When G is a complex reductive algebraic group and G/K is a reductive symmetric space, the decomposition of C[G/K] as a K-module was obtained (in a non-constructive way) by R. Richardson, generalizing the celebrated result of Kostant-Rallis for the linearized problem (the harmonic decomposition of the isotropy representation). To obtain a constructive version of Richardson’s results, this paper ...

Let $G$ be a split reductive group over the ring of integers in $p$ -adic field with residue $\mathbf {F}$ . Fix representation $\overline {\rho }$ absolute Galois an unramified extension {Q}_p$ , valued $G(\mathbf {F})$ We study crystalline deformation for fixed Hodge type that satisfies analog Fontaine–Laffaille condition -valued representations. In particular, we give root theoretic on which...

Let G be a reductive algebraic group over field k and let B Borel subgroup in G. We demonstrate how number of results on the cohomology line bundles ag manifold G/B have had interesting consequences representation theory for G, vice versa. Our focus is case where characteristic positive. In this case, both vanishing behavior modules bundle G-structures nonzero are still very much open problems....

We generalize purity of the Newton stratification to purity for a single break point of the Newton point in the context of local G-shtukas respectively of elements of the loop group of a reductive group. As an application we prove that elements of the loop group bounded by a given dominant coweight satisfy a generalization of Grothendieck’s conjecture on deformations of p-divisible groups with ...

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism...

Let G be a connected reductive algebraic group defined over a global field k and Q a maximal k-parabolic subgroup of G. The constant γ(G, Q, k) attached to (G, Q) is defined as an analogue of Hermite’s constant. This constant depends only on G, Q and k in contrast to the previous definition of generalized Hermite constants ([W1]). Some functorial properties of γ(G, Q, k) are proved. In the case...

The study of spherical representations of the Jacobi group begun in [Sch1] is continued. Using certain index shifting operators, the notion of age of such a representation is introduced, as well as the notion of a local newform. The precise structure of the space of spherical vectors, in particular its dimension, is determined in terms of the age. The age of the spherical principal series repre...

Let L := k̄((ǫ)), where k is a finite field with q elements and ǫ is an indeterminate, and let σ be the Frobenius automorphism. Let G be a split connected reductive group over the fixed field of σ in L, and let I be the Iwahori subgroup of G(L) associated to a given Borel subgroup of G. Let f W be the extended affine Weyl group of G. Given x ∈ f W and b ∈ G(L), we have some subgroup of G(L) that...

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ and Gk (resp. Hk) the set of k-rational points of G (resp. H). The variety Gk/Hk is called a symmetric k-variety. For k = R and C the representation theory of these varieties has been studied extensively. To study t...