Quasisplit Hecke algebras and symmetric spaces

Let (G,K) be a symmetric pair over an algebraically closed field of characteristic different of 2 and let sigma be an automorphism with square 1 of G preserving K. In this paper we consider the set of pairs (O,L) where O is a sigma-stable K-orbit on the flag manifold of G and L is an irreducible K-equivariant local system on O which is "fixed" by sigma. Given two such pairs (O,L), (O',L'), with O' in the closure \bar O of O, the multiplicity space of L' in the a cohomology sheaf of the intersection cohomology of \bar O with coefficients in L (restricted to O') carries an involution induced by sigma and we are interested in computing the dimensions of its +1 and -1 eigenspaces. We show that this computation can be done in terms of a certain module structure over a quasisplit Hecke algebra on a space spanned by the pairs (O,L) as above.Comment: 46 pages. Version 2 reorganizes the explicit calculation of the Hecke module, includes details about computing \bar, and corrects small misprints. Version 3 adds two pages relating this paper to unitary representation theory, corrects misprints, and displays more equations. Version 4 corrects misprints, and adds two cases previously neglected at the end of 7.

An Eigenspace Approach to Isotropic Projections for Data on Binary Trees

The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant under automorphisms of a binary tree. We present a technique by which a slightly relaxed form of the generalized Fourier transform in this case can eventually be computed using only simple tools from linear algebra, which has possible advantages in computational efficiency

The Weitzenb\"ock Machine

In this article we give a unified treatment of the construction of all possible Weitzenb\"ock formulas for all irreducible, non--symmetric holonomy groups. The resulting classification is two--fold, we construct explicitly a basis of the space of Weitzenb\"ock formulas on the one hand and characterize Weitzenb\"ock formulas as eigenvectors for an explicitly known matrix on the other. Both classifications allow us to find tailor--suit Weitzenb\"ock formulas for applications like eigenvalue estimates or Betti number estimates.Comment: 48 page