Hasse diagrams of G/P_1 and G/P_2

in the Paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.5052&rep=rep1&type=pdf at the end, we can see Hasse diagrams for several projective, homogeneous G-varieties for G being a exceptional linear algebraic group. Note that D4/P1 is isomorphic to a six dimensional quadric, that i will denote as Q6. In an unfinished book by Gille, Petrov,N. Semenov and Zainoulline, which can be found … Read more

Norm variety for n=5, p=2 not isomorphic to a quadric

In the paper “Motivic construction of cohomological invariants“, the author displays a list of known norm varieties for several n,p on page 11. For p=2,n=5 it says that a norm variety is given by the variety of singular trace zero lines in a reduced Albert algebra with cohomological invariant f5, which is a twist of … Read more

Local factors of Tamagawa measure

This is a reference request to some computations which I hope can be found in the literature somewhere. Let G⊂GLn be a semisimple linear algebraic group over Q. The Tamagawa measure μ on the group of adelic points G(A) is uniquely determined by the product formula. Nevertheless, it can be written as a product of … Read more

Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let R be a commutative ring and Φ be a finite (also called spherical) reduced irreducible root system of rank ≥2. I will denote by St(Φ,R) the Steinberg group of type Φ over R, i. e. the quotient of the free product ∐α∈ΦXα of root subgroups Xα=⟨xα(a)∣a∈R⟩ modulo Chevalley commutator formulae: [xα(a);xβ(b)]=∏iα+jβ∈Φxiα+jβ(Ni,jα,βaibj), i,j∈N; (here Ni,jα,β … Read more

Automorphisms of unipotent groups

I start with a hopelessly broad question: what is known about the structure of the automorphism group of a (smooth, connected) unipotent group (over a field), and particularly about the structure of diagonalisable subgroup schemes of the automorphism group? It would be nice, but is not essential, if I could assume a non-algebraically closed field … Read more

Quadrics contained in the (complex) Cayley plane

In the paper Ilev, Manivel – The Chow ring of the Cayley plane we can learn, that CH8(X), with X:=E6/P1, denoting the Cayley plane, has three generators with one of them being the class of an 8-dimensional quadric [Q]. We consider CH(−) mod rational equivalence. Now I would like to know more about these quadrics. … Read more

p-adic analogue of self-adjoint operator

Consider the very well-known result that any Hermitian matrix over C, say T, admits a decomposition T=UDU∗ where U is unitary and D is diagonal with real entries. I am looking for a p-adic analogue: take a matrix A with entries in Qp or some algebraic extension. Is there some condition on A that will … Read more

Weyl groups of Levi factors and their cosets

Let G be a connected semisimple algebraic group defined over an algebraically closed field F of characteristic 0. Let B be a minimal parabolic subgroup of G (i.e. a Borel subgroup), let P be a parabolic subgroup of G containing B, and let L be a Levi factor of P. Letting π:P→L be the projection, … Read more