Let G be a finite abelian group with identity element 1G and L = ⊕ g∈G L g be an infinite dimensional G-homogeneous vector space over a field of characteristic 0. Let E = E(L) be the Grassmann algebra generated by L. It follows that E is a G-graded algebra. Let |G| be odd, then we prove that in order to describe any ideal of G-graded identities of E it is sufficient to deal with G′-grading, whe...

We find explicitly the generating functions of the multiplicities in the pure and mixed trace cocharacter sequences of two 4×4 matrices over a field of characteristic 0. We determine the asymptotic behavior of the multiplicities and show that they behave as polynomials of 14th degree. Introduction Let us fix an arbitrary field F of characteristic 0 and two integers n, d ≥ 2. Consider the d gene...

In class, we have proved the important fact that over any field k, a non-solvable connected reductive group containing a 1-dimensional split maximal k-torus is k-isomorphic to SL2 or PGL2. That proof relied on knowing that maximal tori remain maximal after a ground field extension to k, and so relies on Grothendieck’s theorem. But for algebraically closed fields there is no content to Grothendi...

We revisit the Harder-Narasimhan stratification on a minuscule p p -adic flag variety, by theory of modifications alttext="upper G"> G...

Let X be an F -rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F . One may associate toX certain cocharacters of Gwith favorable properties; this is an essential feature of the classification of geometric nilpotent orbits due to Bala-Carter, Pommerening, and, more recently, Premet. Suppose that the Lie algebra has a non-degenerate...

A construction of Bernstein associates to each cocharacter of a split p-adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calcula...

We prove that the Deligne-Lusztig variety associated to minimal length elements in any δ-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in [10]. 1.1 Notations. Let k be an algebraic closure of the finite prime field Fp and G be a connected reductive algebraic group over k with an endomorphism F : G → G such that some power F d of F is the Frobenius endo...