Related papers: Dirac cohomology, branching laws and Wallach modul…
The purpose of this paper is to show the relationship in all dimensions between the structural (diffraction pattern) aspect of tilings (described by \v{C}ech cohomology of the tiling space) and the spectral properties (of Hamiltonians…
The tempered representations of a real reductive Lie group $G$ are naturally partitioned into series associated with conjugacy classes of Cartan subgroups $H$ of $G$. We define partial Dirac cohomology, apply it for geometric construction…
Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as as…
In this paper we study the problem of computing the Dirac cohomology of the special unipotent representations of the real groups Sp(2n,R) and U(p,q).
We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak version of the…
A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the…
Let G be a real reductive Lie group with maximal compact sub- group K. We generalize the usual notion of Dirac index to a twisted version, which is nontrivial even in case G and K do not have equal rank. We compute ordinary and twisted…
We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group…
Let $G$ be a connected real reductive group with maximal compact subgroup $K$ of the same rank as $G$. In the recent paper of Huang, Pand\v{z}i\'{c} and Vogan, it was shown that the admissible $\Theta$--stable parabolic subalgebras…
We present a novel classification of unitarizable supermodules over special linear Lie superalgebras using an algebraic quadratic Dirac operator introduced by Huang and Pand\v{z}i\'c and a corresponding Dirac inequality.
Given a symmetrizable Kac-Moody algebra $\mathfrack{g}$, we study its $\pi$-systems, which are subsets of real roots, the pairwise differences of whose elements are not roots. Such systems arise as simple systems of regular subalgebras of…
We generalize the results of a previous paper of ours to compact Lie groups. Using a recently developed ordinary equivariant homology and cohomology, we define equivariant Poincare complexes with the properties that (1) every compact…
In the setting of a proper, cocompact action by a locally compact, unimodular group $G$ on a Riemannian manifold, we construct equivariant spectral flow of paths of Dirac-type operators. This takes values in the $K$-theory of the group…
The homotopy groups of a commutative algebra in spectra form a commutative algebra in the symmetric monoidal category of graded abelian groups. The grading and the Koszul sign rule are remnants of the structure encoded by anima as opposed…
This paper classifies the equivalence classes of irreducible unitary representations with nonvanishing Dirac cohomology for complex $E_6$. This is achieved by using our finiteness result, and by improving the computing method.
We establish a connection between continuous K-theory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous K-groups vanish in degrees below the negative of the…
The unitary dual of $GL(n, \mathbb{R})$ was classified by Vogan in the 1980s. Focusing on the irreducible unitary representations of $GL(n, \mathbb{R})$ with half-integral infinitesimal characters, we find that Speh representations and the…
We discuss recent developments on branching problems of irreducible unitary representations $\pi$ of real reductive groups when restricted to reductive subgroups. Highlighting the case where the underlying $(g,K)$-modules of $\pi$ are…
In this paper, we generalize the Dirac-dual-Dirac method to Hecke pairs with equivariant coarse embeddings and establish the K-theoretic isomorphisms between the maximal and reduced equivariant Roe algebras. We also extend these results to…
Over a field of characteristic p > 2, the first cohomology of the special linear Lie superalgebra sl(2,1) with coefficients in all \c{hi}-reduced Kac modules and simple modules is determined by use of the weight space decompositions of…