , (resp., , ). That is, there exist common constants , such that By inserting into (5.57) and picking up the coefficients of , we have the following recurrence relations: for all . Here, we assume that if . From (5.65), (5.66), easily we have for all by induction. Therefore, by inserting into (5.65), we have Thus, we have Similarly, if the characteristic roots are , by inserting into (5.57), we have (for all ). From (5.68), and are expressed by Here, for , we define and . Therefore, if we normalize , we have Thus, we have And since , we have Similarly, by using and , if we normalize , we have Therefore, and are expressed as follows: We want the relation between and . We can find the relation by using the regularity of the Shintani function and the asymptotic formula of Gaussian hypergeometric function . Now, since and are regular around and since and must be regular around and must satisfy . Since all hypergeometric functions appearing in the right-hand sides of (5.78) are in the form of , to investigate the behavior of the right-hand sides of (5.78) around , we use the following asymptotic formula.

Formula 1We have Here, is the Euler constant and is defined by

We apply this formula to the right-hand sides of (5.78). Firstly, the coefficient of of equals Therefore, is regular around regardless of the values of , . Next, the coefficient of of equals Since this coefficient must be 0, we have up to (the same) constant multiples. Note that we can easily verify that for these , by using formulas . Now, we have completely determined . We have up to (the same) constant multiples. Here, , , , , . By the same argument we have done when 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 , , and in any case. Summing up, we obtain the following theorem.

Theorem 5.7. Assume that are linearly independent over . Let be a unitary character of defined by (3.1) and the character of . Then, the necessary condition of the existence of the nontrivial Shintani functions attached to the nonspherical principal series representation is that or That is, if the condition above is not satisfied, one has Let and suppose that the condition above is satisfied. Then, (1)if and , one has (2)if and or , one has where is some constant and and are the functions given by (5.85). Especially, in any case, one has The transform gives the result in case of , and the transform gives the result in case of .

Remark 5.8. By using the relation and the formulas of the hypergeometric function we can rewrite , in Theorem 5.7 as functions in . We put , . Then, we have 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.

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