In this paper we prove bounds for ergodic averages for nilflows on general higher-step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become equidistributed at polynomial speed. We analyze the rate of decay which is determined by the number of steps and structure of general nilpotent Lie algebras. Our main result follows from the technique of controlling scaling operators in irreducible representations and measure estimation on close return orbits on general nilmanifolds.

We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker–Fourier coefficients of automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that a smooth admissible representation $\unicode[STIX]{x1D70B}$ has a generalized Whittaker model ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ corresponding to a nilpotent coadjoint orbit ${\mathcal{O}}$ if and only if ${\mathcal{O}}$ lies in the (closure of) the wave-front set $\operatorname{WF}(\unicode[STIX]{x1D70B})$. Previously this was only known to hold for $\mathbb{F}$ non-archimedean and ${\mathcal{O}}$ maximal in $\operatorname{WF}(\unicode[STIX]{x1D70B})$, see Moeglin and Waldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427–452]. We also express ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; Aizenbud et al., Derivatives for representations of$\text{GL}(n,\mathbb{R})$and$\text{GL}(n,\mathbb{C})$, Israel J. Math. 206 (2015), 1–38]. This enables us to extend to $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ several further results by Moeglin and Waldspurger on the dimension of ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ and on the exactness of the generalized Whittaker functor.

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.

Given a semidirect product $G\,=\,N\,\rtimes \,H$ where $N$ is nilpotent, connected, simply connected and normal in $G$ and where $H$ is a vector group for which $ad(\mathfrak{h})$ is completely reducible and $\mathbf{R}$-split, let $\tau $ denote the quasiregular representation of $G$ in ${{L}^{2}}(N)$. An element $\psi \,\in \,{{L}^{2}}(N)$ is said to be admissible if the wavelet transform $f\,\mapsto \,\left\langle f,\,\tau (\cdot )\psi \right\rangle $ defines an isometry from ${{L}^{2}}(N)$ into ${{L}^{2}}(G)$. In this paper we give an explicit construction of admissible vectors in the case where $G$ is not unimodular and the stabilizers in $H$ of its action on $\hat{N}$ are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of ${{L}^{2}}(G)$ into $G$-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.

We study the semilinear equation

$$-{{\Delta }_{\mathbb{H}}}u(\eta )\,+\,u(\eta )\,=\,f(\eta ,\,\,u(\eta )),\,u\in \,\overset{\circ }{\mathop{S}}\,_{1}^{2}(\Omega ),$$

where $\Omega $ is an unbounded domain of the Heisenberg group ${{\mathbb{H}}^{N}},\,N\,\ge \,1$. The space $\overset{\circ }{\mathop{S}}\,_{1}^{2}(\Omega )$ is the Heisenberg analogue of the Sobolev space $W_{0}^{1,\,2}\,\left( \Omega \right)$. The function $f\,:\,\overset{-}{\mathop{\Omega }}\,\,\times \,\mathbb{R}\,\to \,\mathbb{R}$ is supposed to be odd in $u$, continuous and satisfy some (superlinear but subcritical) growth conditions. The operator ${{\Delta }_{\mathbb{H}}}$ is the subelliptic Laplacian on the Heisenberg group. We give a condition on $\Omega $ which implies the existence of infinitely many solutions of the above equation. In the proof we rewrite the equation as a variational problem, and show that the corresponding functional satisfies the Palais–Smale condition. This might be quite surprising since we deal with domains which are far frombounded. The technique we use rests on a compactness argument and the maximum principle.

let $l$ denote a right-invariant sub-laplacian on an exponential, hence solvable lie group $g$, endowed with a left-invariant haar measure. depending on the structure of $g$, and possibly also that of $l, l$ may admit differentiable $l^p$-functional calculi, or may be of holomorphic $l^p$-type for a given $p{\ne} 2$. ‘holomorphic $l^p$-type’ means that every $l^p$-spectral multiplier for $l$ is necessarily holomorphic in a complex neighbourhood of some non-isolated point of the $l^2$-spectrum of $l$. this can in fact only arise if the group algebra $l^1(g)$ is non-symmetric.

assume that $p{\ne} 2$. for a point $\ell$ in the dual $\frak{g}^*$ of the lie algebra $\frak{g}$ of $g$, denote by $\omega(\ell){=}\ad^*(g)\ell$ the corresponding coadjoint orbit. it is proved that every sub-laplacian on $g$ is of holomorphic $l^p$-type, provided that there exists a point $\ell{\in} \frak{g}^*$ satisfying boidol's condition (which is equivalent to the non-symmetry of $l^1(g)$), such that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed. this work improves on results in previous work by christ and müller and ludwig and müller in twofold ways: on the one hand, no restriction is imposed on the structure of the exponential group $g$, and on the other hand, for the case $p{>}1$, the conditions need to hold for a single coadjoint orbit only, and not for an open set of orbits.

it seems likely that the condition that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed could be replaced by the weaker condition that the orbit $\omega(\ell)$ itself is closed. this would then prove one implication of a conjecture by ludwig and müller, according to which there exists a sub-laplacian of holomorphic $l^1$ (or, more generally, $l^p$) type on $g$ if and only if there exists a point $\ell{\in} \frak{g}^*$ whose orbit is closed and which satisfies boidol's condition.

In this paper the Corwin's conjecture is proved, which says that if d is a function analytic near ∞, then the hypoellipticity of the convolution operator Ad, defined by for every u ∊ S'(ℝn), implies that P(x)/ logx → ∞ as x → ∞, where P(x) is the distance from x ∊ ℝn to the set of complex zeros of d.