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.

A PARAMETRIZATION OF δ-SPHERICAL FUNCTIONS ON COMMUTATIVE TRIPLES ASSOCIATED WITH NILPOTENT LIE GROUPS

Let $N$ be a connected and simply connected nilpotent Lie group, $K$ be a compact subgroup of $Aut(N)$, the group of automorphisms of $N$ and $\delta$ be a class of unitary irreducible representations of $K$. The triple $(N,K,\delta)$ is a commutative triple if the convolution algebra $\mathfrak{U}_{\delta}^{1}(N)$ of $\delta$-radial integrable functions is commutative. In this paper, we obtain first a parametrization of $\delta$ spherical functions by means of the unitary dual $\widehat{N}$ and then an inversion formula for the spherical transform of $F\in \mathfrak{U}_{\delta}^{1}(N)$.

Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to RI×C for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra R[G] of a finite group G to its representation algebra R(G,R), via the natural duality of the topological vector space RI to the vector space R(I), for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups.

Covariant functions of characters of compact subgroups

This paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Suppose that $$\xi :H\rightarrow \mathbb {T}$$ ξ : H → T is a character, $$1\le p<\infty$$ 1 ≤ p < ∞ and $$L_\xi ^p(G,H)$$ L ξ p ( G , H ) is the set of all covariant functions of $$\xi$$ ξ in $$L^p(G)$$ L p ( G ) . It is shown that $$L^p_\xi (G,H)$$ L ξ p ( G , H ) is isometrically isomorphic to a quotient space of $$L^p(G)$$ L p ( G ) . It is also proven that $$L^q_\xi (G,H)$$ L ξ q ( G , H ) is isometrically isomorphic to the dual space $$L^p_\xi (G,H)^*$$ L ξ p ( G , H ) ∗ , where q is the conjugate exponent of p. The paper is concluded by some results for the case that G is compact.

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>

Uniqueness of optimal symplectic connections

Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation. We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group. We also prove that when a submersion admits an optimal symplectic connection, it achieves the absolute minimum of a natural log norm functional, which we define.

Automorphic forms and fermion masses

We extend the framework of modular invariant supersymmetric theories to encompass invariance under more general discrete groups Γ, that allow the presence of several moduli and make connection with the theory of automorphic forms. Moduli span a coset space G/K, where G is a Lie group and K is a compact subgroup of G, modded out by Γ. For a general choice of G, K, Γ and a generic matter content, we explicitly construct a minimal Kähler potential and a general superpotential, for both rigid and local $$ \mathcal{N} $$ N = 1 supersymmetric theories. We also specialize our construction to the case G = Sp(2g, ℝ), K = U(g) and Γ = Sp(2g, ℤ), whose automorphic forms are Siegel modular forms. We show how our general theory can be consistently restricted to multi-dimensional regions of the moduli space enjoying residual symmetries. After choosing g = 2, we present several examples of models for lepton and quark masses where Yukawa couplings are Siegel modular forms of level 2.

Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

Let H be a semisimple algebraic group, K a maximal compact subgroup of $$G:=H({{\mathbb {R}}})$$ G : = H ( R ) , and $$\Gamma \subset H({{\mathbb {Q}}})$$ Γ ⊂ H ( Q ) a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space $$\Gamma \backslash G/K$$ Γ \ G / K to corresponding bounds on the arithmetic quotient $$\Gamma \backslash G$$ Γ \ G for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial K-types, yielding such bounds for new classes of automorphic representations. They constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.