We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

We say that a group G of local (maybe formal) biholomorphisms satisfies the uniform intersection property if the intersection multiplicity $(\phi (V), W)$ takes only finitely many values as a function of G for any choice of analytic sets V and W of complementary dimension. In dimension $2$ we show that G satisfies the uniform intersection property if and only if it is finitely determined – that is, if there exists a natural number k such that different elements of G have different Taylor expansions of degree k at the origin. We also prove that G is finitely determined if and only if the action of G on the space of germs of analytic curves has discrete orbits.

This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$, defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of $G$. This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$, $n\geqslant 3$), prompting us to seek necessary and sufficient conditions for non-emptiness.

We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$-regularity of $(G,H)$. This $\mathfrak{a}$-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$. We also provide a classification of the $\mathfrak{a}$-regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_{0}$ at level zero of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_{0}$. We show that if $(V,G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This implies in particular that the real symplectic quotient is graded Gorenstein. In case $K$ is a torus or $\operatorname{SU}_{2}$, we show that these results hold without the hypothesis that $(V,G)$ is $3$-large.

Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$-rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$.

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak{c}$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak{c}\oplus \mathfrak{c}^{\ast }/W\rightarrow V\oplus V^{\ast }/\!\!/\!\!/G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if$(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and some other examples.

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.

We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.

Let $\rho :\,G\,\to \,\text{GL}\left( V \right)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let ${{\sigma }_{1}},\ldots ,{{\sigma }_{n}}$ be a system of generators of the algebra of invariant polynomials $\mathbb{C}{{\left[ V \right]}^{G}}$. We study the problem of lifting mappings $f:\,{{\mathbb{R}}^{q}}\,\supseteq \,U\,\to \,\sigma \left( V \right)\,\subseteq \,{{\mathbb{C}}^{n}}$ over the mapping of invariants $\sigma \,=\,\left( {{\sigma }_{1}},\ldots ,{{\sigma }_{n}} \right):\,V\,\to \,\sigma \left( V \right)$. Note that $\sigma \left( V \right)$ can be identified with the categorical quotient $V//G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $C\subseteq {{C}^{\infty }}$ satisfying some mild closedness properties that guarantee resolution of singularities in $C$, e.g., the real analytic class, then $f$ admits a lift of the same class $C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\text{SB}{{\text{V}}_{\text{loc}}}$, i.e., special functions of bounded variation. If $\rho $ is a real representation of a compact Lie group, we obtain stronger versions.

Let $W$ be a Weyl group, $\sum $ a set of simple reflections in $W$ related to a basis $\Delta $ for the root system $\Phi $ associated with $W$ and $\theta $ an involution such that $\theta (\Delta )\,=\,\Delta $. We show that the set of $\theta $- twisted involutions in $W$, ${{\mathcal{J}}_{\theta }}\,=\,\{w\,\in \,W\,|\,\theta (w)\,=\,{{w}^{-1}}\}$ is in one to one correspondence with the set of regular involutions ${{\mathcal{J}}_{\text{ID}}}$. The elements of ${{\mathcal{J}}_{\theta }}$ are characterized by sequences in $\sum $ which induce an ordering called the Richardson–Springer Poset. In particular, for $\Phi $ irreducible, the ascending Richardson–Springer Poset of ${{\mathcal{J}}_{\theta }}$, for nontrivial $\theta $ is identical to the descending Richardson–Springer Poset of ${{\mathcal{J}}_{\text{ID}}}$.

We consider a class of reductive linear groups defined in terms of weighted oriented graphs of a special sort that we call signed quivers. Each of these yields a symmetric quiver, that is, a quiver endowed with an involutive anti-automorphism together with signs for the vertices and arrows fixed by the involution. The orbits of the groups can be described in terms of the indecomposable symmetric representations of symmetric quivers. We provide a general description of the indecomposable symmetric representations and prove that their dimensions in the finite and tame cases constitute root systems corresponding to certain symmetrizable generalized Cartan matrices.

We explain how A. Givental’s mirror symmetric family [14] to the type A ag variety and its proposed generalization [3] to partial ag varieties by Batyrev, Ciocan-Fontanine, Kim and van Straten relate to the Peterson variety Y ⊂ SLn/B. We then use this theory to describe the totally nonnegative part of Y, extending a result from [30].

Let $\sigma $, $\theta $ be commuting involutions of the connected semisimple algebraic group $G$ where $\sigma$, $\theta $ and $G$ are defined over an algebraically closed field $\underset{\scriptscriptstyle-}{k},$ char $\underline{k}$=0. Let $H:={{G}^{\sigma }}$ and $K:={{G}^{\theta }}$ be the fixed point groups. We have an action $\left( H\,\times \,K \right)\,\times \,G\,\to \,G$, where $\left( \left( h,\,k \right),\,g \right)\,\mapsto \,hg{{k}^{-1}},\,h\,\in \,H$ , $k\,\in \,K,g\,\in \,G$. Let $G\,//\,\left( H\,\times \,K \right)$ denote the categorical quotient Spec $\mathcal{O}{{(G)}^{H\times K}}$ . We determine when this quotient is smooth. Our results are a generalization of those of Steinberg [Ste75], Pittie [Pit72] and Richardson [Ric82] in the symmetric case where $\sigma \,=\,\theta $ and $H\,=K$.

Sun and Wilson defined the notion of infinite determinacy of a smooth function germ singular along a line, and related this notion to some good geometric properties of derived objects related to the given function germ. The paper extends their results to a wider class of smooth function with prescribed non-isolated singularities. For this purpose, it was necessary to study the behaviour of the function germ along a transverse direction of the given singular set, and to relate these properties to geometric properties of the function and derived objects, expressed in terms of relative Łojasiewicz conditions.

We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension 5. The result implies improved universal lower bounds for the volume $\text{vo}{{\text{l}}_{\text{5}}}\left( M \right)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\text{vo}{{\text{l}}_{\text{5}}}\left( M \right)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\text{P}{{\text{S}}_{\Delta }}\text{L}\left( 2,\,\mathbb{H} \right)$ of quaternionic $2\,\times \,2$-matrices with Dieudonné determinant $\Delta$ equal to 1 and isolation properties of $\text{P}{{\text{S}}_{\Delta }}\text{L}\left( 2,\,\mathbb{H} \right)$.

We generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups $G$ and their Lie algebras $\mathfrak{g}$ under the action of a maximal compact subgroup $K$ of $G$. We also introduce a natural partial order on $\mathfrak{g}:\,x\,\le y$ if $f(K\,\cdot \,x)\,\subseteq \,f(K\,\cdot \,y)$ for all $f\,\in \,{{\mathfrak{g}}^{*}}$, the complex dual of $\mathfrak{g}$. This partial order is $K$-invariant and induces a partial order on the orbit space $\mathfrak{g}/K$. We prove that, under some restrictions on $\mathfrak{g}$, the set $f(K\,\cdot \,x)$ is star-shaped with respect to the origin.

Fundamental calculations on singular invariant hyperfunctions on the n ×n square matrix space and on the 2n × 2n alternating matrix space are considered in this paper. By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctions as their Laurent expansion coefficients. The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients. By applying these results, we prove that every quasi-relatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support. In the last section, we give the formula of the Fourier transforms of singular invariant tempered distributions.

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)-\text{space}\,{{\text{H}}^{n+1}}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of ${{\text{H}}^{n+1}}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z})$, the Gaussian modular (Picard) group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}[i])$, and the Eisenstein modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}\left[ \omega \right])$.