Explicit Brauer Induction and Shintani descent

Abstract

<p>This paper demonstrates an application of Explicit Brauer Induction. Snaith (28), introduced this canonical form for Brauer's induction theorem. An overview of this development is given in chapter one with a new proof of Brauer's theorem. In (29-35) Snaith applied Explicit Brauer Induction, primarily in the construction of invariants of representations of finite groups from invariants of one-dimensional characters. Following (28, 29, 30) Chapter two presents proof that the Grothendieck group R₊(G,S¹) is a ring with properties of induction, restriction, inflation, and Frobenius reciprocity. There exists a ring homomorphism b:R₊(G,S¹) → R(G) which was introduced as a footnote in (26, P. 71). We discuss the map T_G, (30 Pp. 454-469), which is a section to b. Deligne (6, Pp. 501-597) has devised generators for Ker b, in the case for G solvable. We examine the case G = D₈. Boltje (1, 2) has devised another section to the map b, termed a_G. This development and examples are given in Chapter three. In Chapter four, conjugacy classes and character values, for the matrix group GL₂F_q are reviewed for all irreducible representations including the cuspidal case (13, Pp. 122). The main result of this paper is contained in Chapter five. We develop the Explicit Brauer Induction formula for a_G(ρ) where G = GL₂F_q and ρ is an irreducible representation of G. This development is used to describe Shintani descent between the irreducible representations of GL₂Fⁿ_q and the irreducible representations of GL₂F_q. The original derivation of Shintani descent (27, Pp. 396-414), uses norms on character values. In the construction given here, the Shintani norm is not used, but rather, the correspondence is obtained by applying Hilbert Theorem 90 to the maximal one dimensional characters which appear in the expression for a_G(ρ).</p>