Whittaker functions from motivic Chern classes (Q2674773)

Let \(G\) be a reductive algebraic group, \(B\) its Borel subgroup containing a maximal torus \(T\). Let \(X=G/B\) be the generalized flag variety, \(W\) the Weyl group of \(G\). For any character \(\lambda\colon T\to \mathbb C^*\) let \(\mathcal L_\lambda\) be the associated line bundle over \(X\). The localization formula for the torus action allows to express the Euler characteristic \(\chi(X,\mathcal L_\lambda)\) as a sum of local contributions coming from the fixed points. If \(\lambda\) is (anti-)dominant we obtain exactly the Weyl character formula. In the present paper the bundle \(\mathcal L_\lambda\) is replaced by \(\mathcal L_\lambda\otimes MC(X(w)^\circ)\), where \(MC(X(w)^\circ)\in K_T(X)\) is the equivariant motivic Chern class of the Schubert cell corresponding to \(w\in W\). (It is an element of the equivariant K-theory of \(X\).) One of the main results says that \(\chi(X,\mathcal L_\lambda\otimes MC(X(w)^\circ))\) can be computed by application of Demazure-Lusztig operations to the monomial \(e^\lambda\). Also, an inductive formula is obtained for normalized \(MC\)-classes by applyication of duality. Both results heavily rely on [\textit{P. Aluffi} et al., ``Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem'', Preprint, \url{arXiv:1902.10101}]. It turns out that the same recurrence holds for the Iwahori-Whittaker functions of the principal series representation of the \(p\)-adic Langlands dual group, see [\textit{B. Brubaker} et al., J. Number Theory 146, 41--68 (2015; Zbl 1366.22008)], hence the Iwahori-Whittaker functions coincide with Euler characteristics of \(\mathcal L_\lambda\otimes MC(X(w)^\circ)\). The appendix contains another nice corollary from the localization theorem for the torus action. Under certain assumptions on the characters of the tangent representations at the fixed points the localization formula expressing the \(\chi_y\)-genus of a smooth \(T\)-variety reduces to one summand. The formula can be applied for smooth Schubert varieties. Here, the \(\chi_y\)-genus coincides with Poincaré polynomial.