On the Finiteness of Quantum K-Theory of a Homogeneous Space

We show that the product in the quantum K-ring of a generalized flag manifold $G/P$ involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the $J$-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

New homogeneous Einstein metrics on quaternionic Stiefel manifolds

We consider invariant Einstein metrics on the quaternionic Stiefel manifold Vpℍn of all orthonormal p-frames in ℍn. This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on Vpℍn ≅ Sp(n)/Sp(n − p), where n = k1 + k2 + k3 and p = n − k3. We view Vpℍn as a total space over the generalized Wallach space Sp(n)/(Sp(k1)×Sp(k2)×Sp(k3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)).

Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds

We obtain an algorithm computing the Chern–Schwartz–MacPherson (CSM) classes of Schubert cells in a generalized flag manifold$G/B$. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure–Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of$G/B$. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold$G/P$. We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting.

Projections of Richardson varieties

While the projections of Schubert varieties in a full generalized flag manifold

INVARIANT EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH TWO ISOTROPY SUMMANDS

Let M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

Product and Puzzle Formulae for $GL_n$ Belkale-Kumar Coefficients

The Belkale-Kumar product on $H^*(G/P)$ is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case $G=GL_n$, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that for $G/P$ a $(d-1)$-step flag manifold, each Belkale-Kumar structure constant is a product of $d\choose 2$ Littlewood-Richardson numbers, for which there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This refines previously known factorizations into $d-1$ factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants. These "BK-puzzles" are related to extremal honeycombs, as in [Knutson-Tao-Woodward '04]; using this relation we give another proof of Ressayre's result. Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always $1$; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension $1$ or $2$.

Berezin quantization on generalized flag manifolds

Let $M=G/H$ be a generalized flag manifold where $G$ is a compact, connected, simply-connected Lie group with Lie algebra $\mathfrak{g}$ and $H$ is the centralizer of a torus. Let $\pi$ be a unitary irreducible representation of $G$ which is holomorphically induced from a character of $H$. Using a complex parametrization of a dense open subset of $M$, we realize $\pi$ on a Hilbert space of holomorphic functions. We give explicit expressions for the differential $d\pi$ of $\pi$ and for the Berezin symbols of $\pi (g)$ ($g\in G$) and $d\pi (X)$ ($X\in \mathfrak{g}$). In particular, we recover some results of S. Berceanu and we partially generalize a result of K. H. Neeb.