The distribution of products of random matrices chosen from fixed spherical classes is determined for classical rank 1 symmetric spaces. It is observed that $n\to\infty$ limit behaves approximately as in the abelian case. A theorem on the rate of convergence to the Haar measure in the case of SU(n) is also established.
A Cartan decomposition for symmetric pairs plays an important role to study not only orbit geometry of the symmetric spaces but also harmonic analysis on them. For non-symmetric reductive pairs, there are examples of generalizations of Cartan decompositions for some spherical complex homogeneous spaces such as complex line bundles over the complexified Hermitian symmetric spaces and triple spaces. This paper provides new examples of a Cartan decomposition for non-symmetric reductive pairs, namely, reductive non-symmetric spherical pairs of rank-one type. We also show that the action of some co…
We prove that for a simply laced group, the closure of the Borel conjugacy class of any nilpotent element of height $2$ in its conjugacy class is normal and admits a rational resolution. We extend this, using Frobenius splitting techniques, to the closure in the whole Lie algebra if either the group has type $A$ or the element has rank $2$.
We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines techniques from semiclassical analysis with harmonic theory on reductive groups, and makes use of new asymptotic bounds for spherical functions that are of independent interest.
Let g be a complex semi-simple Lie algebra and g be a semisimple subalgebra of g. Consider the branching problem of decomposing the simple g-representations V as a sum of simple grepresentations V. When g = g x g, it is the tensor product decomposition. The multiplicity space Mult(V, V) satisfies V = $\oplus$ V Mult(V, V) $\otimes$ V, where the sum runs over the isomorphism classes of simple g-representations. In the case when g is spherical of minimal rank, we describe Mult(V, V) as the intersection of kernels of powers of root operators in some weight space of the dual space V * of V. When g…
We compute the Harish-Chandra $c$-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces $X=G/K$ in terms of Euler $Γ$ functions, proving that it is meromorphic. Compared to the even case, the poles of the $c$-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra series expansion of the spherical superfunctions on $X$. In the case where the multiplicity of the simple root is an even negative number, they have a closed expression as Jacobi polynomials for an unusual choice of parameters.
Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify: (a) all such varieties when G is $\mathrm{SL}(2) \times \mathbb{C}^{\times}$ and (b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and F. Knop's classificat…
We provide $L^p$-versus $L^\infty$-bounds for eigenfunctions on a real spherical space $Z$ of wavefront type. It is shown that these bounds imply a non-trivial error term estimate for lattice counting on $Z$. The paper also serves as an introduction to geometric counting on spaces of the mentioned type. Section 7 on higher rank is new and extends the result from v1 to higher rank. Final version. To appear in Acta Math. Sinica.
Let $G/H$ be a reductive symmetric space of split rank $1$ and let $K$ be a maximal compact subgroup of $G$. In a previous article the first two authors introduced a notion of cusp forms for $G/H$. We show that the space of cusp forms coincides with the closure of the $K$-finite generalized matrix coefficients of discrete series representations if and only if there exist no $K$-spherical discrete series representations. Moreover, we prove that every $K$-spherical discrete series representation occurs with multiplicity $1$ in the Plancherel decomposition of $G/H$.
Let $G$ be a connected complex semisimple Lie group with a fixed maximal torus $T$ and a Borel subgroup $B \supset T$. For an arbitrary automorphism $θ$ of $G$, we introduce a holomorphic Poisson structure $π_θ$ on $G$ which is invariant under the $θ$-twisted conjugation by $T$ and has the property that every $θ$-twisted conjugacy class of $G$ is a Poisson subvariety with respect to $π_θ$. We describe the $T$-orbits of symplectic leaves, called $T$-leaves, of $π_θ$ and compute the dimensions of the symplectic leaves (i.e, the ranks) of $π_θ$. We give the lowest rank of $π_θ$ in any given $θ$-t…
