Abstract:
Gel'fand and Cetlin constructed in the 1950s a canonical basis for a finite-dimensional representation V(\lambda) of U(n,\C) by successive decompositions of the representation by a chain of subgroups. Guillemin and Sternberg constructed in the 1980s the Gel'fand-Cetlin integrable system on the coadjoint orbits of U(n,\C), which is the symplectic geometric version, via geometric quantization, of the Gel'fand-Cetlin construction. (Much the same construction works for representations of SO(n,\R).) A. Molev in 1999 found a Gel'fand-Cetlin-type basis for representations of the symplectic group, using essentially new ideas. An important new role is played by the Yangian Y(2), an infinite-dimensional Hopf algebra, and a subalgebra of Y(2) called the twisted Yangian Y^{-}(2). In this paper we use deformation theory to give the analogous symplectic-geometric results for the case of U(n,\H), i.e. we construct a completely integrable system on the coadjoint orbits of U(n,\H). We call this the Gel'fand-Cetlin-Molev integrable system.

Abstract:
Let G be a compact Lie group and LG its associated loop group. The main result of this manuscript is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient. Our result is a K-theoretic analogue of previous work in rational Borel-equivariant cohomology of Bott, Tolman, and Weitsman. Our proof techniques differ from that of Bott, Tolman, and Weitsman in that they explicitly use the Borel construction, which we do not have at our disposal in equivariant K-theory; we instead directly construct G-equivariant homotopy equivalences to obtain the necessary isomorphisms in equivariant K-theory. The main theorem should also be viewed as a first step toward a similar theorem in K-theory for quasi-Hamiltonian G-spaces and their associated quasi-Hamiltonian quotients.

Abstract:
This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.

Abstract:
In this manuscript we develop the theory of poset pinball, a combinatorial game recently introduced by Harada and Tymoczko for the study of the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. Harada and Tymoczko also prove that in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of the GKM-compatible subspace. Our main contributions are twofold. First we construct an algorithm (which we call the dimension pair algorithm) which yields the result of a successful outcome of Betti poset pinball for any type $A$ regular nilpotent Hessenberg and any type $A$ nilpotent Springer variety, considered as GKM-compatible subspaces of the flag variety $\Flags(\C^n)$. The definition of the algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials which we learned from Erik Insko. Second, in the special case of the type $A$ regular nilpotent Hessenberg varieties specified by the Hessenberg function $h(1)=h(2)=3$ and $h(i) = i+1$ for $3 \leq i \leq n-1$ and $h(n)=n$, we prove that the pinball result coming from the dimension pair algorithm is poset-upper-triangular; by results of Harada and Tymoczko this implies the corresponding equivariant cohomology classes form a $H^*_{S^1}(\pt)$-module basis for the $S^1$-equivariant cohomology ring of the Hessenberg variety.

Abstract:
The main result of this note is a Giambelli formula for the Peterson Schubert classes in the $S^1$-equivariant cohomology ring of a type $A$ Peterson variety. Our results depend on the Monk formula for the equivariant structure constants for the Peterson Schubert classes derived by Harada and Tymoczko. In addition, we give proofs of two facts observed by H. Naruse: firstly, that some constants which appear in the multiplicative structure of the $S^1$-equivariant cohomology of Peterson varieties are Stirling numbers of the second kind, and secondly, that the Peterson Schubert classes satisfy a stability property in a sense analogous to the stability of the classical equivariant Schubert classes in the $T$-equivariant cohomology of the flag variety.

Abstract:
The main contribution of this paper is a generalization of several previous localization theories in equivariant symplectic geometry, including the classical Atiyah-Bott/Berline-Vergne localization theorem, as well as many cases of the localization via the norm-square of the momentum map as initiated and developed by Witten, Paradan, and Woodward. Our version unifies and generalizes these theories by using noncompact cobordisms as in previous work of Guillemin, Ginzburg, and Karshon, and by introducing a more flexible notion of `polarization' than in previous theories. Our localization formulas are also valid for closed 2-forms that may be degenerate. As a corollary, we are able to answer a question posed some time ago by Shlomo Sternberg concerning the classical Brianchon-Gram polytope decomposition,. We illustrate our theory using concrete examples motivated by our answer to Sternberg's question.

Abstract:
We study type $A$ nilpotent Hessenberg varieties equipped with a natural $S^1$-action using techniques introduced by Tymoczko, Harada-Tymoczko, and Bayegan-Harada, with a particular emphasis on a special class of nilpotent Springer varieties corresponding to the partition $\lambda= (n-2,2)$ for $n \geq 4$. First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with $n$ boxes with the alphabet $\{1,2,\ldots,n\}$. Using the adjacent-pair matrix we make more explicit and also extend some statements concerning highest forms of linear operators in previous work of Tymoczko. Second, for a nilpotent operator $N$ and Hessenberg function $h$, we construct an explicit bijection between the $S^1$-fixed points of the nilpotent Hessenberg variety $\Hess(N,h)$ and the set of $(h,\lambda_N)$-permissible fillings of the Young diagram $\lambda_N$. Third, we use poset pinball, the combinatorial game introduced by Harada and Tymoczko, to study the $S^1$-equivariant cohomology of type $A$ Springer varieties $\mathcal{S}_{(n-2,2)}$ associated to Young diagrams of shape $(n-2,2)$ for $n\geq 4$. Specifically, we use the dimension pair algorithm for Betti-acceptable pinball described by Bayegan and Harada to specify a subset of the equivariant Schubert classes in the $T$-equivariant cohomology of the flag variety $\mathcal{F}\ell ags(\C^n)$ which maps to a module basis of $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$ under the projection $H^*_T(\mathcal{F}\ell ags(\C^n)) \to H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. Our pinball module basis is not poset-upper-triangular; this is the first concrete such example in the literature. A consequence of our proof is that there exists a simple and explicit change of basis which transforms our basis to a poset-upper-triangular module basis for $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. We close with open questions for future work.

Abstract:
Let $G$ be a complex semisimple simply connected linear algebraic group. Let $\lambda$ be a dominant weight for $G$ and $\mathcal{I} = (i_1, i_2, \ldots, i_n)$ a word decomposition for an element $w = s_{i_1} s_{i_2} \cdots s_{i_n}$ of the Weyl group of $G$, where the $s_i$ are the simple reflections. In the 1990s, Grossberg and Karshon introduced a virtual lattice polytope associated to $\lambda$ and $\mathcal{I}$, which they called a twisted cube, whose lattice points encode (counted with sign according to a density function) characters of representations of $G$. In recent work, the first author and Jihyeon Yang prove that the Grossberg-Karshon twisted cube is untwisted (so the support of the density function is a closed convex polytope) precisely when a certain torus-invariant divisor on a toric variety, constructed from the data of $\lambda$ and $\mathcal{I}$, is basepoint-free. This corresponds to the situation in which the Grossberg-Karshon character formula is a true combinatorial formula in the sense that there are no terms appearing with a minus sign. In this note, we translate this toric-geometric condition to the combinatorics of $\mathcal{I}$ and $\lambda$. More precisely, we introduce the notion of hesitant $\lambda$-walks and then prove that the associated Grossberg-Karshon twisted cube is untwisted precisely when $\mathcal{I}$ is hesitant-$\lambda$-walk-avoiding.

Abstract:
Let X be a smooth complex projective variety of dimension n equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a completely integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions H_1, ... H_n on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U. Moreover, we show that in many cases, we can construct the integrable system so that the functions H_1, ..., H_n generate a Hamiltonian torus action on U. In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the "moment map" \mu = (H_1, ..., H_n): X to R^n is precisely the Okounkov body \Delta = \Delta(R, v) associated to the homogeneous coordinate ring R of X, and an appropriate choice of a valuation v on R. Our main technical tools come from algebraic geometry, differential (Kaehler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Lojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties X, this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.