We intend to investigated certain$ p$-adic homogeneous spaces together with spherical functions and have number theoretic applications. We have approached this purpose in the following way : 1) to formulate expression formulas of spherical functions which can be widely applied ; 2) to formulate explicit formulas of spherical functions for concrete examples and analyze the spaces ; 3) to obtain number theoretic interesting quantities for the spaces of symmetric forms and hermitian forms.The most important result is to complete the theory of spherical functions on the space of unramified unitary hermitian matrices

About matrix coefficients of semi-simple Lie groups, we obtained a more precise result for the middle discrete series of SU(2, 2) about the asymptotic expansion. We investigated the explicit formula of the matrix coefficients of SU(3, 1)(both are joint works together with T. Hayata and H. Koseki). Utilizing the asymptotic expansion, we obtained the explicit formula of the c-functions of certain P_J-principal series representations of Sp(2, R) explicitly(joint work with M. Iida). We push forward the investigation of 0-cells of the fundamental domain of the Siegel modular group of genus 2

In the period of the project, we got the following results.1.Explicit formula for Whittaker functions of SL(3,R) belonging to non-spherical principal series. (Published in 2004).2.Given a pair of arithmetic quotients (X,Y), where X,Y are either type IV quotients, or quotients of the complex hyperball, and when Y is a divisor of X, we constructed the Green current of Y as an automorphic form. (Published in 2003).3.We proved the confluence from Siegel-Whittaker functions to Whittaker functions for P_J principal series representations of Sp(2,R). (In press).4.Together with Miki Hirano of Ehime University, we obtained the integral and power series expressions of the principal series Whittaker functions on GL(3,C) with minimal K-type. The project is still in progress.5.The project to get explicit formula for P_J principal series representations of Sp(3,R) : The contents are finished. We are preparing a joint paper together with Miki Hirano and Taku Ishii of Chiba Inst. of Technology.6.We maintained the monthly seminar on Automorphic Forms at Komaba

We investigate Sp_2 as a spherical homogeneous Sp_2×(Sp_1)^2-space intensively. First we give explicit formulas of spherical functions explicitly as an application of the previous general formula which was given by the author. Then we determine the Hecke module structure of the space of Schwartz-Bruhat functions on Sp_2 and parametrization of all spherical functions, where the space of spherical functions attached to the same parameter has dimension 4.Extending the calculation of functional equations of spherical functions on Sp_2, we give a general method to obtain functional equations of spherical functions on certain spherical homogeneous p-adic spaces. This is based on the uniqueness of the relatively invariant distributions on a homogeneous space, and it guarantees the existance of functional equations attached to elements of Weyl group.As a joint research with F.Sato, we give an integral representation of the p-oart of the Siegel series, which is the main part of the Fourier coefficients of Siegel Eisenstein series. By using this representation, Siegel series is expresses as a finite sum of spherical functions on SO(n, n)/S(O(n)×O(n)) with respect to Siegel parabolic subgroup, which gives us a new proof of functional equations of Siegel series. The explicit calculation is now in progress

In this research project, we investigated zeta functions of prehomogeneous vector spaces from the view points(1)relations between functional equations and representations of general linear groups,(2)relations to automorphic L-functions,(3)generalization of the theory to non-regular prehomogeneous vector spaces.(1)We showed that the functional equations of zeta functions are closely related to intertwining operators between degenerate principal series representations of general linear groups, and, using the relation, we obtained an integral expression of Eulerian type of the gamma matrices of functional equations. This enables us to identify the variable change in functional equations as an action of an element in the Weyl group of a general linear group, and to decompose functional equations into a product of more elementary functional equations. There exists a similar results for p-adic local zeta functions. As an application of p-adic theory, we investigated the Fourier coefficients of Elsenstein series of Sp(n) and GL(n) and the theory of spherical transforms on certain spherical homogeneous spaces.(2)We identified the Koecher-Maass series of real analytic Siegel Eisenstein series with a zeta function associated with a certain prehomogeneous vector space on which the Siegel parabolic subgroup of SO(n, n) acts. It is quite probable that this result can be extended to other classical groups. A considerable progress has been made in explicit calculation of zeta functions. We obtained an explicit expression of zeta functions in terms of the Riemann zeta function and the Mellin transforms of the Cohen Eisenstein series for more than 70 percent of irreducible regular reduced prehomogeneous vector spaces.(3)For non-regular prehomogeneous vector spaces, we developed a general theory of integral representations and the functional equation of the zeta integrals, which is a formal generalization of the theory for regular prehomogeneous vector spaces. We also gave the first example of explicit functional equations for non-regular spaces

In these three years, the principal researcher of this subject organized a conference at Oberwolfach in Germany, three autumn workshops and two mini conferences in Japan, hence 6 conferences in total, and studied effectively with many foreign researchers and cooperative researchers in Japan. In particular in three autumn workshops, we took three themes, Koecher Maass series, Eisenstein series, dimension formulae of automorphic forms, as initially planned. There we studied the foundation and development of arithmetic zeta functions, and obtained satisfactory results, and besides we published three volumes of proceedings of 640 total pages which can be regarded as fundamental references of the research of this direction. More concretely, the principal researcher defined Koecher-Maass series for automorphic forms of any tube domains, proved their analytic continuation and gave functional equations. Explicit forms of Koecher-Maass series for modular forms such as Eisenstein series, or those closely related to liftings were also obtained (jointly with Katsurada) and established the meaning of Koecher-Maass series as "arithmetic" zeta functions. Also, he obtained a lifting conjecture on Siegel modular forms of half integral weights (joint with Hayashida), developed theories on modular forms of rational weights and modular varieties, and applied differential operators to the study of structures of vector valued Siegel modular forms. As you can see from the list of their papers, the other researchers of our project also actively studied on spherical functions and zeta functions of prehomogeneous vector spaces, their explicit forms and convergence, Jacobi forms, liftings, explicit density formulae, adele geometry, real analysis on automorphic forms, Siegel modular forms mod p, fundamental lemma, the Fourier expansion of modular forms on bounded symmetric domains of non-tube type and other various researches. These are all related to the development of our research project and we think our project was very successful. Now we think we are in the stage to go from fundamental research to various applications, and expecting and planning a project of next stage such as explicit theories of graded rings of automorphic forms and vertex operator algegras

1.Sato, the head investigator, studied the weakly spherical homogeneous spaces (WSHS for short) obtained from the prehomogeneous vector spaces (PV for short) (SpinィイD210ィエD2 × GLィイD23ィエD2, half-spin 【cross product】 ΛィイD21ィエD2), (SLィイD25ィエD2 × GLィイD23ィエD2, ΛィイD22ィエD2 【cross product】 ΛィイD21ィエD2) and proved the functional equations satisfied by Eisenstein series attached to the spaces, which is a joint work with T. Kimura of Tsukuba Univ. and his students. As an application we can calculate explicitly the Fourier transforms of the complex powers of relative invariants of the PV's above. This shows that the theory of WSHS is very fruitful, since the structure of these two PV's is very complicated and even the method of micro local calculus can not be applied to them. We also made several attempts to generalize the theory to WSHS of reductive groups other than the general linear group. The result is still unsatisfactory ; however several suggestive partial results have been obtained.2.Hironaka succeeded in calculating the explicit formula for spherical functions of spherical homogeneous spaces over p-adic fields in a rather general setting. Using the formula, she constructed the theory of spherical functions of the space of hermitian forms over unramified quadratic extensions of the base p-adic field and gave an explicit formula for local densities of hermitian forms. Moreover, jointly with Sato, she calculated local densities of quadratic forms over nondyadic p-adic fields explicitly in the most general setting ; the results can be extended to the space of hermitian forms over ramified quadratic extensions.3.Arakawa studied mainly Koecher-Maass zeta functions attached to Siegel modular forms and obtained some results including the following : (a) a generalization of Koecher-Maass zeta functions to Jacobi forms, (b) an explicit expression of the Koecher-Maass zeta function attached to the Nagaoka Eisenstein series of weight 1 of degree 2, which is obtained from p-adic Siegel Eisenstein series

Let G be an algebraic group defined over a p-adic field k, X a homogeneous space of G and X = X (k).Let H(G,K) be the Hecke algebra of G = G(k) with respect to a maximal compact subgroup K of G.An H(G,K)-common eigenfunction on X is called a spherical function, which is an intereted object both in Number Theory and Representation Theory.Under certain condition on orbits by a parabolic subgroup in X, we have given a method to obtain explicit formulas of spherical functions by means of functional equations and spherical functions of groups. We can apply it to the spaces of symmetric forms and hermitian forms, which are interesting objects in Number theory. In these spaces, spherical functions can be viewed as generating functions of local densities of representation, so they are closely related. It is a classical problem in Number Theory to obtain explicit formulas of local densities of representations for symmetric forms and hermitian forms.For the space X of unramified hermitian forms, we have determined an explicit formula for spherical functions by the above method and the structure of the space of Schwartz-Bruhat functions S(K＼X) as Hecke module. We have also obtained two kinds of explicit formulas of local densities by using spherical functions on X.By a joint research with F. Sato, we have obtained a complete explicit formula of local densities of symmetric forms for p ≠ 2, in general sizes. The method is to classify symmetric forms by the action of Iwahori subgroup and calculate certain Gaussian sums.Extending the above method to the the case of hermitina forms, we have obtained an explicit formula of local densities of hermitian forms for p ≠ 2, in general sizes

(1) Research on loop group bundlesi. Characteristic map of a loop group bundle is realized as a matrix valued function.ii. Lifting and descent of a vector bundle or a loop group bundle over M to a loop group bundle or a vector bundle over OMEGAM or MXS^1 are defined.(2) Research on characteristic classes of loop group bundles (string classes)i. Differential geometric definition of string classes is given.ii. Relations between string classes and Chern classes of a bundle and its lifting or descent are computed.iii. String classes are expressed as WZNW terms of characteristic maps.(3) By using above results and non-abelian de Rham theory, relations between Chern-Simons gauge theory and topological field theory are studied.(4) To extend above results for loop groups over exceptiohal groups, concrete realizations of exceptional groups are done.(5) To get more advanced information in this direction, Eisenstein serieses of nurmber theoretical symmetric spaces are studied.(6) Research on current group bundles , I.Noncommutative connections. The notion of noncommutative connection is introduced and several results such as reduction to U_1-bundles and noncommutative Poincare lemma, are get.(7) Research on current group bundles, II.Connections with respect to the Dirac operator. Connections with respect to the Dirac operator is defined. They give families of Dirac operator and their n-functions give bundle invariants

本研究は多様体、特に代数多様体を対象の中心にすえて、その構造、それに働く代数的作用、解析的作用等を数学の各分野、特に代数的、解析的、位相幾何学的分野から、総合的に究明する点にあった。より具体的には代数多様体をめぐる種々の研究、位相的不変量の計算とそのG-構造、解析的方程式の基本解の構成、スペクトル分解とその作用素で生成される半群の研究、基本解の特異領域の幾何学的構造等がそれである。これらの点に関しては、それぞれに一定の成果があげられている。高次微分の作用に関する不変環の理論、エルミートあるいは対称空間の球関数、球変換の理論、例外型リー群の対合自己同型と不変群の実現、非アーベル的ドラーム理論の構成等顕著な成果がみられる(研究発表の項を参照されたい)。また本研究を発展させる上で重要なポイントの一つとなる他大学における研究者との交流も科研費を利用して計画通り行われ、意見・情報の交換、ディスカッションを行なうことが出来た。また、本大学での、解析学、代数学、幾何学の各分野で、他大学の研究者を交えてのセミナー、シンポジウム等がもたれたことも特筆すべきことである。

1.実2次体の量指標のL-関数のある特殊値のLambert型Dirichlet級数のある極における留数を用いた興味深いexpressionを得た. また二重ガンマ関数と上記Lambert型Dirichlet級数との間の関数等式を得た(荒川).
2.2元2次形式のゼータ関数を核関数として球フーリエ変換の有理数体上の類似物を構成し, 有理2元2次形式の空間上のシャワルツ関数の空間のGL(2)のヘッケ環加群としての構造を決定した. H.Maassによって考えられたKoecherのゼータ函数の球関数係数への拡張を概均質ベクトル空間の枠組で一般化した(佐藤)
3.局所体上のエルミート形式および対称形式の空間Xに球関数の類似を定義し, これらを核関数として, X上のSchwartz-Bruhat空間上の球-Fourier変換を与えた. 球関数の関数等式を研究することにより, 球Fourier変換像の情報を提供した. 特に, サイズ2の形式については, 具体的に球関数のparametrization,Fourier逆変換や, Hecke環加群としてのS-B空間の構造の決定等を行った(小林).
4.P進体上の交代行列の空間の球函数論, 概均質ベクトル空間の理論の応用として, 交代行列の表現の局所密度の公式を得た. (佐藤, 小林)
5.Riemann Zeta関数の零点の分布についてのいくつかの興味ある結果を導いた. 零点の虚数部分の一様分布性に関する結果を用いて, L-関数の平均値定理を得た. また, 零点の分布についてのGramの法則に関する定理を得, 予想を作った(藤井).

交付された補助金は、研究代表者及び研究分担者の各地で開催されるシンポジウム、研究集会、及び各地の大学の研究者との研究打合せ資料収集等に充当する予定であったが、本年度は代数学、環論、及びトポロジ-関係のシンポジウムが夏期に札幌、福島の遠隔地で開催されたため、それらに出席するための旅費に大部分が使われるという結果になってしまい,各地の大学の研究者との研究打合せのための旅費が、予定に比べて大幅に縮小し、資料収集のみを目的とした旅費に充当することが不可能になってしまった。具体的には、代数学シンポジウムが7月に北海道大学学術交流会館で、トポロジ-シンポジウムが7月に福島大学で、環論シンポジウムが8月に北海道大学百年記念会館で開催され、トポロジ-シンポジウムには、研究分担者の向井、阿部が出席し、代数学シンポジウムと環論シンポジウムには研究代表者の西川、研究分担者の二宮、岸本、小林が出席し、これら全員の旅費を賄うことが出来た。また、環論シンポジウムには二宮が「Blocks of p-solvable groups with 2 or 3 simple modules」なる題目で講演した。これらのシンポジウムに手分けして出席したことにより,研究代表者及び分担者それぞれが、Hopf対数の構造論との関連をもつ種々の分野における第一線の研究の状況、具体的には有限群のmodular表現論、環の拡大理論、射影空間のHomotopy論、可微分多様体の構造理論、Hecke代数と対称空間の表現論等の分野の研究の現状についての理解が進み、代表者、分担者それぞれの研究の準備の進展及び論文作成等の効果を得ることが出来た

標数Oの局所体k上のn次非退化対称行列のなす空間Xには、G=GLu(k)が自然に作用している。X上の、K=GLu(Ok)不変でコンパクトな台をもつ関数の空間をS(K＼X)とすると,これには、Hecke環H(G,K)が作用している。kが奇素数pについてpーfieldである場合に、H(G,K)の表現空間としてS(K|X)の構造を調べることを以前した。今回は2ーfieldの場合を考察した。以上の対称空間の構造を調べる上でも、局所理論は是非しておくべきことである。X上の帯球関数を定義し、それを用いてS(K|X)上に球Fowier変換Fを定義し、これを通してS(K＼X)のH(G,K)ー加群としての構造を調べる。size nの帯球関数は、size n未満の帯球関数と,行列の表現の局所密度で表わすことができ、帯球関数の研究と局所密度の研究は密接に結びついている。より具体的には、i)Fの単射性,ii)Fの像,iii)Fの逆変換,iv)S(K＼X)上のPlruckerel測度,v)H(G,K)ー同時固有関数,の決定を問題にする。size zの時は完全に決定された。p≠2との著しい違いは、S(K＼X)のH(G_1K)ー自由加群としての階数で64(p≠2のときは16),一般にも、P=2なら8^n,P≠2なら4^nと思われる。一般のsizeについても、単射性と、関数等式については決定した。以上と同様の考察を、2ーfield上のHevmete行列のなす空間についても考察し、まとめた。補助金により、他大学の研究者と交流できたのは、大変有意義であった。又、数論関係の書籍も購入できた

有限群の正標数の体上の群多元環の構造についての研究の観点から、その根基の巾零指数と群の構造との関連について考察することを当面の目標とした。体Kの標数をpとし、群Gのシローp部分群の位数をP^aとする。群多元環KGの根基の巾零指数をt(G)で表すことにする。Gがp可解群のとき、t(G)はp^a以下であることが知られている。さらに、p^<a-1>≦t(G)≦p^aをみたす群Gの構造は完全に決定されている。そこで、不等式p^<a-2>≦t(G)<p^<a-1>をみたす群Gを決定するべく研究を行った。Gがp群の場合、この研究を遂行するためには、指数p^<a-2>の群をすべて決定する必要があるが、これについては、Millerによる分類があり、そのような群の同型類の個数が得られている。このMillerの論法を精密化することにより、指数p^<a-2>のp群すべてについて、それらの生成元と基本関係による表示を与えることができた。この結果を用いることにより、Gがp群の場合には、不等式p^<a-2>≦t(G)<p^<a-1>をみたす群を完全に決定することができた。さらに、この問題をp可解群の場合に解決するべく研究中であるが、いくつかの例外的な群をのぞいて、ほぼ解決の見通しが立つ状況になった。さらに、この研究の過程において、永年、研究者の間で予想されていた不等式t(G)≦t(P)(Gはp可解群で、PはGのシローp部分群)に対する反例を見つけることができた。このことは、この方面の研究者の注目を集めると思われる。この反例が特異なものなのかどうかについて、さらに上記不等式t(G)≦t(P)をみたす群の特徴付けについて、研究を深める必要があると考えている

対称空間のある種の作用素の同時固有関数族として得られる関数は球関数と呼ばれ、対称空間の表現論で重要な役割をになうものである。実または複素数での理論やp-進reductive群の球関数の理論はすでに研究されている。p-進体上の等質空間の球関数の理論を構成し、さらには、代数体上定義された対称空間、等質空間の調和解析の理論を構成することは、興味深い問題である。以下は標数0の非アルキメデス的局所体上定義された等質空間を考え、ヘッケ環の同時固有関数を球関数と呼ぶ。まず、球等質空間上の球関数についての表現論的な考察をし、一定の仮定の下で、球関数の公式を与えた。これを不分岐エルミート形式のなす空間について適用し、球関数の理論を構成した。具体的には、標準的な球関数をうまく定義し、その良い具体的表示式を与えること;球関数を核関数とするフーリエ変換の像、および逆変換の決定この空間上の急減少関数のなす空間のヘッケ環加群としての構造の決定すべての球関数をうまくパラメトライズすることまた、球関数は局所密度μ_p(B,A)達の母関数とみなすことができる。したがって、球関数の明示式を用いて、局所密度μ_pを引き出すことができる。実際、局所密度および原始的局所密度の、組合わせ論的量を用いた明示式を得た

As planned in our application, at Tokyo and at Kobe we had a series of monthly seminars. Since we have already reported about them in the annual reports, here we do not write about them. We also omit the details about the workshops at RIMS, Kyoto University and the summer schools. The proceedings on these meetings are available. The head investigator Oda in a joint work with Takahiro HAYATA (Yamagata Univ.) and Harutaka KOSEKI (Mie University) obtained an explicit formula for matrix coefficients of the middle discrete series of SU(2,2). Added to this he also obtained explicit formulae of matrix coefficients for the large discrete series of SU(2,2) and Sp(2,R). Moreover in a joint paper with Masao TSUZUKI (Sophia Univ.) he constructed Green functions of modular divisors on arithmetic quotients of certain bounded symmetric domains, as automorphic forms. We also explain the outline of the fruits of the research by the investigators joined to this plan. This result has not only have applications for the dimension of spaces of automorphic forms, but also theoretically significance. Ibukiyama and Saito had remarkable results on the evaluation of the special values of the zeta functions of prehomogeneous vector spaces. This result has not only application to the explicit dimension formulae of the spaces of automorphic forms, but also theoretically very important meaning. The investigation of p-adic spherical functions by Sato and Hironaka made much progress. The new point among others is that they can handle also the case of "ramified" spherical functions. The joint work of Murase and Sugano also advanced. Their work of the primitive theta function of SU(2,1) is one interesting result. But also the is a progress in the theory of automorphic L-functions on orthogonal groups A joint work with Shin-ichi Kato (Kyoto Univ.) on p-adic spherical functions is one of fruits. Watanabe together with Masanori Morishita (Kanazawa Univ.) pushed forward the investigation of Hermite constants for algebraic groups, Which is so to speak a non-abelian version of "Geometry of Numbers". The result of Katsurada is also interesting by regarding it as an investigation of ramified p-adic spherical functions

In these four years, the principal researcher of this subject organized a conference at Oberwol-fach, three Spring Confecences, two Autumn Workshops (one of which is Japan-Germany Seminar supported by JSPS), Mini-workshop on Number Theory, and Meeting on Modular Forms and Zeta Functions, hence 8 conferences in total, and studied effectively with many foreign researchers and cooperative researchers in Japan, In particular, in Spring Conference, we focused to communicate with wide areas arround modular forms, and took Graded rings of modular forms, Vertex operator algebras in first and second one, and more mixed areas in the third one. We published five Proceedings of 970 total pages. More concretely, the principal researcher proposed a conjecture on Shimura type correspondence between Siegel modular forms of integral and half integral weight, studied structures of scalar valued and vector valued Siegel modular forms and differential operators, Borcherds product expression, holonomic system coming from differential operators on modular forms, positivity of eta products, Koecher-Maass series of forms obtained by lifting, modular forms of rational weight. As it can be seen from the list of their papers, the other researchers of our project also actively studied on conformal field theory, spherical and Whittaker functions, prehomogenous vector spaces, modular forms on unitary groups, elliptic root systems, p-adic modular forms, L-packet, adele geometry, converse theorem, vertex operator algebras, combinatorial designs, weakly spherical spaces, inverse Galois problem and CAP forms

The main problems we investigated in this research project are(1) To identify the zeta functions associated with prehomogeneous vector spaces with some kind of zeta functions attached to automorphic forms(2) To construct a theory of(local) functional equations which is not covered by the theory of prenomogeneous vector spaces. The results we obtained are as follows :(1) According to the classification theory due to Sato and Kimura, irreducible regular prehomogeneous vector spaces are classified into 5 series of classical type and 24 spaces of sporadic type. We identified the zeta functions associated with 4 series of prehomogeneous vector spaces of classical type with the standard L-functions or Koecher-Maass zeta functions of certain real analytic Eisenstein series. One of the results which are necessary for the proof of these results is a new integral representation of the Siegel series (= p-part of the Fourier coefficients of Eisenstein series). As another application of the new integral representation, we proved a formula which connects the Siegel series to spherical functions on a p-adic semisimple symmetric space of the orthogonal groups.(2) We proved that, given a pair of homogeneous polynomials on Ra satisfying a local functional equation and a pair of nondegenerate dual quadratic mappings of R^m to R^n, then, the pull backs of the polynomials by the quadratic mappings also satisfy a local functional equation. This generalizes a result due to Faraut-Koranyi-Clerc and we can construct many examples of functional equations which are not covered by the theory of prehomogeneous vector spaces. We also classified nondegenerate dual quadratic mappings over quadratic spaces and proved that such quadratic mappings are in one to one correspondence to representations of a tensor product of 2 Clifford algebras

Theory of prehomogeneous vector spaces gives a systematic method of constructing zeta functions from polynomial invariants of prehomogeneous group actions. We investigated zeta functions of prehomogeneous vector spaces from the following 3 view points : (1) relations to Eisenstein-periods, (2) relations to the Koecher-Maass zeta functions of automorphic forms, (3) zeta functions of invariants of non-prehomogeneous group-actions. The most important result is the construction of zeta functions of non-degenerate quadratic mappings, which include zeta functions of certain non-prehomogeneous polynomials of degree 4