arXiv subjects

Robin van Haastrecht

Publications and source records attributed to Robin van Haastrecht.

6 recordsLinked to original sources

arXiv2026

Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces

Let $G/K$ be a Hermitian symmetric space and $V_τ$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_τ)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, τ)} V_τ$ over $G/K$ defined by a finite-dimensional representation $(V_τ, τ)$ of $K$. We classify irreducible representations $(V_τ, τ)$ such that $\mathcal D^G(G/K, V_τ)$ is commutative. We construct eigenfunctions for the differential operators and study the invariance property of the eigenvalues under the Weyl group for the restricted real root system of $G$.

math.RT↗

arXiv2026

On the degenerate principal series of $G_{2(2)}$ induced from a Heisenberg parabolic subgroup

We study degenerate principal series representations of the split real group $G_{2(2)}$ induced from a character of a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. Using the Lie algebra action on the space of $K$-finite vectors, we find the points of reducibility and the complementary series. The minimal representation and a limit of discrete series are identified as kernel of the corresponding Knapp-Stein intertwining operator. Moreover, we show that some quaternionic discrete series representations occur as the subrepresentation on which the family of intertwining operators vanishes of order two.

math.RT↗

arXiv2025

Wehrl inequalities for matrix coefficients of holomorphic discrete series

We prove Wehrl-type $L^2(G)-L^{p}(G)$ inequalities for matrix coefficients of vector-valued holomorphic discrete series of $G$, for even integers $p=2n$. The optimal constant is expressed in terms of Harish-Chandra formal degrees for the discrete series. We prove the maximizers are precisely the reproducing kernels.

math.RT↗

arXiv2024

Functional calculus of quantum channels for the holomorphic discrete series of $SU(1,1)$

The tensor product of two holomorphic discrete series representations of $SU(1,1)$ can be decomposed as a direct sum of infinitely many discrete series. I shall introduce equivariant quantum channels for each component of the direct sum, mapping bounded operators on one factor of the tensor product to operators on the component. Next I prove a limit formula for the trace of the functional calculus and I prove that the limit can be expressed using generalized Husimi functions or using Berezin transforms.

math.RT↗

arXiv2024

Limit formulas for the trace of the functional calculus of quantum channels for $SU(2)$

Lieb and Solovej \cite{liebsolBloch} studied traces of quantum channels, defined by the leading component in the decomposition of the tensor product of two irreducible representations of $SU(2)$, to establish a Wehrl-type inequality for integrals of convex functions of matrix coefficients. It is proved that the integral is the limit of the trace of the functional calculus of quantum channels. In this paper, we introduce new quantum channels for all the components in the tensor product and generalize their limit formula. We prove that the limit can be expressed using Berezin transforms.

math.RT↗

arXiv2020

Gelfand Pairs of Complex Reflection Groups

In this article the zonal spherical functions of the Gelfand pair $(G(r,d,n), S_n)$ of complex reflection groups will be calculated. After this, a product formula for these spherical functions and a discrete analog of the Laplace operator which has the spherical functions as eigenfunctions will be given.

math.RT↗