Browsing by Author "Su, Changjian"

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    From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes
    Aluffi, Paolo; Mihalcea, Leonardo C.; Schürmann, Jörg; Su, Changjian (American Mathematical Society, 2024-01-01)
    The equivariant motivic Chern class of a Schubert cell in a complete flag manifold X = G/B is an element in the equivariant K-theory ring of X to which one adjoins a formal parameter y. In this paper we prove several folklore results about motivic Chern classes, including finding specializations at y = −1 and y = 0; the coefficient of the top power of y; how to obtain Chern-Schwartz-MacPherson (CSM) classes as leading terms of motivic classes; divisibility properties of the Schubert expansion of motivic Chern classes. We collect several conjectures on the positivity, unimodality, and log concavity of CSM and motivic Chern classes of Schubert cells, including a conjectural positivity of structure constants of the multiplication of Poincar´e duals of CSM classes. In addition, we prove a ‘star duality’ for the motivic Chern classes, showing how they behave under the involution taking a vector bundle to its dual. We use the motivic Chern transformation to define two equivariant variants of the Hirzebruch transformation, which appear naturally in the Grothendieck-Hirzebruch-Riemann-Roch formalism. We utilize the Demazure-Lusztig recursions from the motivic Chern class theory to find similar recursions giving the Hirzebruch classes of Schubert cells, their Poincar´e duals, and their Segre versions. We explain the functoriality properties needed to extend the results to partial flag manifolds G/P.
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    Motivic Chern Classes of Schubert Cells, Hecke Algebras, and Applications to Casselman's Problem
    Aluffi, Paolo; Mihalcea, Leonardo C.; Schuermann, Joerg; Su, Changjian (Société Mathematique de France, 2024-04-02)
    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 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.