sense, since it can be viewed as the identity component of GL(1,R) ≅ R*. The problem of classifying the real reductive groups largely reduces to classifying the simple Lie groups. These are classified by their Satake diagram; or one can just refer to the list of simple Lie groups (up to finite coverings). Useful theories of admissible representations and unitary representations have been developed for real reductive groups in this generality. The main differences between this definition and the definition of a reductive algebraic group have to do with the fact that an algebraic group G over R may be connected as an algebraic group while the Lie group G(R) is not connected, and likewise for simply connected groups. For example, the projective linear group PGL(2) is connected as an algebraic group over any field, but its group of real points PGL(2,R) has two connected components. The identity component of PGL(2,R) (sometimes called PSL(2,R)) is a real reductive group that cannot be viewed as an algebraic group. Similarly, SL(2) is simply connected as an algebraic group over any field, but the Lie group SL(2,R) has fundamental group isomorphic to the integers Z, and so SL(2,R) has nontrivial covering spaces. By definition, all finite coverings of SL(2,R) (such as the metaplectic group) are real reductive groups. On the other hand, the universal cover of SL(2,R) is not a real reductive group, even though its Lie algebra is reductive, that is, the product of a semisimple Lie algebra and an abelian Lie algebra. For a connected real reductive group G, the quotient manifold G/K of G by a maximal compact subgroup K is a symmetric space of non-compact type. In fact, every symmetric space of non-compact type arises this way. These are central examples in Riemannian geometry of manifolds with nonpositive sectional curvature. For example, SL(2,R)/SO(2) is the hyperbolic plane, and SL(2,C)/SU(2) is hyperbolic 3-space. For a reductive group G over a field k that is complete with respect to a discrete valuation (such as the p-adic numbers Qp), the affine building X of G plays the role of the symmetric space. Namely, X is a simplicial complex with an action of G(k), and G(k) preserves a CAT(0) metric on X, the analog of a metric with nonpositive curvature. The dimension of the affine building is the k-rank of G. For example, the building of SL(2,Qp) is a tree. Representations of reductive