Minimal idempotents on solvable groups

Abstract

In this paper, we begin to develop a theory of character sheaves on an affine algebraic group G defined over an algebraically closed field k of characteristic p > 0 using the approach developed by Boyarchenko and Drinfeld for unipotent groups. Let l be a prime different from p. Following Boyarchenko and Drinfeld ([BD08]), we define the notion of an admissible pair on G and the corresponding idempotent in the Q l -linear triangulated braided monoidal category D G (G) of conjugation equivariant Q l -complexes (under convolution with compact support) and study their properties. In the spirit of [BD08], we aim to break up the braided monoidal category

Character sheaves on neutrally solvable groups

“…We shall continue to use all the standard notation and conventions from [BD], [De2]. Hence for example if G is a (perfect quasi-) algebraic group, then D G (G) denotes the Q -linear triangulated ribbon r-category of conjugation equivariant constructible Q -complexes.…”

“…It is often more convenient to pass to the perfectizations of algebraic groups and schemes. Hence continuing our convention from [De2], by algebraic group we actually mean perfect quasi-algebraic group over k (i.e., the perfectization of an algebraic group). We refer to [BD,§1.9] for more about this convention.…”

“…In this paper we will define a set CS(G) of isomorphism classes of some special objects in D G (G) known as character sheaves on G. If C ∈ CS(G), then we will see that C is a simple object in D G (G), in the sense that Hom D G (G) (C, C) = Q . In [De2], we studied minimal idempotents in the braided monoidal category D G (G) and proved that each such minimal idempotent is in fact locally closed (cf. [De2,§2.3.2]) and can be obtained from an admissible pair for G. Let G denote the set of (isomorphism classes of) minimal idempotents in D G (G).…”

“…In [De2], we studied minimal idempotents in the braided monoidal category D G (G) and proved that each such minimal idempotent is in fact locally closed (cf. [De2,§2.3.2]) and can be obtained from an admissible pair for G. Let G denote the set of (isomorphism classes of) minimal idempotents in D G (G). For each e ∈ G, we will first define the set CS e (G) of character sheaves in the full subcategory eD G (G) ⊆ D G (G).…”

“…In [De2] we embarked upon the journey towards developing a theory of character sheaves on a general affine algebraic group G defined over an algebraically closed field k of characteristic p > 0. The study of character sheaves was initiated by Lusztig for reductive groups in his works [L].…”