Minimal length elements of finite Coxeter groups

Abstract

Abstract. We give a geometric proof that minimal length elements in a (twisted) conjugacy class of a finite Coxeter group W have remarkable properties with respect to conjugation, taking powers in the associated Braid group and taking centralizer in W .

Lifting of elements of Weyl groups

“…This is is proved in [7], [6] and [9] via case-by-case analyses, and a general proof is in [10]. In fact, we may choose a good element having minimal length in the conjugacy class.…”

Minimal length elements of extended affine Weyl groups

“…Property (2) was first conjectured and verified by Lusztig [23, 1.2] for untwisted classical groups and proved in general in [15].…”

“…The key step is Proposition 2.5, which reduces the study of arbitrary alcove to an alcove that intersects the affine subspace associated to the Newton point ofw. We then use the "partial conjugation" method introduced in [11] to reduce to a similar problem for a finite Coxeter group, which is proved in [8] and [9] via a case-by-case analysis and later in [15] by a case-free argument. This leads to a case-free proof which works for all cases, including the exceptional affine Weyl groups, which seems very difficult via the approach in [12].…”

“…The following result is proved in [8], [7] and [11] via a case-by-case analysis with the aid of computer for exceptional types. A case-free proof which does not rely on computer calculation was recently obtained in [15]. …”

Centers and cocenters of 0-Hecke algebras

“…The following result is proved in [4, Theorem 1.1], [3, Theorem 2.6] and [7,Theorem 7.5] (see also [9] for a case-free proof). …”