The characters of each family are indexed by conjugacy classes of pairs ("x",σ) where "x" is in one of the groups Z/2Z"n", "S"3, "S"4, "S"5 respectively, and σ is a representation of its centralizer. (The family of size 39 only occurs for groups of type "E"8, and the family of size 21 only occurs for groups of type "F"4.) The families are in turn indexed by special representations of the Weyl group, or equivalently by 2-sided cells of the Weyl group. For example, the group "E"8(F"q") has 46 families of unipotent characters corresponding to the 46 special representations of the Weyl group of "E"8. There are 23 families with 1 character, 18 families with 4 characters, 4 families with 8 characters, and one family with 39 characters (which includes the 13 cuspidal unipotent characters). Suppose that "q" is an odd prime power, and "G" is the algebraic group "SL"2. We describe the Deligne–Lusztig representations of the group "SL"2(F"q"). (The representation theory of these groups was well known long before Deligne–Lusztig theory.) The irreducible representations are:  There are two classes of tori associated to the two elements (or conjugacy classes) of the Weyl group, denoted by "T"(1) (cyclic of order "q"−1) and "T"("w") (cyclic of order "q" + 1). The non-trivial element of the Weyl group acts on the characters of these tori by changing each character to its inverse. So the Weyl group fixes a character if and only if it has order 1 or 2. By the orthogonality formula, "R"θ("w") is (up to sign) irreducible if θ does not have order 1 or 2, and a sum of two irreducible representations if it has order 1 or 2. The Deligne-Lusztig variety "X"(1) for the split torus is 0-dimensional with "q"+1 points, and can be identified with the points of 1-dimensional projective space defined over F"q". The representations "R"θ(1) are given as follows:  The Deligne-Lusztig variety "X"("w") for the non-split torus is 1-dimensional, and can be identified with the complement of "X"(1) in 1-dimensional projective space.