arXiv subjects

Devadatta G. Hegde

Publications and source records attributed to Devadatta G. Hegde.

5 recordsLinked to original sources

arXiv2026

A simple construction of the automorphic residual spectrum

We consider the spherical Borel Eisenstein series induced from the trivial representation for a split semisimple linear algebraic group over a number field. We prove that its regularization at the special point corresponding to half the weighted marking of a distinguished coadjoint nilpotent orbit in the Langlands dual Lie algebra is nonzero and square-integrable. Our proof follows the philosophy of Kazhdan and Okounkov. We give a geometric interpretation of Langlands' square-integrability criterion in this setting and, using the equivariant integration formula, prove that the regularization satisfies this criterion. As an immediate consequence, we obtain a simple and uniform proof of Arthur's unitarity conjecture, without case-by-case analysis or machine computation.

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arXiv2026

On Franke's theorem in the simplest case

For level one spherical automorphic forms on the upper half-plane, we prove directly that every automorphic form is a sum of a cusp form and a linear combination of Laurent coefficients of the standard Eisenstein series. This is the simplest instance of Franke's general theorem, which asserts that automorphic forms on a reductive group are spanned by Laurent coefficients of Eisenstein series induced from cuspidal automorphic forms on Levi subgroups. Unlike Franke's general argument, ours does not invoke Langlands' construction of the discrete automorphic spectrum from cuspidal Eisenstein series. It rests instead on basic analytic properties of automorphic forms and Green's identity.

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arXiv2026

Rethinking the work of Langlands on Eisenstein series

Chapter $7$ of Langlands' monograph "On the functional equations satisfied by Eisenstein series" employs a sophisticated residue scheme to construct a portion of the discrete automorphic spectrum. We show, by examples, applications, and heuristics, that this construction is a straightforward regularization of cuspidal Eisenstein series at distinguished points, and that the regularization must track BOTH the zeros and the poles of these Eisenstein series. Unlike the one-variable case, the zero set and pole set of a several-complex-variable meromorphic function can intersect at a point. The distinguished points supporting the discrete spectrum are typically of this kind. The zeros of cuspidal Eisenstein series - largely invisible in the rank-one case - begin to play a starring role in higher rank situations, on equal footing with the poles. We redo Langlands' famous $G_{2}$ calculation and show that once zeros are tracked, the calculation reduces to elementary algebra. Drawing on rank-two examples, we introduce a program to re-think Langlands' construction from first principles, giving zeros and poles of cuspidal Eisenstein series equal standing from the very beginning. The program has the advantage of making the underlying phenomenon transparent, though carrying it out in full generality will require substantial further work.

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arXiv2024

Poles of unramified degenerate Eisenstein series

We determine the poles of maximal unramified degenerate Eisenstein series of a split semisimple algebraic group over a number field using a straightforward global argument, avoiding delicate analysis of intertwining operators.

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arXiv2021

Schwartz functions, Hadamard products, and the Dixmier-Malliavin theorem

In this paper we show that functions of the form $\prod_{n\ge1}\frac{1}{\left(1+\frac{x^{2}}{a_{n}^{2}}\right)}$ where $a_{n}>0$ and $\sum_{n\ge1}\frac{1}{a_{n}^{2}}<\infty$ are in the Schwartz space of the real line, answering a question raised by Casselman. As a consequence we obtain substantial simplifications in the proofs of Dixmier and Malliavin of their theorem that every test function on a Lie group is a finite linear combination of convolutions of two test functions, and an analogue of this for Fréchet space Lie group representations.

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