Let $\Omega \subset \mathbb{R}^d$ with $d\geq 2$ be a bounded domain of class ${\mathcal C}^{1,\beta }$ for some $\beta \in (0,1)$. For $p\in (1, \infty )$ and $s\in (0,1)$, let $\Lambda ^s_{p}(\Omega )$ be the first eigenvalue of the mixed local–nonlocal operator $-\Delta _p+(-\Delta _p)^s$ in Ω with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber–Krahn-type inequality for $\Lambda _{p}^s(\cdot )$ under polarization. As an application of this strict inequality, we obtain the strict monotonicity of $\Lambda _{p}^s(\cdot )$ over the annular domains and characterize the rigidity property of the balls in the classical Faber–Krahn inequality for $-\Delta _p+(-\Delta _p)^s$.

A nine-dimensional exponential Lie group G and a linear form ℓ on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at ℓ the canonically associated unitary representation ρ=ρ(ℓ,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

It is well known that every finite subgroup of GLd(Q[ell ]) is conjugate to a subgroup of GLd(Z[ell ]). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if [ell ]>d+1, and G is a subgroup of Sp2d(Q[ell ]) of inertia type, then G is conjugate in GL2d(Q[ell ]) to a subgroup of Sp2d(Z[ell ]). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime [ell ], isogeny classes of Abelian varieties all of whose polarizations have degree divisible by [ell ]2. We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.