We investigate the existence of hyperbolic, spherical or Euclidean structure on cone-manifolds whose underlying space is the three-dimensional sphere and singular set is a given two-bridge knot. For two-bridge knots with not more than 7 crossings we present trigonometrical identities involving the lengths of singular geodesics and cone angles of such cone-manifolds. Then these identities are used to produce exact integral formulae for the volume of the corresponding cone-manifold modeled in the hyperbolic, spherical and Euclidean geometries.

Let k be an algebraically closed field of characteristic zero and B a factorial affine k-domain equipped with a locally nilpotent derivation δ. We investigate B when there exists an element z ∈ B such that δ(z) = αp for p ≥ 1 and a prime element α of A = Ker δ. One such example of particular interest is the coordinate ring of an affine pseudo-n-space, which is defined as a smooth affine variety X equipped with a faithfully at morphism q : X → 𝔸1 such that q–1(𝔸 $$ {}_{\ast}^1 $$ ) ≅ 𝔸 $$ {}_{\ast}^1 $$ × 𝔸 n–1 and q*(0) is irreducible and reduced where 𝔸 $$ {}_{\ast}^1 $$ = 𝔸1 \ {0}. In [9], a criterion was given for an affine pseudo-3-space to be isomorphic to a hypersurface of form xpy – g(x; t; z) = 0 in 𝔸4. Such a hypersurface is called a hypersurface of Danielewski type and studied in [17], [16]. In this article, under the condition that A/αA is factorial, we describe B in terms of equivariant affine modification developed by Kaliman and Zaidenberg [15] and give a criterion for B to be isomorphic to the residue ring A[Y;Z]/(αpY – g(Z)) for p ≥ 1 and an irreducible polynomial g(Z) ∈ A[Z]\A. As a consequence, we obtain a criterion for an affine pseudo-n-space to be isomorphic to a hypersurface of Danielewski type for n ≥ 3.

A $GL_d$-pseudocharacter is a function from a group $\Gamma$ to a ring $k$ satisfying polynomial relations which make it "look like" the character of a representation. When $k$ is an algebraically closed field, Taylor proved that $GL_d$-pseudocharacters of $\Gamma$ are the same as degree-$d$ characters of $\Gamma$ with values in $k$, hence are in bijection with equivalence classes of semisimple representations $\Gamma \rightarrow GL_d(k)$. Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group $H$ over an algebraically closed field $k$ of characteristic 0 and for any group $\Gamma$, there exists an infinite collection of functions and relations which are naturally in bijection with $H^0(k)$-conjugacy classes of semisimple representations $\Gamma \rightarrow H(k)$. In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG-algebra. We then define generating sets and generating relations for these objects and show that, for all $H$ as above, the corresponding FFG-algebra is finitely presented. Hence we can always define $H$-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of representations, following Larsen.

Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to H of all irreducible representations of G realized in spaces of sections of homogeneous line bundles on X.

An involutive diffeomorphism σ of a connected smooth manifold M is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs (M, σ), where M is an irreducible connected symmetric space, not necessarily Riemannian, and σ is a dissecting involutive automorphism. In particular, we show that the only irreducible, connected and simply connected Riemannian symmetric spaces with dissecting isometric involutions are $$ {\mathbbm{S}}^n $$ and ℍn, where the corresponding fixed point spaces are $$ {\mathbbm{S}}^{n-1} $$ and ℍn − 1, respectively.

Following a question of Vinberg, a general method to construct monomial bases for finite-dimensional irreducible representations of a reductive Lie algebra was developed in a series of papers by Feigin, Fourier, and Littelmann. Relying on this method, we construct monomial bases of multiplicity spaces associated with the restriction of the representation to a reductive subalgebra. As an application, we produce monomial bases for representations of the general linear and symplectic Lie algebras associated with natural chains of subalgebras. We also show that our basis in type A is related to both the Gelfand-Tsetlin basis and the Littelmann basis via triangular transition matrices which implies that the triangularity property extends to the matrix connecting the Gelfand-Tsetlin and canonical bases. A similar relationship holds between our basis in type C and a suitably modified version of the basis constructed earlier by the first author.

For a Weyl group $W$ of classical type, we present a formula to calculate the restriction of (graded) Springer representations of $W$ to a maximal parabolic subgroup $W'$ where the types of $W$ and $W'$ are in the same series. As a result, we obtain recursive formulas for the Betti numbers of Springer fibers for classical types.

We establish rank-finiteness for the class of G-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.

A Tits polygon is a bipartite graph in which the neighborhood of each vertex is endowed with an “opposition relation” satisfying certain axioms. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. Every Tits polygon has a distinguished set of circuits. A Tits quadrangle is a Tits polygon in which these circuits all have length 8. There is a standard construction that produces a Tits polygon from certain pairs (∆, T), where ∆ is an irreducible spherical building and T is a Tits index of relative rank 2. We call a Tits quadrangle exceptional if it arises from such a pair (∆, T) for ∆ the spherical building associated to the group of rational points of an exceptional algebraic group. In this paper, we characterize the exceptional Tits quadrangles as extensions of orthogonal Tits quadrangles in a suitable sense.

Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector bundles on X with Nori's Tannakian approach to principal G-bundles, to give an equivalence of categories between T-equivariant principal G-bundles on X and certain compatible ∑-filtered algebras associated to X, when the characteristic of K is 0.