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We obtain a Lie theoretic intrinsic characterization of the connected and simply connected solvable Lie groups whose regular representation is a factor representation. When this is the case, the corresponding von Neumann algebras are…

Given a complex, separable Hilbert space $\cH$, we consider differential expressions of the type $\tau = - (d^2/dx^2) + V(x)$, with $x \in (a,\infty)$ or $x \in \bbR$. Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in…

We study asymptotics of traces of (noncommutative) monomials formed by images of certain elements of the universal enveloping algebra of the infinite-dimensional unitary group in its Plancherel representations. We prove that they converge…

It is shown that the $F_4$ rational and trigonometric integrable systems are exactly-solvable for {\it arbitrary} values of the coupling constants. Their spectra are found explicitly while eigenfunctions by pure algebraic means. For both…

We prove that the additive group $(E^\ast,\tau_k(E))$ of an $\mathscr{L}_\infty$-Banach space $E$, with the topology $\tau_k(E)$ of uniform convergence on compact subsets of $E$, is topologically isomorphic to a subgroup of the unitary…

We say that a finite almost simple $G$ with socle $S$ is admissible (with respect to the spectrum) if $G$ and $S$ have the same sets of orders of elements. Let $L$ be a finite simple linear or unitary group of dimension at least three over…

Let $K$ be an algebraically closed field with $\operatorname{char}(K)\neq 2$, and $A$ a skewed-gentle $K$-algebra. In this case, Crawley-Boevey's description of the indecomposable $A$-modules becomes particularly easy. This allows us to…

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the…

Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd characteristic $p$, and $G$ be the unramified unitary group $U(2, 1)(E/F)$. For an irreducible smooth representation $\pi$ of $G$ over…

J. Pevtsova and the author constructed a ``universal $p$-nilpotent operator" for an infinitesimal group scheme $G$ over a field $k$ of characteristic $p > 0$ which led to coherent sheaves on the scheme of 1-parameter subgroups of $G$…

Let $X$ be a locally compact Hausdorff space, let $A$ be a partially ordered algebra, and let $\pi\colon \mathrm{C}_{\mathrm c}(X)\to A$ be a positive algebra homomorphism. Under conditions on $A$ that are satisfied in a good number of…

Admissible vectors for unitary representations of locally compact groups are the basis for group-frame and covariant coherent state expansions. Main tools in the study of admissible vectors have been Plancherel and central integral…

Let V be a standard subspace in the complex Hilbert space H and U : G \to U(H) be a unitary representation of a finite dimensional Lie group. We assume the existence of an element h in the Lie algebra of G such that U(exp th) is the modular…

The Oscillator Groups,$\G_\lambda,$ are the only solvable, non commutative, simply connected Lie groups to admit a Lorentzian bi-invariant metric. For these groups, we give sufficient conditions for a left-invariant pseudo-Riemannian metric…

Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(\alpha,a)…

Let G be a connected simple linear Lie group and H in G a symmetric subgroup such that the corresponding symmetric space G/H is non-compactly causal. We show that any irreducible unitary representation of G leads naturally to a net of…

Given a minuscule representation of a simple Lie algebra, we find an algebraic model for the action of a regular element and show that these models can be glued together over the adjoint quotient, viewed as the set of all regular conjugacy…

Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for $G = \mathrm{GL}_n$, we consider polynomial representations of $G_r T$ for an arbitrary closed…

In this article, we give a general construction of spectral triples from certain Lie group actions on unital C*-algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple…

Let $G/K$ be a Hermitian symmetric space and $V_\tau$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_\tau)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, \tau)} V_\tau$…