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 characteristic We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra. Euler characteristic of certain affine flag varieties The purpose of this note is to prove, as Lusztig stated, that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nil-elliptic elementnt istcl wherel is the rank of the associated finite type Lie algebra. For the case of positive characteristic we use the classification of finite irreducible groups generated by pseudoreflections due to Kantor, Wagner, Zalesski? and Sere?kin. We also compute the Euler characteristic of the space of partial flags containingnt and give a connection with hyperplane arrangements. The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic ≠ 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. This is true regardless of the characteristic of the field or of the order of the parameterq in the definition ofHn. Computing invariants of reductive groups in positive characteristic This paper gives an algorithm for computing invariant rings of reductive groups in arbitrary characteristic. Morse Theory and Euler Characteristic of Sections of Spherical Varieties We generalize the formula for the Euler characteristic of a hypersurface in the torus (C*)d, due to D. Whether the corresponding results hold in positive characteristic is not known. Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. This interpretation involves an Euler characteristic χ built from Ext groups between integral Weyl modules. The algorithm presented here computes a geometric characteristic of this action in the case where G is connected and reductive, and $\rho$ is a morphism of algebraic groups: The algorithm takes as input the Let k be a field of characteristic zero, let a,b,c be relatively prime positive integers, and define a Let k be an algebraically closed field of characteristic p ≥ 0. When the characteristic of k is 0, it is known that the invariants of d vectors, d ≥ n, are obtained from those of n vectors by polarization. Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero.

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