The monoid generated by projections in an algebraic group

Abstract

Let G be a simple algebraic group, g(G) the set of all Bore1 subgroups of G, and A the set of all parabolic subgroups of G. For PEA, let proj,,:g(G)-+&?(G) be given by proj.(B)=(PnB)R,(P)cP, where R,(P) is the unipotent radical of P. Tits [9] has given a purely combinatorial (i.e., in terms of the Tits building A) definition of proj,. In this paper we are interested in the monoid M = M(G) generated (with respect to composition) by proj., PEA. Let PEA, P # G, L a Levi factor of P. Let H, denote the group of automorphisms of &I(L) induced by N,(L). Thus H, is a finite extension of L’ = L/C(L), where C(L) denotes the center of L. The following results are established: (1) The maximal subgroup of A4 with identity element proj. is isomorphic to H,, and moreover every maximal subgroup of A4 with identity element e # 1 is isomorphic to some H,. (2) M is a fundamental regular monoid. (3) The idempotent set E( M ) = { proj p 0 proj o 1 P, Q E A >. (4) The set of principal right ideals of M is in one to one correspondence with A. (5) The set A with all the Bore1 subgroups identified is in one to one correspondence with the set of all principal left ideals of M. (6) The principal ideals of M are in one to one correspondence with the conjugacy classes in W of W, (Is S ), where W is the Weyl group of G with fundamental generating set S. (7) If r = IS(, then for all a EM, ar lies in a subgroup of M.