(pp. 163-172)

The situation in odd characteristic is complicated by the quadratic character *χ*₂: the ! hypergeometric sheaf of type (7, 1)

$ \cal{F}(\chi_2, \, k) \, := \, (A^{-7})^{deg} \, \otimes \, \cal{H} (\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi_2, \, \chi_2, \, \chi_2, \, \chi_2; \, \chi_2), $

though lisse on $ \mathbb{G}_m $ , is not pure of weight zero, nor is its *G*_{geom} the group *G*₂. One knows [Ka-ESDE, 8.4.7] that this sheaf sits in a short exact sequence

$ 0 \, \rightarrow \, V \, \otimes \, \cal{L}_{\chi_2} \, \rightarrow \, \cal{F}(\chi_2, \, k) \, \rightarrow \, (A^{-7})^{deg} \, \otimes \, Kl(\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi_2, \, \chi_2, \, \chi_2) (-1) \, \rightarrow \, 0, $

with

$ V \, := \, (A^{-7})^{deg} \, \otimes \, H^1_c(\mathbb{G} _m / \overline {k}, \, Kl(\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi_2, \, \chi_2, \, \chi_2)). $

The rank 6 quotient $ (A^{-7})^{deg} \, \otimes \, Kl(\mathbb{1}, \, \mathbb{1}, \, \mathbb{1}, \, \chi_2, \, \chi_2, \, \chi_2)(-1) $ is pure of weight 0, and the rank one subobject $ V \, \otimes \, \cal{L}_{\chi_2} $ is pure of weight −4; in fact *Frob*_{k}|*V* = *A*^{−}⁴, cf. [Ka-GKM, 4.0].

In order to construct the objects *N*(*a*, *k*), in odd characteristic we will first construct an...