Abstract: We define a Fourier transform for functions on a reductive p-adic group, which transforms by a nondegenerate character of a maximal unipotent subgroup, and with compact support modulo this unipotent subgroup. This transformation, as well as wave packets are studied using a theory of the constant term. Then, a result of V. Heiermann is used to characterize the image of this Fourier transform.

Abstract: The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry.

In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I plan to give an exposition on the recent developments on the question about the global natures of locally non-Riemannian homogeneous spaces, with emphasis on the existence problem of compact forms, rigidity and deformation.

Abstract: Given affine Hecke algebras H₁, H₂ we introduce the notion of a "spectral transfer map" from H₁ to H₂. Such a map is not given by an algebra homomorphism from H₁ to H₂ but rather by a homomorphism from the center Z₂ of H₂ to the center Z₁ of H₁ which is required to be "compatible" in a certain way with the Harish-Chandra μ-functions on the centers Z₁ and Z₂.

The main property of such a transfer map is that it induces a correspondence between the tempered spectra of H₁ and H₂ which respects the canonical spectral measures ("Plancherel measures"), up to a locally constant factor with values in the nonzero rational numbers.

The category of smooth, unramified representations of a connected split simple p-adic group of adjoint type G(F) is Morita equivalent, via Bernstein's functor, to a direct sum R of affine Hecke algebras. It is a remarkable fact that R admits an essentially unique spectral transfer map to the Iwahori-Matsumoto Hecke algebra of G which is equivariant with respect the natural action of the center Z(\hat{G}) of the Langlands dual group \hat{G}.

Using these facts and results of joint work with Solleveld we discuss the combinatorial structure of unramified L-packets of classical split groups.

Abstract: We explain a classification of Fuchsian systems on the Riemann sphere together with Katz's middle convolution, Yokoyama's extension and their relation to a Kac-Moody root system discovered by Crawley-Boevey.

Then we present a beautifully unified connection formula for the solution of the Fuchsian ordinary differential equation without moduli and apply the formula to the harmonic analysis on a symmetric space.