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We show that the strongly symmetric spectral convex compact sets are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. Spectrality is the property that every state has a convex…

Generalized parallelizable spaces allow a unified treatment of consistent maximally supersymmetric truncations of ten- and eleven-dimensional supergravity in generalized geometry. Known examples are spheres, twisted tori and hyperboloides.…

Petrecca and R\"oser (2018, \cite{Petrecca2019}), and Schueth (2017, \cite{Schueth2017}) had shown that for a generic $G$-invariant metric $g$ on certain compact homogeneous spaces $M=G/K$ (including symmetric spaces of rank 1 and some Lie…

Let G be a connected reductive group. Recall that a G-variety X is called spherical if X is normal and a Borel subgroup of G has an open orbit on X. To a spherical homogeneous G-space one assigns certain combinatorial invariants: the weight…

For a connected reductive group $ G $ defined over a number field $ k $, we construct the Schwartz space $ \mathcal{S}(G(k)\backslash G(\mathbb{A})) $. This space is an adelic version of Casselman's Schwartz space $…

Let $G$ be a simple simply connected complex algebraic group and let $\mathfrak{g}_*$ be a $\mathbf{Z}/m$-grading on its Lie algebra $\mathfrak{g}$. In a recent series of articles, G. Lusztig and Z. Yun, studied the classification of simple…

A symplectic reductive homogeneous space is a pair $(G/H,\Omega)$, where $G/H$ is a reductive homogeneous $G$-space and $\Omega$ is a $G$-invariant symplectic form on it. The main examples include symplectic Lie groups, symplectic symmetric…

We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over $\tF_{2^n}$. Let $G$ be an adjoint algebraic group of type $B,C$ or $D$ defined over an algebraically closed field…

For a Gelfand pair $(G,K)$ with $G$ a Lie group of polynomial growth and $K$ a compact subgroup, the "Schwartz correspondence" states that the spherical transform maps the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash G/K)$…

A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary…

Let $F$ be a $p$-adic field of characteristic zero and odd residual characteristic. Let $\mathbf{Sp}_{2n}(F)$ denote the symplectic group defined over $F$, where $n\geq 2$. We prove that the Speh representations $\mathcal{U}(\delta,2)$,…

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier-Saito theorem for solvable Lie groups. We then give a…

We give explicit computations of the $\Gamma$-Euler characteristic of several families of orbit space definable translation groupoids. These include the translation groupoids associated to finite-dimensional linear representations of the…

We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R}) $ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally…

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion…

Let $G$ be a linear connected non-compact real simple Lie group and let $K\subset G$ be a maximal compact subgroup of $G$. Suppose that the centre of $K$ isomorphic to $\mathbb{S}^1$ so that $G/K$ is a global Hermitian symmetric space. Let…

Let G_2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a generic, smooth, connected, projective curve over $\mathbb{C}$ of genus at least 2. For a complex Lie group G, let H^0(M(G),L^k) be the space of…

Let $G$ be a connected semisimple group over an algebraically closed field $k$ of characteristic 0. Let $Y=G/H$ be a spherical homogeneous space of $G$, and let $Y'$ be a spherical embedding of $Y$. Let $k_0$ be a subfield of $k$. Let $G_0$…

By adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on…

If $(G,K)$ is a Gelfand pair, with $G$ a Lie group of polynomial growth and $K$ a compact subgroup of $G$, the Gelfand spectrum $\Sigma$ of the bi-$K$-invariant algebra $L^1(K\backslash G/K)$ admits natural embeddings into ${\mathbb R}^n$…