A Fuchsian group is a discrete subgroup of PSL(2,R). As such it acts discontinuously on H (the upper half plane model of the hyperbolic plane) by fractional linear transformations. This action induces an action on the real line. It is well known that if an isometry of H fixes a point of the real line then the point is one of a pair, in the case that the isometry is hyperbolic or the isometry in...

The elliptic elements of M, each with two conjugate complex fixed points, are precisely the conjugates of nontrivial powers of A and B. The parabolic elements, each with a single real fixed point, are precisely the conjugates of nontrivial powers of C = AB: z ~ z + 1. The remaining nontrivial elements of M are hyperbolic, each with two real fixed points. A subgroup S of M is torsionfree (and th...

With every nontrivial connected algebraic group G we associate a positive integer gtd(G) called the generic transitivity degree of G and equal to the maximal n such that there is a nontrivial action of G on an irreducible algebraic variety X for which the diagonal action of G on Xn admits an open orbit. We show that gtd(G) 6 2 (respectively, gtd(G) = 1) for all solvable (respectively, nilpotent...

Definition 1. Let G be a permutation group on a set  and x be an element of . Then Gx  g ∈ G ∣ gx  x is called the stabilizer of x and consists of all the permutations of G that produce group fixed points in x. Definition 2. A vector subspace S ⊂ V is isotropic if for any v,w ∈ S, the symmetric bilinear form satisfies: Bv,w  0 Definition 3. A maximal parabolic subgroup in an orthogon...

We classify the split simple affine algebraic groups G of types A and C over a field with the property that the Chow group of the quotient variety E/P is torsion-free, where P ⊂ G is a special parabolic subgroup (e.g., a Borel subgroup) and E is a generic G-torsor (over a field extension of the base field). Examples of G include the adjoint groups of type A. Examples of E/P include the Severi-B...

For a homogeneous space G/P , where P is a parabolic subgroup of a complex semisimple group G, an explicit Kähler–Einstein metric on it is constructed. The Einstein constant for the metric is 1. Therefore, the intersection number of the first Chern class of the holomorphic tangent bundle of G/P coincides with the volume of G/P with respect to this Kähler–Einstein metric, thus enabling us to com...

Let ψ be a smooth character of a closed subgroup, H, of a reductive p-adic group G. If P is parabolic subgroup of G such that PH is open in G, we define the constant term of every smmoth fubction on G which transforms by ψ under the right action of G. The example of mixed models is given: it includes symmetric spaces and Whittaker models. In this case a notion of cuspidal function is defined an...

We will restrict out attention to the simplest possible case, namely G = SL2(R), Γ = SL2(Z), and right K = SO(2)-invariant functions on Γ\G. That is, we neglect the finite primes and holomorphic automorphic forms. Let N be the subgroup of G consisting of upper-triangular unipotent matrices, and P the parabolic subgroup consisting of all upper-triangular matrices. For simplicity we give K total ...

Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this paper, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup. Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal toru...

Let G be a not necessarily split reductive group scheme over a commutative ring R with 1. Given a parabolic subgroup P of G, the elementary group EP (R) is defined to be the subgroup of G(R) generated by UP (R) and UP−(R), where UP and UP− are the unipotent radicals of P and its opposite P −, respectively. It is proved that if G contains a Zariski locally split torus of rank 2, then the group E...