(2, 3, k)-generated groups of large rank by Lucchini A.

By Lucchini A.

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It is a combinatorial object which provides us with a language to study representations of GL,(F). (For more information on root systems see [a]). Let IR be the field of real numbers, and consider the space R" with the standard basis e l , . . , en. With respect to this basis, we shall identify every element IC in IRn with an n-tuple ( ~ 1 ,. ,IC,). The group S, acts on R" by permuting the entries of ( X I , . . ,x,). This action preserves the inner product n i= 1 Let a Note that the group S, preserves R.

EK. Clearly, Lectures o n Representations of p-adic Groups 41 If a 2 1, then n is in K1 and ~ ( n=) 1. On the other hand, recall that ~(n= ' ) -1. It follows that f(&) Thus, f ( A a ) proved. = = f(n&L) = f(W)= -f(U 0 if a >_ 1, and f must be a multiple of e K . The lemma is 0 We can now finish the proof of proposition. 2 implies that there exists an irreducible quotient V' of V. Let P be the projection from V onto V'. 3 there is a splitting s : V' -+ V such that P o s = I d v f . But s0 PE HOrnGO (V,V )= c Idv ' so s o P must be equal to I d v .

Moreover, it is called unramified, if it is trivial on Zp”. Every unramified character x is completely determined by its value ~ ( p )In. particular, the group of unramified characters is isomorphic to C X . Any other p a d i c field F is simply a finite extension of Q p (see [3]). The integral closure of Z, will be denoted by 0. It has a unique maximal ideal ( a )which , is the radical of the ideal ( p ) . We shall normalize the absolute value on F so that *1 Jwl = -. 4 3. Structure of G L , ( F ) Let F be a p a d i c field, and G = GL,(F).

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