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Let X be an irreducible reduced complex space on which a connected compact Lie group K acts by holomorphic automorphisms. Let G be the complexification of K and g the Lie algebra of G. Following the theory of algebraic transformation…

Let $G = GL(V)$ for an N-dimensional vector space $V$ over an algebraically closed field k, and $G^{\theta}$ the fixed point subgroup of $G$ under an involution $\theta$ on $G$. In the case where $G^{\theta} = O(V)$, the generalized…

Let $\mathbb D=G/K$ be a complex bounded symmetric domain of tube type in a Jordan algebra $V_{\mathbb C}$, and let $D=H/L =\mathbb D\cap V$ be its real form in a Jordan algebra $V\subset V_{\mathbb C}$. The analytic continuation of the…

Let G be a solvable Lie group endowed with right Haar measure. We define and study a dense Frechet *-subalgebra S of L1(G), consisting of smooth functions rapidly-decreasing at infinity on G. When G is nilpotent, we recover the classical…

Given a symmetric triple $(G,K,\sigma)$ of compact type, with $G^{\sigma} = K$, the well known Cartan embedding $\hat{\Phi}: G/K \to G$ homothetically embeds the symmetric space $M = G/K$ as a totally geodesic submanifold of $G$. In this…

We study closed subspaces of $L^2(X)$, where $(X, \mu)$ is a $\sigma$-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group $\Gamma$ on $X$. We provide…

A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action…

Let $\mathbb{H}_n$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a closed subgroup of $U(n)$ acting on $\mathbb{H}_n$ by automorphisms such that $(K,\mathbb{H}_n)$ is a Gelfand pair. Let $G=K\ltimes\mathbb{H}_n$ be the semidirect…

It is well-known that the Harish-Chandra transform, $f\mapsto\mathcal{H}f,$ is a topological isomorphism of the spherical (Schwartz) convolution algebra $\mathcal{C}^{p}(G//K)$ (where $K$ is a maximal compact subgroup of any arbitrarily…

For a locally compact group $G$ and compact subgroup $K$, we consider a Delsarte-type extremal problem for $G$-invariant positive definite kernels on the homogeneous space $G/K$, generalising a certain Tur\'an problem for isotropic positive…

Let $G_k$ be a connected reductive group over an algebraically closed field $k$ of char $\neq 2$. Let $\theta_k$ be an algebraic group involution of $G_k$ and denote the fixed point subgroup by $K_k$. We construct an integral model for the…

We calculate the R(G)-algebra structure on the reduced equivariant K-groups of two-dimensional spheres on which a compact Lie group G acts as involutions. In particular, the reduced equivariant K-groups are trivial if G is abelian, which…

Given a compact Gelfand pair (G,K) and a locally compact group L, we characterize the class P_K^\sharp(G,L) of continuous positive definite functions f:G\times L\to \C which are bi-invariant in the G-variable with respect to K. The…

A strong Gelfand pair $(G, H)$ is a finite group $G$ and a subgroup $H$ where every irreducible character of $H$ induces to a multiplicity-free character of $G$. We determine the strong Gelfand pairs of the sporadic groups, their…

We deduce a proof of the isomorphism theorem for certain closed subspace $\mc S^p_\Gamma(X)$ of the $L^p$-Schwartz class functions $(0< p \leq 2) $ on a Riemannian symmetric space $X$ where $\Gamma$ is a finite subset of $\what{K}_M$. The…

For a connected quasi-split reductive algebraic group $G$ over a field $k$, which is either a finite field or a non-archimedean local field, $\theta$ an involutive automorphism of $G$ over $k$, let $K =G^\theta$. Let $K^1=[K^0,K^0]$, the…

Consider the (Helgason-) Fourier transform on a Riemannian symmetric space G/K. We give a simple proof of the L^p-Schwartz space isomorphism theorem (0 <p \le 2) for K-finite functions. The proof is a generalization of J.-Ph. Anker's proof…

This work investigates analytic Hilbert modules $\mathcal{H}$, over the polynomial ring, consisting of holomorphic functions on a $G$-space $\Omega \subset \mathbb{C}^m$ that are homogeneous under the natural action of the group $G$. In a…

Let $G$ be a compact, connected Lie group and $T \subset G$ a maximal torus. Let $(M,\omega)$ be a monotone closed symplectic manifold equipped with a Hamiltonian action of $G$. We construct a module action of the affine nil-Hecke algebra…

This paper is a follow-up contribution to our work [20] where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of [20] to the context of n-tuples of bounded linear operators on…