Generalizing the proof – by Hecht and Schmid – of Osborne’s conjecture we prove an Archimedean (and weaker) version of a theorem of Colette Moeglin. The result we obtain is a precise Archimedean version of the general principle – stated by the second author – according to which a local Arthur packet contains the corresponding local $L$-packet and representations which are more tempered.

We extend a monotonicity result of Wang and Gong on the product of positive definite matrices in the context of semisimple Lie groups. A similar result on singular values is also obtained.

We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context of semisimple Lie groups.

Let $G$ be a connected real semi-simple Lie group and $H$ a closed connected subgroup. Let $P$ be a minimal parabolic subgroup of $G$. It is shown that $H$ has an open orbit on the flag manifold $G/P$ if and only if it has finitely many orbits on $G/P$. This confirms a conjecture by T. Matsuki.

We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $SU(p,q)$ (resp. $S{O}^{*}\left(2n\right)$, $n$ odd) and the surface group is maximal in some $S\left(U\right(p,p)\times U(q-p\left)\right)\subset SU(p,q)$ (resp. $S{O}^{*}(2n-2)\times SO\left(2\right)\subset S{O}^{*}\left(2n\right)$). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.

In this note, we study formal deformations of derived representations of the principal series representations of $\mathrm{SL}(2,ℝ)$. In particular, we recover all the representations of the derived principal series by deforming one of them. Similar results are also obtained for $\mathrm{SL}(2,ℂ)$.

For the scalar holomorphic discrete series representations of $\mathrm{SU}(2,2)$ and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside $\mathrm{SU}(2,2)$. We construct a Cayley transform between the Ol’shanskiĭ semigroup having $\mathrm{U}(1,1)$ as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for $\mathrm{SU}(2,2)$. This allows calculating the composition series in terms of harmonic analysis on $\mathrm{U}(1,1)$. In particular we show that the Ol’shanskiĭ Hardy space for $\mathrm{U}(1,1)$ is different...

We generalize Jacobi forms of an arbitrary degree and construct torus bundles over abelian schemes whose sections can be identified with such generalized Jacobi forms.

In the first section of this paper we give a characterization of those closed convex cones (wedges) $W$ in the Lie algebra $sl(2,\mathbf{R}{)}^n$ which are invariant under the maximal compact subgroup of the adjoint group and which are controllable in the associated simply connected Lie group $Sl(2,\mathbf{R}{)}^{n^{\sim }}$, i.e., for which the subsemigroup $S=(expW)$ generated by the exponential image of $W$ agrees with the whole group $G$ (Theorem 13). In Section 2 we develop some algebraic tools concerning real root decompositions with respect to compactly...

We develop the L² harmonic analysis for (Dirac) spinors on the real hyperbolic space Hⁿ(ℝ) and give the analogue of the classical notions and results known for functions and differential forms: we investigate the Poisson transform, spherical function theory, spherical Fourier transform and Fourier transform. Very explicit expressions and statements are obtained by reduction to Jacobi analysis on L²(ℝ). As applications, we describe the exact spectrum of the Dirac operator, study the Abel transform...