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.