ABSTRACT An explicit formula for the radial part of matrix coefficients of the discrete series of... more ABSTRACT An explicit formula for the radial part of matrix coefficients of the discrete series of Gel'fand–Kirillov dimension 5 of the special unitary group SU(2, 2) of index (2+, 2−), is obtained in terms of Gaussian hypergeometric functions plus some formulae of binomial coefficients.
We consider Whittaker model for generalized principal series representations of the real sympleti... more We consider Whittaker model for generalized principal series representations of the real sympletic group of degree 2. We obtain an integral formula for the radial part of the vector of with an extreme K-typQ in the Whittaker model. Introduction. In our previous papers [O], [M-O], we investigated Whittaker functions of the large discrete series representations, and of the principal series representations of the real symplectic group Sp(2; R) of rank 2, respectively. In this paper we shall obtain explicit integral formulae for the radial part of the Whittaker functions on G = Sp(2; R), belonging to the principal series representations associated with the Jacobi parabolic subgroup P t of G. Let (π, H π) be an irreducible admissible representation. Denote by N a maximal unipotent subgroup of G. For a continuous character η: N^>C* of N, let C™(N\G) be the space of complex-valued C 00-functions f on G satisfying f{ng) = η(n) f(g) for any neN, geG. Consider C™(N\G) as a (g, Λ^)-module via the right regular action of G. Then the intertwining space Hom (9 , X) (i/ π , C?(N\G)) is the space of algebraic Whittaker vectors. When π is a principal series representation with a generic parameter μ of α£, the dimension of the above space is known and equals the order of the (little) Weyl group, i.e. 8 in our case (cf. Kostant [Kos, §5]). Here α£ is the dual of the complexification of the Lie algebra a of A. Choose a K-type (τ, V τ \ τeK, which occurs with multiplicity one in 7/ π , and let /: V τ CL_> H n be an injective ΛMiomomorphism which is unique up to nonzero scalar multiple. Then we call the elements of the image of the restriction map Whittaker functions with K-type τ* belonging to the representation π. Now consider the standard maximal parabolic subgroup P ί of G associated to the long simple root. In this paper we call this parabolic subgroup the Jacobi The first named author is supported by JSPS Research Fellowships for Young Scientists.
Publisher Summary This chapter describes hodge numbers of a Kummer covering of P2 ramified along ... more Publisher Summary This chapter describes hodge numbers of a Kummer covering of P2 ramified along a line configuration. It presents Hodge type of H2(α) for Generic α.
Starting with a datum of Hodge structure given in Section 3.7, in this section we construct an ab... more Starting with a datum of Hodge structure given in Section 3.7, in this section we construct an abelian variety. The method of this construction is a slight generalization of that of Satake [40], Kuga-Satake [20], and Deligne [10], More precisely speaking, we consider the case where the given Hodge structure has a totally real algebraic number field as an endomorphism ring, in place of the trivial one Q.
ABSTRACT An explicit formula for the radial part of matrix coefficients of the discrete series of... more ABSTRACT An explicit formula for the radial part of matrix coefficients of the discrete series of Gel'fand–Kirillov dimension 5 of the special unitary group SU(2, 2) of index (2+, 2−), is obtained in terms of Gaussian hypergeometric functions plus some formulae of binomial coefficients.
We consider Whittaker model for generalized principal series representations of the real sympleti... more We consider Whittaker model for generalized principal series representations of the real sympletic group of degree 2. We obtain an integral formula for the radial part of the vector of with an extreme K-typQ in the Whittaker model. Introduction. In our previous papers [O], [M-O], we investigated Whittaker functions of the large discrete series representations, and of the principal series representations of the real symplectic group Sp(2; R) of rank 2, respectively. In this paper we shall obtain explicit integral formulae for the radial part of the Whittaker functions on G = Sp(2; R), belonging to the principal series representations associated with the Jacobi parabolic subgroup P t of G. Let (π, H π) be an irreducible admissible representation. Denote by N a maximal unipotent subgroup of G. For a continuous character η: N^>C* of N, let C™(N\G) be the space of complex-valued C 00-functions f on G satisfying f{ng) = η(n) f(g) for any neN, geG. Consider C™(N\G) as a (g, Λ^)-module via the right regular action of G. Then the intertwining space Hom (9 , X) (i/ π , C?(N\G)) is the space of algebraic Whittaker vectors. When π is a principal series representation with a generic parameter μ of α£, the dimension of the above space is known and equals the order of the (little) Weyl group, i.e. 8 in our case (cf. Kostant [Kos, §5]). Here α£ is the dual of the complexification of the Lie algebra a of A. Choose a K-type (τ, V τ \ τeK, which occurs with multiplicity one in 7/ π , and let /: V τ CL_> H n be an injective ΛMiomomorphism which is unique up to nonzero scalar multiple. Then we call the elements of the image of the restriction map Whittaker functions with K-type τ* belonging to the representation π. Now consider the standard maximal parabolic subgroup P ί of G associated to the long simple root. In this paper we call this parabolic subgroup the Jacobi The first named author is supported by JSPS Research Fellowships for Young Scientists.
Publisher Summary This chapter describes hodge numbers of a Kummer covering of P2 ramified along ... more Publisher Summary This chapter describes hodge numbers of a Kummer covering of P2 ramified along a line configuration. It presents Hodge type of H2(α) for Generic α.
Starting with a datum of Hodge structure given in Section 3.7, in this section we construct an ab... more Starting with a datum of Hodge structure given in Section 3.7, in this section we construct an abelian variety. The method of this construction is a slight generalization of that of Satake [40], Kuga-Satake [20], and Deligne [10], More precisely speaking, we consider the case where the given Hodge structure has a totally real algebraic number field as an endomorphism ring, in place of the trivial one Q.
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Papers by Takayuki Oda