The Weyl Group as Xed Point Set of Smooth Involutions
نویسنده
چکیده
We show that the Weyl group W = M 0 =M of a noncompact semisimple Lie group is obtained by taking xed point sets of smooth involutions in K=M. More precisely, one considers rst the xed point set X of the involutions deened on K=M by the elements of order 2 in exp ia. The Weyl group is either X , or the xed point set of the involutions deened on X by special elements of order 4 in exp ia.
منابع مشابه
Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In t...
متن کاملCOUPLED FIXED POINT THEOREMS FOR RATIONAL TYPE CONTRACTIONS VIA C-CLASS FUNCTIONS
In this paper, by using C-class functions, we will present a coupled xed problem in b-metric space for the single-valued operators satisfying a generalized contraction condition. First part of the paper is related to some xed point theorems, the second part presents the uniqueness and existence for the solution of the coupled xed point problem and in the third part we...
متن کاملA fixed point result for a new class of set-valued contractions
In this paper, we introduce a new class of set-valued contractions and obtain a xed point theoremfor such mappings in complete metric spaces. Our main result generalizes and improves many well-known xed point theorems in the literature.
متن کاملFully Commutative Elements in the Weyl and Affine Weyl Groups
Let W be a Weyl or affine Weyl group and let Wc be the set of fully commutative elements in W . We associate each w ∈ Wc to a digraph G(w). By using G(w), we give a graph-theoretic description for Lusztig’s a-function on Wc and describe explicitly all the distinguished involutions of W . The results verify two conjectures in our case: one was proposed by myself in [15, Conjecture 8.10] and the ...
متن کاملInvolutions in Weyl Groups
Let G be a split real group with Weyl group W . Let E be an irreducible representation of W . Let V be the stable Lie algebra version of the coherent continuation representation of W . The main result of this paper is a formula for the multiplicity of E in V . The formula involves the position of E in Lusztig’s set ∐ M(G). The paper treats all quasi-split groups G as well.
متن کامل