Multiplicities of principal series representations of finite groups with split (B, N)-pairs

Representations of generic algebras and finite groups of Lie type

“…where a is a certain one-dimensional character of IF(<5) which is described geometrically (see (5.2)) and which reduces to the sign character of W in the case of principal series. Thus our theorem generalizes the results of Curtis [3] who studied the case of principal series (1#) and McGovern [12], who treated the case of generalized principal series, where L is empty but 8 ^ 1. These two authors also proved results corresponding to the comparison theorem in their respective situations, as did Lusztig [11] for the case where 8 is a unipotent cuspidal representation.…”

“…(iv) Two special cases of (5.9) have been dealt with by other authors: the case of principal series (L empty, 5=1) was treated by Curtis in [3] while McGovern [12] proved (5.9) in the generalized principal series case (L empty, 8 ¥= 1). The same remark applies to the duality theorem below (Theorem (7.5)).…”

Disconnected groups of Lie type as Galois groups.

“…An important aid in the computation of the missing character values is given by the result of Howlett and Lehrer (comparison theorem, [13], Theorem 5.9, see also [25], Theorem 3.4). It describes the decomposition of Harish-Chandra induced unipotent characters from Levi subgroups.…”

Character value estimates for groups of Lie type

“…We have ^r e (?nZ(G) < Z(G ; ). Since r doesn't divide |Z(G')|, we have \O r ((v) To prove the theorem, it suffices to show that \ψ(h)/ψ(l)\ < 6/q for any nonlinear irreducible character ψ of H. Using Lemma 1.9, Our work will be based on the following "comparison theorem" of McGovern [17]. By abuse, we will use the same symbol λ to denote the restriction of λ to the diagonal subgroup of G. …”

“…We essentially get the best possible absolute upper bound for the multiplicities. The theory developed in [13] and [17] is used to bound these multiplicities in terms of corresponding multiplicities in the Weyl group W of G. We obtain only a crude absolute upper bound for the number of distinct linear constituents of χ Pj . Our bound involves the indices of certain large reflection subgroups of W in their normalizers.…”