We define a map from the set of conjugacy classes of a Weyl group W to the representation ring of W tensored with the ring of polynomials in one variable.
Let $mathsf G$ be a connected reductive linear algebraic group defined over $mathbb R$, and let $C: mathsf Grightarrow mathsf G$ be a fundamental Chevalley involution. We show that for every $gin mathsf G(mathbb R)$, $C(g)$ is conjugate to $g^{-1}$ i
n the group $mathsf G(mathbb R)$. Similar result on the Lie algebras is also obtained.
A {it $k$-involution} is an involution with a fixed point set of codimension $k$. The conjugacy class of such an involution, denoted $S_k$, generates $text{Mob}(n)$-the the group of isometries of hyperbolic $n$-space-if $k$ is odd, and its orientatio
n preserving subgroup if $k$ is even. In this paper, we supply effective lower and upper bounds for the $S_k$ word length of $text{Mob}(n)$ if $k$ is odd, and the $S_k$ word length of $text{Mob}^+(n)$, if $k$ is even. As a consequence, for a fixed codimension $k$ the length of $text{Mob}^{+}(n)$ with respect to $S_k$, $k$ even, grows linearly with $n$ with the same statement holding in the odd case. Moreover, the percentage of involution conjugacy classes for which $text{Mob}^{+}(n)$ has length two approaches zero, as $n$ approaches infinity.
Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichle
t series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a metaplectic representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check the remarkable fact that their formulas continue to define a group action for general (unspecialized) parameters. In the first part of paper we give a conceptual explanation of this fact, by giving a uniform and elementary construction of the metaplectic representation for generic Hecke algebras as a suitable quotient of a parabolically induced affine Hecke algebra module, from which the associated Chinta-Gunnells Weyl group action follows through localization. In the second part of the paper we extend the metaplectic representation to the double affine Hecke algebra, which provides a generalization of Cheredniks basic representation. This allows us to introduce a new family of metaplectic polynomials, which generalize nonsymmetric Macdonald polynomials. In this paper, we provide the details of the construction of metaplectic polynomials in type A; the general case will be handled in the sequel to this paper.
In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.
Let $G$ be a connected complex semisimple Lie group with a fixed maximal torus $T$ and a Borel subgroup $B supset T$. For an arbitrary automorphism $theta$ of $G$, we introduce a holomorphic Poisson structure $pi_theta$ on $G$ which is invariant unde
r the $theta$-twisted conjugation by $T$ and has the property that every $theta$-twisted conjugacy class of $G$ is a Poisson subvariety with respect to $pi_theta$. We describe the $T$-orbits of symplectic leaves, called $T$-leaves, of $pi_theta$ and compute the dimensions of the symplectic leaves (i.e, the ranks) of $pi_theta$. We give the lowest rank of $pi_theta$ in any given $theta$-twisted conjugacy class, and we relate the lowest possible rank locus of $pi_theta$ in $G$ with spherical $theta$-twisted conjugacy classes of $G$. In particular, we show that $pi_theta$ vanishes somewhere on $G$ if and only if $theta$ induces an involution on the Dynkin diagram of $G$, and that in such a case a $theta$-twisted conjugacy class $C$ contains a vanishing point of $pi_theta$ if and only if $C$ is spherical.