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Unitary representations attached to parabolic subgroups: the case of abelian unipotent radical
Dan Ciubotaru
https://arxiv.org/abs/2510.11862 https://arxiv…

Unitary representations attached to parabolic subgroups: the case of abelian unipotent radical
We classify the unitary representations with integral infinitesimal character in Lusztig's category of unipotent representations in the case when the geometric parameter space comes from the action of a Levi subgroup on the abelian nilradical of a (maximal) parabolic subalgebra. We organise the unitary representations into microlocal Arthur packets. This is a test case for investigating a conjectural description of unitary representations with integral infinitesimal character.

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Spectral Equivalence of Solvable Lie Algebras with Nilradical I: General and Heisenberg Cases
Gary Hu
https://arxiv.org/abs/2509.21418 https://arxiv.org/pd…

Spectral Equivalence of Solvable Lie Algebras with Nilradical I: General and Heisenberg Cases
Building on the work of Yang, Key, and Muller, we investigate the spectral theory of solvable Lie algebras through their characteristic polynomials and associated spectral invariants. We begin by studying the general case: we compare two notions of spectral equivalence, establish the first known lower bound for the spectral invariant $k(L)$ via the weight decomposition of the nilradical, and introduce a new notion of spectral rigidity to study spectral equivalence within parameterized families.…

On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term
Mohammud Foondun, Davar Khoshnevisan, Eulalia Nualart
https://arxiv.org/abs/2510.00214

On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term
We consider a parabolic stochastic partial differential equation (SPDE) on $[0\,,1]$ that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an $L\log L$ growth condition. We prove that the SPDE is well posed when the initial data is in $L^2[0\,,1]$. This solves a strong form of an open problem.

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