By American Mathematical Society

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In a different context, the adjective “irreducible" can have a completely different meaning when it is applied to a group. 12. 15) Remark. If H is a subgroup of GL( , C) that is not irreducible (that is, if H is reducible), then, after a change of basis, we have H⊂ GL(k, C) 0 ∗ , GL(n − k, C) for some k with 1 ≤ k ≤ n − 1. Similarly for GL( , R). 16) Corollary. If H is a nonabelian, closed, connected, irreducible subgroup of SL( , C), then H is semisimple. Proof. Suppose A is a connected, abelian, normal subgroup of H.

36 CHAPTER 3. INTRODUCTION TO SEMISIMPLE LIE GROUPS • A subset H of SL( , R) is Zariski closed if there is a subset Q of R[x1,1 , . . , x , ], such that H = Var(Q). ) • The Zariski closure of a subset H of SL( , R) is the (unique) smallest Zariski closed subset of SL( , R) that contains H. The Zariski closure is sometimes denoted H. 30) Example. 1) SL( , R) is Zariski closed. Let Q = ∅. 2) The group of diagonal matrices in SL( , R) is Zariski closed. Let Q = { xi,j | i ≠ j }. 3) For any A ∈ GL( , R), the centralizer of A is Zariski closed.

521]. 40 CHAPTER 3. INTRODUCTION TO SEMISIMPLE LIE GROUPS References [Bor4] A. , Springer, New York, 1991. [Car] É. Cartan: Les groupes réels simples finis et continus, Ann. Sci. École Norm. Sup. 31 (1914) 263–355. [Hel2] S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, 1978. ) [Hum1] J. E. Humphreys: Introduction to Lie Algebras and Representation Theory. Springer, Berlin Heidelberg New York, 1972. [Mos1] G. D. Mostow: Self adjoint groups, Ann. Math.