we consider the semigroup $s$ of highest weights appearing in tensor powers $v^{otimes k}$ of a finite dimensional representation $v$ of a connected reductive group. we describe the cone generated by $s$ as the cone over the weight polytope of $v$ intersected with the positive weyl chamber. from this we get a description for the asymptotic of the number of highest weights appearing in $v^{otime...

We consider the semigroup $S$ of highest weights appearing in tensor powers $V^{otimes k}$ of a finite dimensional representation $V$ of a connected reductive group. We describe the cone generated by $S$ as the cone over the weight polytope of $V$ intersected with the positive Weyl chamber. From this we get a description for the asymptotic of the number of highest weights appearing in $V^{otime...

Let $G$ be a tamely ramified reductive $p$-adic‎ ‎group‎. ‎We study distinction of a class of irreducible admissible representations‎ ‎of $G$ by the group of fixed points $H$ of an involution‎ ‎of $G$‎. ‎The representations correspond to $G$-conjugacy classes of‎ ‎pairs $(T,phi)$‎, ‎where $T$ is a‎ ‎tamely ramified maximal torus of $G$ and $phi$ is a quasicharacter‎ ‎of $T$ whose restriction t...

Recall that an algebraic group G is called (linearly) reductive if any its rational (i.e., algebraic) representation is completely reducible. The finite groups, the group GLn and the products GLn1 × . . .GLnk are reductive. Below G denotes a reductive algebraic group and X is an affine algebraic variety equipped with an (algebraic) action of G. Results explained below in this section can be fou...

We state a conjecture on the reduction modulo defining characteristic of unipotent representation finite reductive group.

In this paper, we extend the work in Morita’s Theory for the Symplectic Groups [7] to split reductive groups. We construct and study the holomorphic discrete series representation and the principal series representation of a split reductive group G over a p-adic field F as well as a duality between certain sub-representations of these two representations.

We will drop the compactness hypothesis on G in the results of §6, doing this in such a way that problems can be reduced to the compact case. This involves the notions of reductive Lie groups and algebras and Cartan involutions. Let © be a Lie algebra. A subalgebra S c © is called a reductive subaU gebra if the representation ad%\® of ίΐ on © is fully reducible. © is called reductive if it is a...

We show that a mod-ℓ-representation of p-adic group arising from the analogue Yu's construction is supercuspidal if and only it arises representation finite reductive group. This has been previously shown by Henniart Vigneras under assumption second adjointness holds.

Abstract Consider a reductive linear algebraic group G acting linearly on polynomial ring S over an infinite field; key examples are the general group, symplectic orthogonal and special with classical representations as in Weyl’s book: For consider direct sum of copies standard representation dual; other cases, take representation. The invariant rings respective cases determinantal rings, defin...