It is shown how new integral-geometric formulae can be obtained from the existing formulae of Crofton type. In particular, for classical Crofton formulae in which the answer depends on the Riemannian volume, we obtain generalizations in terms of the mixed Riemannian volume defined in the paper. The method is based on the calculations in the ring of normal densities constructed in the previous work of the authors.

On 12 February 2019, the prominent Russian mathematician Arkady L’vovich Onishchik passed away. He was born in Moscow on 14 November 1933. In 1956 he graduated from the Faculty of Mechanics and Mathematics of Moscow State University and began postgraduate studies in the Department of Higher Algebra. His research supervisor was E.B. Dynkin, whose area of interest at that time was the theory of Lie groups and algebras. Onishchik obtained his first research results already in his student years. Some of them were included in the long paper [1] with Dynkin on the global structure of compact Lie groups, published in the journal Uspekhi Matematicheskikh Nauk. After completing his postgraduate studies, he was retained in the Department of Higher Algebra, then headed by A. G. Kurosh. In 1960 he defended his Ph.D. thesis “On transitive Lie groups of transformations”, and in 1970 his D.Sc. thesis “ Compact homogeneous spaces and decompositions of Lie groups”. However, he had no real chance of getting a professorship at Moscow State University, since in 1968 he was one of the signers of the “letter of the ninety-nine” in defense of the mathematician A. S. Yesenin-Vol’pin, who had been forcibly committed to a mental hospital in connection with his dissident activities. Therefore, in 1975 Onishchik accepted an offer to go to the newly created Yaroslavl State University (YarSU), where he received a professorship and worked until 2016. In 1961 Onishchik and È. B. Vinberg, also a student of Dynkin, organised a seminar on Lie groups (subsequently, a seminar on Lie groups and invariant theory) in the Faculty of Mechanics and Mathematics of Moscow State University. This seminar was a successor of Dynkin’s seminar and kept many of its traditions. Onishchik was one of the leaders of the seminar in all the years that followed, even after his move to YarSU. The 1967/68 seminar notes formed the basis of the book Seminar on Lie groups and algebraic groups by Vinberg and Onishchik [21], which has become indispensable for specialists in this area and was translated into English.

We suggest an algorithm allowing to obtain some new integral-geometric formulae from the existing formulae of Crofton type. These new formulae are applied to get smooth versions of BKK theorem. The algorithm is based on the calculations in the ring of normal densities on a manifold.

Let X be an n-dimensional manifold and \({V_{1}, . . . ,{V}_{n} \subset {C}^\infty(X,\mathbb{R})}\) finite-dimensional vector spaces with Euclidean metric. We assign to each Vi a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of the cotangent bundle to X. We prove that the average number of isolated common zeros of \({{f}_{1} \in {V}_{1}, . . . , {f}_{n} \in {V}_{n}}\) is equal to the mixed symplectic volume of these Finsler ellipsoids. If X is a homogeneous space of a compact Lie group and all vector spaces Vi together with their Euclidean metrics are invariant, then the average numbers of zeros satisfy the inequalities, similar to Hodge inequalities for intersection numbers of divisors on a projective variety. This is applied to the eigenspaces of Laplace operator of an invariant Riemannian metric. The proofs are based on a construction of the ring of normal densities on X, an analogue of the ring of differential forms. In particular, this construction is used to carry over the Crofton formula to the product of spheres.

On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$ eigenfunctions of the Laplace operator $\Delta $ with eigenvalue $\lambda$. If $M$ is homogeneous under a compact Lie group preserving the metric then we prove that the average number of common zeros of $n$ eigenfunctions does not exceed $c(n)\lambda^{n/2}{\rm vol}\,M$, the expression known from the celebrated Weyl's law. Moreover, if the isotropy representation is irreducible, then the estimate turns into equality. The constant $c(n)$ is explicitly given. The method of proof is based on the application of Crofton's formula for the sphere.

We consider the eigenfunctions of the Laplace operator \(\Delta \) on a compact Riemannian manifold M of dimension n. For M homogeneous with irreducible isotropy representation and for a fixed eigenvalue \(\lambda \) of \(\Delta \) we find the average number of common zeros of n eigenfunctions. It turns out that, up to a constant depending on n, this number equals \(\lambda ^{n/2}\mathrm{vol}\,M\), the expression known from the celebrated Weyl’s law. To prove this we compute the volume of the image of M under an equivariant immersion into a sphere.

We find new sufficient conditions for the commutator map of a real semisimple Lie algebra to be surjective. As an application we prove the surjectivity of the commutator map for all simple algebras except $\mathfrak su_{p,q}$ ($p$ or $q$ >1), $\mathfrak so_{p,p+2}$ ($p$ odd or $p=2$), $\mathfrak u^*_{2m+1}(\mathbb H)$ ($m\ge 1$) and $EIII$.

With each antiholomorphic involution [Formula: see text] of a connected complex semisimple Lie group [Formula: see text] we associate an automorphism [Formula: see text] of its Dynkin diagram. The definition of [Formula: see text] is given in terms of the Satake diagram of [Formula: see text]. Let [Formula: see text] be a self-normalizing spherical subgroup. If [Formula: see text] then we prove the uniqueness and existence of a [Formula: see text]-equivariant real structure on [Formula: see text] and on the wonderful completion of [Formula: see text].

We study equivariant real structures on spherical varieties. We call such a structure canonical if it is equivariant with respect to the involution defining the split real form of the acting reductive group G. We prove the existence and uniqueness of a canonical structure for homogeneous spherical varieties G/H with H self-normalizing and for their wonderful embeddings. For a strict wonderful variety we give an estimate of the number of real form orbits on the set of real points.

In this survey, we gather together various results on the action of a real form of a complex semisimple Lie group on its flag manifolds. We start with the finiteness theorem of J.Wolf implying that at least one of the orbits is open. We give a new proof of the converse statement for real forms of inner type, essentially due to F.M.Malyshev. Namely, if a real semisimple Lie group of inner type has an open orbit on an algebraic homogeneous space of the complexified group then the homogeneous space is a flag manifold. To prove this, we recall, partly with proofs, some results of A.L.Onishchik on the factorizations of reductive groups. Finally, we discuss the cycle spaces of open orbits and define the crown of a symmetric space of non-compact type. With some exceptions, the cycle space agrees with the crown. We sketch a complex analytic proof of this result, due to G.Fels, A.Huckleberry and J.Wolf.

The paper is a survey of recent results in geometric representation theory describing group actions which induce multiplicity-free representations in the spaces of holomorphic functions. For connected compact Lie groups of automorphisms of Stein manifolds we characterize such actions in terms of antiholomorphic involutions. Some proofs are given and some results are new. For example, spherical complex spaces are defined for arbitrary real forms of complex reductive groups. Their properties, which we prove here, were known only for compact real forms. We also show that a complex compact homogeneous manifold of a complex reductive group is spherical if and only if the fiber of its Tits fibration is a complex torus.

It is proved that a Stein manifold acted on by a connected compact Lie group is spherical if and only if there exists an antiholomorphic involution preserving each orbit of the action. This involution can be chosen equivariant with respect to a Weyl involution of the group.

Let X be a holomorphically separable irreducible reduced complex space, K a connected compact Lie group acting on X by holomorphic transformations, theta : K -> K a Weyl involution, and mu : X -> X an antiholomorphic involution map satisfying mu(kx) = theta(k) mu(x) for x in X and k in K. We show that if the holomorphic functions on X form a multiplicity free K-module then mu maps every K-orbit onto itself. For a spherical affine homogeneous space X=G/H of the reductive group G (the complexification of K) we construct an antiholomorphic map mu with these properties.

We find some eigenvalues of the Laplacian on the Cayley graph of a Coxeter group with respect to its Coxeter generators and give an upper bound for the minimal positive eigenvalue.

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 groups, we call the complex space X spherical if X is normal and its tangent space at some point is generated by the vector fields from a Borel subalgebra b or g. We give several characterizations of spherical Stein spaces. In particular, we prove that a connected Stein manifold X is spherical if and only if the algebra of K-invariant differential operators on X is commutative.