Representations of generic algebras and finite groups of Lie type

Abstract

Abstract. The complex representation theory of a finite Lie group G is related to that of certain "generic algebras". As a consequence, formulae are derived ("the Comparison Theorem"), relating multiplicities in G to multiplicities in the Weyl group W of G. Applications include an explicit description of the dual (see below) of an arbitrary irreducible complex representation of G.

Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow $2$-subgroup conjecture

“…It is known from the work of Howlett and Lehrer (see [HL80,HL83]) that the members of Irr G|R G L (λ) are in bijection with characters η ∈ Irr(W (λ)), where W (λ) is the so-called relative Weyl group with respect to λ. (We note that in fact, the original bijection required a "twist" by a certain 2-cycle, but that it was shown later that one could take this 2-cycle to be trivial -see [Gec93].)…”

“…We denote by H K the K-algebra K ⊗ A H, and for any ring homomorphism f : A → C, we obtain the C-algebra H f := C ⊗ A H with basis {1 ⊗ a w : w ∈ W (λ)}. By [HL83,Proposition 4.7], when H f is semisimple, this yields a bijection η → η f between K-characters of simple H K -modules and characters of simple…”

Restriction of Odd Degree Characters and Natural Correspondences

“…Then χ is a unipotent character of G tensored with a linear character; in particular, χ belongs to the principal series. By the Comparison Theorem [HL,Theorem 5.9] (see also [C,Theorem 5.1] for the case of the principal series), the computation of * R G L (χ) can be replaced by the computation of (χ λ 1 ) S n−1 , where we identify S n , respectively S n−1 , with the Weyl group of G, respectively of L. (See also [FS,Proposition (1C)] for the explicit formula in the case of G.) Applying Theorem 1.2 and (2.2), we are done in this case.…”

Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs

“…We take XG t o ^e the character in S(G/L, XL) corresponding to the alternating character of stabw(xL). In parti¬ cular, nlL<x0(T£xG) = XM is irreducible for Me £f G L and n{L<XL)(TLGxG) = XL by [31], The¬ orem 5.9. Finally suppose that there exists an indecomposable projective CG-lattice YG such that n(LxO(YG)= X G h a s c h a r a c t e r xG-Setting XM = n(LtXL)(TfixG)for ME^G,L and denoting the projective cover XM by YM, we can apply Theorem 4.15 to get a projective restriction system &£%(XG, YL).…”