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We present two classes of examples of Hopf algebroids associated with noncommutative principal bundles. The first comes from deforming the principal bundle while leaving unchanged the structure Hopf algebra. The second is related to…

Given a topological group G, its orbit category Orb_G has the transitive G-spaces G/H as objects and the G-equivariant maps between them as morphisms. A well known theorem of Elmendorf then states that the category of G-spaces and the…

Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$ and suppose that $V$ is a polar $G$-representation. We prove the existence of a radial parts map $\mathrm{rad}: \mathcal{D}(V)^G\to A_{\kappa}$ from the $G$-invariant…

Let P be a parabolic subgroup of a simple affine algebraic group G defined over C and X a compact connected K\"ahler manifold. L. \'Alvarez-C\'onsul and O. Garc\'ia-Prada associated to these a quiver Q and representations of Q into…

We formulate and prove a new variant of the Segal Conjecture describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group G to the classifying space of a compact Lie group K as the p-adic…

Let $G/K$ be a Hermitian symmetric space and $V_\tau$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_\tau)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, \tau)} V_\tau$…

We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialization, recover the topological mirror symmetry conjecture of…

We describe the $C_2$-equivariant homotopy type of the space of commuting n-tuples in the stable unitary group in terms of Real K-theory. The result is used to give a complete calculation of the homotopy groups of the space of commuting…

Using Morse-theoretic techniques, we show that the moduli space of U*(2n)-Higgs bundles over a compact Riemann surface is connected.

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and…

Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $\tau$ we compare the notions of $\tau$-representation equivalence with $\tau$-isospectrality. We exhibit infinitely many $K$-types…

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on…

To the integral symplectic group Sp(2g,Z) we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the…

The central result of this paper is an identification of the shifted Cartier dual of the moduli stack $\mathcal{M}_{\mathfrak{g}}(C)$ of $\widetilde{G}$-Higgs bundles on $C$ of arbitrary degree (modulo shifts by $Z(\widetilde{G})$) with a…

We calculate the characteristic classes for flat SL(n,R) and Sp(2m,R)-bundles over a compact surface as functions of the spectral data in the Higgs bundle description, which consists of the points of order 2 in an abelian variety. Using…

We determine an explicit presentation by generators and relations of the cohomology algebra $H^*(\mathbb P^2\setminus C,\mathbb C)$ of the complement to an algebraic curve $C$ in the complex projective plane $\mathbb P^2$, via the study of…

Let $A$ be a unital simple separable exact C$^*$-algebra which is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed non-compact Cuntz class has vanishing homotopy groups. Combined…

The moduli space of stable bundles of rank 2 and degree 1 on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of…

Let w be an elliptic element of the Weyl group of a connected reductive group G. Let X be the set of pairs (g,B) where g is an element of G, B is a Borel subgroup of G and B,gBg^{-1} are in relative position w. Then G acts naturally on X.…

We consider infinite tensor product actions of $G = \mathbb{Z}/p\mathbb{Z}$ on the UHF-algebra $D = \text{End}(V)^{\otimes \infty}$ for a finite-dimensional unitary $G$-representation $V$ and determine the equivariant homotopy type of the…