Introduction to Bicategories by Benabou J.

By Benabou J.

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We shall therefore consider the structure of this endomorJ phism algebra. We first recall the situation in the special case when PJ = Band ,p = 1. Then End (1gJ) is the Hecke algebra H(GF , BF ). 3 that this algebra has dimension IWFI and basis Tw, WE WF. WF is a Coxeter group with Coxeter generators SJ corresponding to the F-orbits J on the Dynkin diagram of G. The multiplication of the basis elements is determined by the relations T. T. Jw+(PJ-1)Tw ifl(sJw)=I(w)-1 where W E WF, i is the length function on WF, and PJ = IUF n (UF)WOSJI.

The mapping ~ -+ ~* is also an isometry of generalized characters. Thus one has (~*, ,,*) = (~, ,,) for any two generalized characters ~, " of GF • We mention two examples of the effect of this duality operation. In the first place we have 1* = St. Thus the dual of the principal character is the Steinberg character. It follows of course that St* = 1. Secondly we take a Deligne-Lusztig generalized character R T ,6 of GF • Then we have R},6 = BGBT R T ,6' This result was proved by Deligne and Lusztig in [2].

Given a subalgebra of L(G) isomorphic to sI 2 (K) there is a subgroup of G isomorphic to SL 2 (K) or to PGL 2 (K) whose Lie algebra is the given subalgebra. Let S be a maximal torus of this subgroup. Then SeT for some maximal torus T of G. Let t = L(T) and Ke~ = L(X~) for each root subgroup X~ of G. Then we have L( G) = t EB L