Equivariant tautological integrals on flag varieties (Q6564186)

From the introduction: ``Let \(\mathbb P^n\) be the complex projective space of dimension \(n\) and \(x\in H^2 (\mathbb P^n; \mathbb Z)\) be hyperplane class. Then \(H^* (\mathbb P^n; \mathbb Z)= \mathbb Z [x]/ \langle x^{n+1}\rangle\). A general element in \(H^* (\mathbb P^n; \mathbb Z)\) can be written as \(Q(x)\) for some polynomial \(Q\) with coefficients in \(\mathbb Z\), and we have\N\[\N\int_{\mathbb P^n} Q(x) = \operatorname{Res}_{z=0} \frac{Q(z)}{z^{n+1}} dz. \N\]\NNote that there is a natural action of the torus \(T = ({\mathbb C}^*)^{n+1}\) on \(\mathbb P^n\) as follows: \N\[\NT\times \mathbb P^n \rightarrow \mathbb P^n, \quad ((t_0, \dots,t_n),[x_0,\dots, x_n])\rightarrow [t_0 x_0,\dots, t_n x_n]. \N\]\NWe can also consider integrals of \(T\) -equivariant cohomology classes. Let \(H\) be the equivariant hyperplane class, \(\mathfrak{t}\) be the Lie algebra of \(T\) and \({\alpha}_i = 2\pi \sqrt{-1} u_i,\, i_1,\dots, n+1\) be the weights of \(T\) defined by \({\alpha}_i (X_1,\dots, X_{n+1})=X_i\). Then for any polynomial \(Q\) we have \N\[ \N\int_{\mathbb P^n} Q(H) = \operatorname{Res}_{z=\infty} \frac{-Q(z)}{\prod_{1\leq i \leq n+1}(u_i + z)} dz. \N\]\Nwhere \(\operatorname{Res}_{z=\infty}\) is the residue at infinity.\N\NIt is natural to ask for the similar formulas for general flag varieties. In [Ann. Math. (2) 175, No. 2, 567--629 (2012; Zbl 1247.58021)], \textit{G. Bérczi} and \textit{A. Szenes} wrote the equivariant integrals as iterated residues at infinity. The main idea of [loc. cit.] is to apply the Atiyah-Bott-Berline-Vergne formula to express the equivariant integrals as sums over fixed points and then play combinatorics. \N\NIn this paper, we follow the idea of Bérczi and Szenes to derive the equivariant residue formulas for generalized flag varieties of types \(A, B, C, D\).''\N\NIn particular, the author recovers the formulas expressing the integrals as iterated residues at infinity, which were first obtained by \textit{M. Zielenkiewicz} [``The Gysin homomorphism for homogeneous spaces via residues - PhD Thesis'', Preprint, \url{arXiv:1702.04123}] using symplectic reduction.