2-Local subgroups of finite groups by Kondratev A.S.

By Kondratev A.S.

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Z (Z is the center of G ) is of finite index in G. 4. Smooth representations Let V be a vector space over the field of complex numbers @. A smooth representation of G on V is a homomorphism 7r : G + GL(V) G. Sawin 24 such that every vector v in V is fixed by a (sufficiently small) congruence subgroup. We shall refer to V as a (smooth) G-module. A representation is admissible if V K zis finite dimensional for every i. It turns out that any irreducible smooth module is in fact admissible. We shall give a proof of this fact for GL2(F).

Show that the kernel of 01, is precisely I n d : ( & , ( N ) . 2. Q w ( f ) . The proposition is proved. 0 We now note a very important consequence of the above result. If x # is called regular) then the above exact sequence splits. 3: The following gives a complete answer to decomposing principal series representations of G L 2 ( F ) . 0 The induced representation I n d g ( X ) is irreducible if ~ 1 1 x # 2 Otherwise, the composition series is of length two, with one of the subfactors a one-dimensional representation.

It has a filtration by the principal G. Savin 22 congruence subgroups Ki of G defined by Ki = {g E G I g == 1 mod wz}. These groups are normal in K , and provide a fundamental system of neighborhoods of 1 in G, defining a topology of G. 1. Cartan decomposition Let A E Z" be the group of diagonal matrices where ml, . . , m, are integers. Let A' be the subset of A consisting of diagonal matrices such that ml 2 . . 2 m,. Then G = KA'K. 3 . 2 . Bruhat- Tits decomposition Let B denote the group of upper triangular matrices in G, and N the subgroup of B consisting of matrices with 1 on the diagonal.

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