Typical representations via fixed point sets in Bruhat–Tits buildings

For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence is a set of broadly applicable tools for addressing the branching rules of π \pi and the unicity of [ G , π ] G [G,\pi ]_G -types.

Intertwining maps between 𝑝-adic principal series of 𝑝-adic groups

In this paper we study p p -adic principal series representation of a p p -adic group G G as a module over the maximal compact subgroup G 0 G_0 . We show that there are no non-trivial G 0 G_0 -intertwining maps between principal series representations attached to characters whose restrictions to the torus of G 0 G_0 are distinct, and there are no non-scalar endomorphisms of a fixed principal series representation. This is surprising when compared with another result which we prove: that a principal series representation may contain infinitely many closed G 0 G_0 -invariant subspaces. As for the proof, we work mainly in the setting of Iwasawa modules, and deduce results about G 0 G_0 -representations by duality.

Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathit{\boldsymbol{\mathrm{G}}} $\end{document}</tex-math></inline-formula> be a semisimple linear algebraic group defined over rational numbers, <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{K} $\end{document}</tex-math></inline-formula> be a maximal compact subgroup of its real points and <inline-formula><tex-math id="M3">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> be an arithmetic lattice. One can associate a probability measure <inline-formula><tex-math id="M4">\begin{document}$ \mu_{ \mathrm{H}} $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M5">\begin{document}$ \Gamma \backslash \mathrm{G} $\end{document}</tex-math></inline-formula> for each subgroup <inline-formula><tex-math id="M6">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M7">\begin{document}$ \mathit{\boldsymbol{\mathrm{G}}} $\end{document}</tex-math></inline-formula> defined over <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{Q} $\end{document}</tex-math></inline-formula> with no non-trivial rational characters. As G acts on <inline-formula><tex-math id="M9">\begin{document}$ \Gamma \backslash \mathrm{G} $\end{document}</tex-math></inline-formula> from the right, we can push forward this measure by elements from <inline-formula><tex-math id="M10">\begin{document}$ \mathrm{G} $\end{document}</tex-math></inline-formula>. By pushing down these measures to <inline-formula><tex-math id="M11">\begin{document}$ \Gamma \backslash \mathrm{G}/ \mathrm{K} $\end{document}</tex-math></inline-formula>, we call them homogeneous. It is a natural question to ask what are the possible weak-<inline-formula><tex-math id="M12">\begin{document}$ * $\end{document}</tex-math></inline-formula> limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of <inline-formula><tex-math id="M13">\begin{document}$ \Gamma \backslash \mathrm{G}/ \mathrm{K} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M14">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general <inline-formula><tex-math id="M15">\begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document}</tex-math></inline-formula> with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on <inline-formula><tex-math id="M16">\begin{document}$ {\text{SL}}_n $\end{document}</tex-math></inline-formula> proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. <b>193</b> words.</p>

Cycle Intersection for SOp,q-Flag Domains

A real form G0 of a complex semisimple Lie group G has only finitely many orbits in any given compact G-homogeneous projective algebraic manifold Z=G/Q. A maximal compact subgroup K0 of G0 has special orbits C which are complex submanifolds in the open orbits of G0. These special orbits C are characterized as the closed orbits in Z of the complexification K of K0. These are referred to as cycles. The cycles intersect Schubert varieties S transversely at finitely many points. Describing these points and their multiplicities was carried out for all real forms of SLn,ℂ by Brecan (Brecan, 2014) and (Brecan, 2017) and for the other real forms by Abu-Shoga (Abu-Shoga, 2017) and Huckleberry (Abu-Shoga and Huckleberry). In the present paper, we deal with the real form SOp,q acting on the SO (2n, C)-manifold of maximal isotropic full flags. We give a precise description of the relevant Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number. The results in the case of G/Q for all real forms will be given by Abu-Shoga and Huckleberry.

A Manin–Mumford Theorem for the Maximal Compact Subgroup of a Universal Vectorial Extension of a Product of Elliptic Curves

We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin–Mumford-type statement. This answers some questions posed by Corvaja–Masser–Zannier, which arose in connection with their investigation of the intersection of an algebraic curve with the maximal compact subgroup of various algebraic groups. In particular they proved that these intersections are finite for universal vectorial extensions of elliptic curves. Using Khovanskii’s zero-estimates combined with a stratification result of Gabrielov–Vorobjov and recent work of the authors, we obtain effective bounds for this intersection that only depend on the degree of the algebraic variety and the dimension of the group. As a corollary, we obtain new uniform results of Manin–Mumford type for additive extensions of certain abelian varieties.

The Parameter Space of Orbits of a Maximal Compact Subgroup Acting on a Flag Manifold

The orbits of a real form G of a complex semisimple Lie group GC and those of the complexification KC of its maximal compact subgroup K acting on Z=GC/Q, a homogeneous, algebraic, GC-manifold, are finite. Consequently, there is an open G-orbit. Lower-dimensional orbits are on the boundary of the open orbit with the lowest dimensional one being closed. Induced action on the parameter space of certain compact geometric objects (cycles) related to the manifold in question has been characterized using duality relations between G- and KC-orbits in the case of an open G-orbit and more recently lower-dimensional G-orbits. We show that the parameter space associated with the unique closed G-orbit in Z agrees with that of the other orbits characterized as a certain explicitly defined universal domain.

The topology of Baumslag–Solitar representations

Let [Formula: see text] be a Baumslag–Solitar group and let [Formula: see text] be a complex reductive algebraic group with maximal compact subgroup [Formula: see text]. We show that, when [Formula: see text] and [Formula: see text] are relatively prime with distinct absolute values, there is a strong deformation retraction of Hom([Formula: see text]) onto Hom([Formula: see text]).

Asymptotics for the Radon transform on hyperbolic spaces

Let G/H be a hyperbolic space over R; C or H; and let K be a maximal compact subgroup of G. Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D. For any L^2-Schwartz function f on G/H we prove that the Abel transform A(Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K-finite and K∩H-invariant functions.

Branching Rules for n-fold Covering Groups of SL2 over a Non-Archimedean Local Field

Let G be the n-fold covering group of the special linear group of degree two over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of G to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

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]