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Register a new userThe special linear representations of compact Lie groups
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The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special linear representation for next using.
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Let $G$ be a real classical group of type $B$, $C$, $D$ (including the real metaplectic group). We consider a nilpotent adjoint orbit $check{mathcal O}$ of $check G$, the Langlands dual of $G$ (or the metaplectic dual of $G$ when $G$ is a real metaplectic group). We classify all special unipotent representations of $G$ attached to $check{mathcal O}$, in the sense of Barbasch and Vogan. When $check{mathcal O}$ is of good parity, we construct all such representations of $G$ via the method of theta lifting. As a consequence of the construction and the classification, we conclude that all special unipotent representations of $G$ are unitarizable, as predicted by the Arthur-Barbasch-Vogan conjecture. We also determine precise structure of the associated cycles of special unipotent representations of $G$.
In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called {em PSH-algebras}. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bi-algebra that contains a large PSH-algebra that extends Zelevinskys algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.
A special symplectic Lie group is a triple $(G,omega, abla)$ such that $G$ is a finite-dimensional real Lie group and $omega$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $ abla$. In this paper starting from a special symplectic Lie group we show how to ``deform the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure $ abla$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.
Let $G$ be an almost linear Nash group, namely, a Nash group which admits a Nash homomorphism with finite kernel to some $GL_k(mathbb R)$. A homology theory (the Schwartz homology) is established for the category of smooth Fre representations of $G$ of moderate growth. Frobenius reciprocity and Shapiros lemma are proved in this category. As an application, we give a criterion for automatic extensions of Schwartz homologies of Schwartz sections of a tempered $G$-vector bundle.
Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank not greater then 3 are explicitly studied. We derive the polynomials of simple Lie groups B_3 and C_3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.
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