IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

We relate resolvent and scattering kernels for the Laplace operator on Riemannian symmetric spaces of rank one via boundary values in the sense of KashiwaraOshima. From this, we derive that the poles of the corresponding meromorphic ¯ continuations agree in a half-plane, and the residues correspond to each other under the boundary value map, so in particular the multiplicities agree as well. In the opposite half-plane, which is the square root of the resolvent set, the resolvent has no poles, whereas the scattering poles agree with the poles of the standard Knapp– Stein intertwiner. As a by-product of the underlying ideas, we obtain a new and self-contained proof of Helgason’s conjecture for distributions in the case of rank one symmetric spaces.

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International Mathematics Research Notices

Resonances and Scattering Poles in Symmetric Spaces of Rank One

In this note, we address a question raised by B. Krötz on the classification of G-invariant domains of holomorphy for irreducible admissible Banach representations of connected non-compact simple real linear Lie groups G. When G is not of Hermitian type, we give a complete description of such G-invariant domains for irreducible admissible uniformly bounded representations on reflexive Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine such G-invariant domains only when the representations considered are highest or lowest weight representations. © 2017, Springer Science+Business Media, LLC, part of Springer Nature.

Transformation Groups

DOMAINS OF HOLOMORPHY FOR IRREDUCIBLE ADMISSIBLE UNIFORMLY BOUNDED BANACH REPRESENTATIONS OF SIMPLE LIE GROUPS

Let G be a connected reductive complex algebraic group with split real form (G, σ). Consider a strict wonderful G-variety X equipped with its σ-equivariant real structure, and let X be the corresponding real locus. Further, let E be a real differentiable G-vector bundle over X. In this paper, we introduce a distribution character for the regular representation of G on the space of smooth sections of E given in terms of the spherical roots of X, and show that on a certain open subset of G of transversal elements it is locally integrable and given by a sum over fixed points. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Mathematische Nachrichten

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Consider a <a href="https://www.sciencedirect.com/topics/mathematics/riemannian-symmetric-space">Riemannian symmetric space</a> of non-compact type, where G is a connected, real, <a href="https://www.sciencedirect.com/topics/mathematics/semisimple">semisimple</a> Lie group, and K a maximal compact subgroup. Let be its Oshima <a href="https://www.sciencedirect.com/topics/mathematics/compactifications">compactification</a>, and the left-regular representation of G on . In this paper, we examine the <a href="https://www.sciencedirect.com/topics/mathematics/convolution-operator">convolution operators</a> for rapidly decaying functions f on G, and characterize them within the framework of totally characteristic <a href="https://www.sciencedirect.com/topics/mathematics/pseudodifferential-operator">pseudodifferential operators</a>, describing the singular nature of their kernels. As a consequence, we obtain asymptotics for heat and <a href="https://www.sciencedirect.com/topics/mathematics/resolvent">resolvent</a> kernels associated to strongly <a href="https://www.sciencedirect.com/topics/mathematics/elliptic-operator">elliptic operators</a> on . As a further application, a regularized trace for the operators can be defined, yielding a distribution on G which can be interpreted as a global character of π, and is given by a fixed point formula analogous to the Atiyah–Bott character formula for an induced representation of G.