Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-ﬁnite compactly supported smooth functions on X is characterized. Contents 1. Introduction 2. Notation 3. The Paley-Wiener space. Main theorem 4. Pseudo wave packets 5. Generalized Eisenstein integrals 6. Induction of Arthur-Campoli relations 7. A property of the Arthur-Campoli relations 8. Proof of Theorem 4.4 9.
Characteristic classes for oriented pseudomanifolds can be deﬁned using appropriate self-dual complexes of sheaves. On non-Witt spaces, self-dual complexes compatible to intersection homology are determined by choices of Lagrangian structures at the strata of odd codimension. We prove that the associated signature and L-classes are independent of the choice of Lagrangian structures, so that singular spaces with odd codimensional strata, such as e.g. certain compactiﬁcations of locally symmetric spaces, have well-deﬁned L-classes, provided Lagrangian structures exist.
In this paper we give a geometric version of the Satake isomorphism [Sat]. As such, it can be viewed as a ﬁrst step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classiﬁcation by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come ˇ in pairs.