Yumiko Hironaka

Local densities of alternating forms

Abstract We give an explicit formula for local densities of integral representations of non-degenerate alternating matrices with entries in the ring of p -adic integers in terms of elementary divisors. The proof of the formula is based on the local functional equation satisfied by the zeta function on the space of alternating forms and some properties of spherical functions.

Eisenstein series on reductive Symmetrie spaces and representations of Hecke algebras

0.1. Let G be a reductive algebraic group defined over Q and σ an involutive Qautomorphism of G. Denote by H the subgroup of G of elements fixed by σ. We consider the Symmetrie space X = G/H. A connected component XQ of the set X(i?) of real points is an affine Symmetrie space of the identity component of G (i?). If X0 is Riemannian, then for an arithmetic subgroup Γ of G, one can define the Selberg-Langlands Eisenstein series, which is a real analytic function on Γ\Χ0 (cf. [L], [HC]). In general, X(i?) may contain a connected component XQ that is not a Riemannian Symmetrie space. The action of an arithmetic subgroup Γ on such a component X0 is not properly discontinuous and we can no longer expect to define real analytic Eisenstein series on F\XQ.

Spherical functions on the space of p-adic unitary hermitian matrices

We are interested in the harmonic analysis on

Spherical functions on p-adic homogeneous spaces

p

Zeta functions of finite groups by enumerating subgroups

-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space

Spherical functions and local densities on hermitian forms

X

Spherical Functions and Local Densities of Alternating Forms

of unitary hermitian matrices of odd size over a

Local Densities of Representations of Quadratic Forms over p-Adic Integers (The Non-Dyadic Case)

{\mathfrak p}

Local Zeta Functions on Hermitian Forms and Its Application to Local Densities

-adic field of odd residual characteristic, which is a continuation of our previous paper where we have studied for even size matrices. First we give the explicit representatives of the Cartan decomposition of

SPHERICAL FUNCTIONS OF HERMITIAN AND SYMMETRIC FORMS III

X