Moduli of filtered quiver representations

Equivariant motivic Hall algebras

“…with the vertical identity maps omitted, where = (1) is the integral in degree 1. In order to compare the two compositions, we take the fibre product on the left, as follows.…”

“…In order to circumvent the above convergence issues, we need to apply more sophisticated inversion formulae, like those due to Zagier [69] and Laumon-Rapoport [39], the latter based on the idea of Kottwitz to apply the Langlands lemma from the theory of Eisenstein series. In particular, (6.10) applies in the case where ω : E = rep K (Q) → D = 0, as well as to Example 6.16 (1), but for instance not to ω : vect X → D = 0. Proof.…”

“…cf. [1]). Now assume that D is split (proto-)exact, as we want to think of it rather as a collection of structure groups.…”

“…For η = ϕ(pr D (τ )), this yieldsχ c (M ss [N ],α ) = τ ([N ],α) (−1) m−1 t i<j χ op (αi,αj) Ind Aut(N ) Pη(K) (χ c (M [N1],α1 ) • • • χ c (M [Nm],αm )).Of course, the analogous statement holds for any equivariant motivic measure (Example 4.5). Now consider the situation of Example 6.16(1). Then by (6.5), we obtain from the aboveχ c (π 0 M ss α ) = τ α (−1) m−1 t i<j χ op (αi,αj ) Ind GLr(K) Pη (K) (χ c (GL η1 /P α1 ) • • • χ c (GL ηm /P αm )).But the Euler-Poincaré characteristic of the flag variety of type α is given byχ c (GL r /P α ) = (t r − 1) • • • (t − 1) (t δ1 − 1) • • • (t − 1) • • • (t δn − 1) • • • (t − 1) =…”