The c-functions, related to a reductive symmetric space G/H and a fixed representation τ of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions.

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteris...

We regard the shearlet group as a semidirect product group and show that its standard representation is,typically, a quasiregu- lar representation. As a result we can characterize irreducible as well as square-integrable subrepresentations of the shearlet group.

This definition goes back to J.Dixmier, see [Di, 11.1.6]. He considered index because of its importance in Representation Theory. The problem of computing the index may also be treated as part of Invariant Theory. For, if q is an algebraic Lie algebra and Q is a corresponding algebraic group, then ind q equals the transcendence degree of the field of Q-invariant rational functions on q. If q is...

The question whether P 6= NP is widely regarded to be one of the most important outstanding problems in modern mathematics [39]. I work in an emerging intersection of three areas of mathematics which recent developments suggest may provide new test cases and perhaps new insight into this problem. These three areas are computational representation theory, polyhedral geometry, and theoretical com...

suppose $g$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$f,$ $p=mn$ is a maximal parabolic subgroup of $g,$ $frak{n}$ is‎ ‎the lie algebra of the unipotent radical $n.$ under the adjoint‎ ‎action of its stabilizer in $m,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

If (π, V ) is an admissible representation of a p-adic reductive group G and P = MN is a parabolic subgroupwith unipotent radicalN , its Jacquet module VN is the universalN -trivial quotientH0(N,V ) of V by the span of vectors π(n)v − v (n ∈ N ). One fundamental property is that V VN is an exact functor. This construction plays an important role in homomorphisms into representations induced fro...

We introduce a concept and develop a theory of Galois subalgebras in skew semigroup rings. Proposed approach has a strong impact on the representation theory, first of all the theory of Harish-Chandra modules, of many infinite dimensional algebras including the Generalized Weyl algebras, the universal enveloping algebras of reductive Lie algebras, their quantizations, Yangians etc. In particula...

Let $k'/k$ be a finite purely inseparable field extension and let $G'$ reductive $k'$-group. We denote by $G=\R_{k'/k}(G')$ the Weil restriction of across $k'/k$, pseudo-reductive group. This article gives bounds for exponent geometric unipotent radical $\RR_{u}(G_{\bar{k}})$ in terms invariants starting with case $G'=\GL_n$ applying these results to where is simple