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

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Let k be the algebraic closure of a finite field F q of characteristic p. Let G be a connected unipotent group over k equipped with an F q -structure given by a Frobenius map F : G −→ G. We will denote the corresponding algebraic group defined over F q by G 0 . Character sheaves on G are certain special objects in the triangulated braided monoidal category D G (G) of bounded conjugation equivariant Q l -complexes (where l = p is a prime number) on G. Boyarchenko has proved that the "trace of Frobenius" functions associated with F -stable character sheaves on G form an orthonormal basis of the space of class functions on G 0 (F q ) and that the matrix relating this basis to the basis formed by the irreducible characters of G 0 (F q ) is block diagonal with "small" blocks. In particular, there is a partition of the set of character sheaves as well as a partition of the set of irreducible characters of G 0 (F q ) into "small" families know as Lpackets. In this paper we describe these block matrices relating character sheaves and irreducible characters corresponding to each L-packet. We prove that these matrices can be described as certain "crossed S-matrices" associated with each L-packet. We will also derive a formula for the dimensions of the irreducible representations of G 0 (F q ) in terms of certain modular categorical data associated with the corresponding L-packet. In fact we will formulate and prove more general results which hold for possibly disconnected groups G such that G • is unipotent. To prove our results, we will establish a formula (which holds for any algebraic group G) which expresses the inner product of the "trace of Frobenius" function of any F -stable object of D G (G) with any character of G 0 (F q ) (or of any of its pure inner forms) in terms of certain categorical operations.

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