Related papers: The Index Theorem for Homogeneous Differential Ope…
We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the…
In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group $\mathbb{H}(G):=G \times \widehat{G} \times \mathbb{T},$ where $G$ a locally compact abelian group with its…
We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Q_k of quasi-invariants of a given multiplicity is not, in general, an algebra but a module over the…
Let $G$ be a compact Lie group that acts smoothly on a closed manifold $M$. Using a general Simonenko principle, we derive a novel criterion for the Fredholm property of $G$-pseudodifferential operators acting on Sobolev spaces of sections…
We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras,…
We consider the Weyl algebra A (=A_n(k)) and its Rees algebra B with respect to the Bernstein filtration. The homogenisation of a differential operator in A is an element in B. In this paper we establish the validity of the division theorem…
In this paper, we survey recent results on index defects of elliptic operators on manifolds with boundary. Index defects are similar to the Hirzebruch signature defects in topology, where the defects appear as the correction terms to the…
The integral formulae pertaining to the unitary group $\mathsf{U}(d)$ have been comprehensively reviewed, yielding fresh results and innovative proofs. Central to the derivation of these formulae lies the employment of Schur-Weyl duality, a…
Antiunitary representations of Lie groups take values in the group of unitary and antiunitary operators on a Hilbert space H. In quantum physics, antiunitary operators implement time inversion or a PCT symmetry, and in the modular theory of…
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of…
We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann…
We compute the K-theory of the C*-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group G.…
We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups $G$ to rank one subgroups $G_1$. For this we use the realizations of complementary…
Given a suitable ordering of the positive root system associated with a semisimple Lie algebra, there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module…
The homotopy groups of the (stabilized) group of invertible pseudodifferential operators of order zero acting on a closed manifold X are computed in terms of the K-theory of the cosphere bundle S*X. At the same time, we show that the…
Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main…
The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov-Lawson for generalised Dirac operators as well as…
The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) K\"ahler structure, famously used to realize the group's irreducible representations in holomorphic sections of appropriate line bundles (Borel-Weil…
For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This…
For an arbitrary pseudo-differential operator $A:\mathcal{S}(\mathbb{R}% ^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$ with Weyl symbol $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$, we consider the pseudo-differential operators…