In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces.We introduce a motivic Chern class transformation mCy: K 0 (var/X) → G 0 (X) ⊗ Z[y], which generalizes the total λ-class λy(T * X) of the cotangent bundle to singular spaces. Here K 0 (var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G 0 (X) is the Grothendieck group of coherent sheaves of O X -modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito.We define a natural transformation Ty * : K 0 (var/X) → H * (X) ⊗ Q[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty * is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = −1), the Todd class 1 2 J.-P. Brasselet, J. Schürmann & S. Yokura transformation in the singular Riemann-Roch theorem of Baum-Fulton-MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1).We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov-Libgober and the stringy Chern classes of Aluffi and De Fernex-Lupercio-Nevins-Uribe.All our results can be extended to varieties over a base field k of characteristic 0.We introduce characteristic homology class transformations mC y and T y * on K 0 (var/X) related by the following commutative diagram:by taking fiberwise the (topological) Euler characteristic with compact support. Note that the homomorphisms e and mC 0 are surjective. Remark 0.1. Note that in the algebraic context the Chern-Schwartz-MacPherson transformation c * of [44] is defined only on spaces X embeddable into a smooth space (e.g. for quasi-projective varieties). But using the technique of "Chow envelopes" as in [31, Sec. 18.3], this transformation can uniquely be extended to all (reduced) separated schemes of finite type over spec(k). 4 J.-P. Brasselet, J. Schürmann & S. YokuraFor X a compact complex algebraic variety, we can also construct the following commutative diagram of natural transformations:4) with L * the homology L-class transformation of Cappell-Shaneson [21] (as reformulated by Yokura [81]). Here Ω(X) is the Abelian group of cobordism classes of self-dual constructible complexes. To make Ω(·) covariant functorial (correcting [81]), we use the definition of a "cobordism" given by Youssin [89] in the more general context of triangulated categories with duality. Note that the homology L-class transformation of Cappell-Shaneson [21] is defined o...

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Motivic Chern classes are elements in the K-theory of an algebraic variety X, depending on an extra parameter y. They are determined by functoriality and a normalization property for smooth X. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory of flag manifolds G/B. We show that the motivic class of a Schubert cell is determined recursively by the Demazure-Lusztig operators in the Hecke algebra of the Weyl group of G, starting from the class of a point. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Fehér, Rimányi and Weber. The Hecke algebra action on the K-theory of the Langlands dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the Poincaré duals of the motivic Chern classes to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman's basis. We apply this correspondence to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations and holomorphy properties of the coefficients in the transition matrix between the standard and the Casselman's basis. 1 The class SM Cy(Ω) := D(M Cy (Ω)) λy(T * (G/B)) may be regarded as the motivic analogue of the Segre-MacPherson class c * (1 1 Ω ) c(T (G/B)) .

Generalizing a theorem of Macdonald, we show a formula for the mixed Hodge structure on the cohomology of the symmetric products of bounded complexes of mixed Hodge modules by showing the existence of the canonical action of the symmetric group on the multiple external self-products of complexes of mixed Hodge modules. We also generalize a theorem of Hirzebruch and Zagier on the signature of the symmetric products of manifolds to the case of the symmetric products of symmetric parings on bounded complexes with constructible cohomology sheaves where the pairing is not assumed to be non-degenerate.