Invariant measures on nilpotent orbits associated with holomorphic discrete series

Let G R G_\mathbb R be a real form of a complex, semisimple Lie group G G . Assume G R G_\mathbb R has holomorphic discrete series. Let W \mathcal W be a nilpotent coadjoint G R G_\mathbb R -orbit contained in the wave front set of a holomorphic discrete series. We prove a limit formula, expressing the canonical measure on W \mathcal W as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the positive chamber defined by the Borel subalgebra associated with holomorphic discrete series.

Kähler–Ricci flow on homogeneous toric bundles

Assume that [Formula: see text] is a homogeneous toric bundle of the form [Formula: see text] and is Fano, where [Formula: see text] is a compact semisimple Lie group with complexification [Formula: see text], [Formula: see text] a parabolic subgroup of [Formula: see text], [Formula: see text] is a surjective homomorphism from [Formula: see text] to the algebraic torus [Formula: see text], and [Formula: see text] is a compact toric manifold of complex dimension [Formula: see text]. In this note, we show that the normalized Kähler–Ricci flow on [Formula: see text] with a [Formula: see text]-invariant initial Kähler form in [Formula: see text] converges, modulo the algebraic torus action, to a Kähler–Ricci soliton. This extends a previous work of Zhu. As a consequence, we recover a result of Podestà–Spiro.

Expanding Cone and Applications to Homogeneous Dynamics

Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

Irreducible Unitary Representations Concerning Homogeneous Holomorphic Line Bundles Over Elliptic Orbits

In this paper we consider a homogeneous holomorphic line bundle over an elliptic adjoint orbit of a real semisimple Lie group, and set a continuous representation of the Lie group on a certain complex vector subspace of the complex vector space of holomorphic cross-sections of the line bundle. Then, we demonstrate that the representation is irreducible unitary.

ON THE GROWTH OF TORSION IN THE COHOMOLOGY OF ARITHMETIC GROUPS

Let $G$ be a semisimple Lie group with associated symmetric space $D$, and let $\unicode[STIX]{x1D6E4}\subset G$ be a cocompact arithmetic group. Let $\mathscr{L}$ be a lattice inside a $\mathbb{Z}\unicode[STIX]{x1D6E4}$-module arising from a rational finite-dimensional complex representation of $G$. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup $H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}$ as $\unicode[STIX]{x1D6E4}_{k}$ ranges over a tower of congruence subgroups of $\unicode[STIX]{x1D6E4}$. In particular, they conjectured that the ratio $\log |H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}|/[\unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6E4}_{k}]$ should tend to a nonzero limit if and only if $i=(\dim (D)-1)/2$ and $G$ is a group of deficiency $1$. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including $\operatorname{GL}_{n}(\mathbb{Z})$ for $n=3,4,5$ and $\operatorname{GL}_{2}(\mathscr{O})$ for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron–Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron–Venkatesh conjecture.

Ergodic boundary representations

We prove a von Neumann-type ergodic theorem for averages of unitary operators arising from the Furstenberg–Poisson boundary representation (the quasi-regular representation) of any lattice in a non-compact connected semisimple Lie group with finite center.

Coadjoint geometry for discretely decomposable restrictions of certain series of representations of indefinite unitary groups

Zuckerman’s derived functor module of a semisimple Lie group [Formula: see text] yields a unitary representation [Formula: see text] which may be regarded as a geometric quantization of an elliptic orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. We highlight a certain family of those irreducible unitary representations [Formula: see text] of the indefinite unitary group [Formula: see text] and a family of subgroups [Formula: see text] of [Formula: see text] such that the restriction [Formula: see text] is known to be discretely decomposable and multiplicity-free by the general theory of Kobayashi (Discrete decomposibility of the restrictions of [Formula: see text] with respect to reductive subgroups, II, Ann. of Math. 147 (1998) 1–21; Multiplicity-free representations and visible action on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005) 497–549), where [Formula: see text] is not necessarily tempered and [Formula: see text] is not necessarily compact. We prove that the corresponding moment map [Formula: see text] is proper, determine the image [Formula: see text], and compute the Corwin–Greenleaf multiplicity function explicitly.

Associated symmetric pair and multiplicities of admissible restriction of discrete series

Let [Formula: see text] be a symmetric pair for a real semisimple Lie group [Formula: see text] and [Formula: see text] its associated pair. For each irreducible square integrable representation [Formula: see text] of [Formula: see text] so that its restriction to [Formula: see text] is admissible, we find an irreducible square integrable representation [Formula: see text] of [Formula: see text] which allows us to compute the Harish-Chandra parameter of each irreducible [Formula: see text]-subrepresentation of [Formula: see text] as well as its multiplicity. The computation is based on the spectral analysis of the restriction of [Formula: see text] to a maximal compact subgroup of [Formula: see text]