Functional calculus on non-homogeneous operators on nilpotent groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator $$\mathcal {L}$$ L on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator $$\mathcal {L}_0$$ L 0 on the ‘local’ contraction $$G_0$$ G 0 of G, as well as of the corresponding Rockland operator $$\mathcal {L}_\infty $$ L ∞ on the ‘global’ contraction $$G_\infty $$ G ∞ of G. We provide asymptotic estimates of the Riesz potentials associated with $$\mathcal {L}$$ L at 0 and at $$\infty $$ ∞ , as well as of the kernels associated with functions of $$\mathcal {L}$$ L satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for $$\mathcal {L}$$ L which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with $$\mathcal {L}$$ L from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.

Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups

We analyze the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by Meckes for the Abelian case. We show that for regular sequences of groups, the limiting distribution of eigenvalues (respectively singular values) is a mixture of eigenvalue (respectively singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular, for the sequence of symmetric groups, the limiting distributions are just the circular and quarter circular laws, whereas e.g. for the dihedral groups, the limiting distributions have unbounded supports but are different than in the Abelian case. We also prove that under additional assumptions on the sequence of groups (in particular, for symmetric groups of increasing order) families of stochastically independent random convolution operators converge in moments to free circular elements. Finally, in the Gaussian case, we provide Central Limit Theorems for linear eigenvalue statistics.

Random permutations and related topics

This article considers some topics in random permutations and random partitions highlighting analogies with random matrix theory (RMT). An ensemble of random permutations is determined by a probability distribution on Sn, the set of permutations of [n] := {1, 2, . . . , n}. In many ways, the symmetric group Sn is linked to classical matrix groups. Ensembles of random permutations should be given the same treatment as random matrix ensembles, such as the ensembles of classical compact groups and symmetric spaces of compact type with normalized invariant measure. The article first describes the Ewens measures, virtual permutations, and the Poisson-Dirichlet distributions before discussing results related to the Plancherel measure on the set of equivalence classes of irreducible representations of Sn and its consecutive generalizations: the z-measures and the Schur measures.

Plancherel Distribution of Satake Parameters of Maass Cusp Forms on GL3

We prove an equidistribution result for the Satake parameters of Maass cusp forms on $\operatorname{GL}_{3}$ with respect to the p-adic Plancherel measure by using an application of the Kuznetsov trace formula. The techniques developed in this paper deal with the removal of arithmetic weight $L(1,F,\operatorname{Ad})^{-1}$ in the Kuznetsov trace formula on $\operatorname{GL}_{3}$.

Transfer of Plancherel Measures for Unitary Supercuspidal Representations betweenp-adic Inner Forms

LetFbe ap-adic field of characteristic 0, and letMbe anF-Levi subgroup of a connected reductiveF-split group such thatGLnifor positive integersrand ni. We prove that the Plancherel measure for any unitary supercuspidal representation ofM(F) is identically transferred underthe local Jacquet–Langlands type correspondencebetweenMand itsF-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups SO4nand Sp4nunderthe local Jacquet–Langlands correspondence. It can be applied to a simply connected simpleF-group of typeE6orE7, and a connected reductiveF-group of typeAn,Bn,CnorDn.

LIMIT MULTIPLICITIES FOR PRINCIPAL CONGRUENCE SUBGROUPS OF AND

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups $\text{GL}(n)$ and $\text{SL}(n)$ we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur’s trace formula obtained in [Finis, Lapid, and Müller, Ann. of Math. (2) 174(1) (2011), 173–195; Finis and Lapid, Ann. of Math. (2) 174(1) (2011), 197–223], which allows us to show that for $\text{GL}(n)$ and $\text{SL}(n)$ the contribution of the continuous spectrum is negligible in the limit.