Singularities of orbit closures in loop spaces of symmetric varieties

Abstract

We study the singularities of closures of Iwahori orbits on loop spaces of symmetric varieties extending the celebrated work of Lusztig-Vogan to the affine setting. We show that the IC-complexes of orbit closures (with possible non-trivial coefficients) are pointwise pure and satisfy a parity vanishing property. We apply those geometric results to study the affine Lusztig-Vogan modules and obtain fundational results about them including the positivity properties of the affine Kazhdan-Lusztig-Vogan polynomials. Along the way, we construct conical transversal slices inside loop spaces of symmetric varieties generalizing the work of Mars-Springer in the finite dimensional setting. Our results answer a question of Lusztig. We deduce results for singularities of spherical orbit closures and provide applications to relative Langlands duality including the positivity for the relative Kostka-Foulkes polynomials and the formality conjecture.