SMALLEST DEGREES OF REPRESENTATIONS OF EXCEPTIONAL GROUPS OF LIE TYPE

Abstract

where m ∈ N and q is an arbitrary power of a prime p. For a group G in this list let G := G(q)sc be a corresponding finite group of Lie type arising as group of fixed points under a Frobenius map of a simple simply-connected algebraic group. Up to a finite number of exceptions G is the universal covering group of G and G ∼= G/Z(G). So, in this case, the smallest degrees of non-trivial projective representations of G are equal to the smallest non-trivial degrees of representations of G (which is a perfect group). As the main result of this note we determine in Section 2 the first few smallest non-trivial degrees of complex representations of the groups G, together with their multiplicities. We get these as an application of Deligne-Lusztig theory and Lusztig’s classification of irreducible characters of finite groups of Lie type. The groups with exceptional universal coverings (as well as the Tits group F4(2) ′) are listed and dealt with in Section 3. This completes the determination of d0(G) for all groups in the above list. In Section 4 we collect for the first five types of groups some values dl(G) for l a prime not equal to p. The information is complete in the first three cases. This improves the known lower bounds for dl(G), l 6= p, given by Landazuri, Seitz and Zalesskii in [LS74] and [SZ93].