Exotic Symmetric Space Over a Finite Field, Iii

Abstract

Let X = G/H × V , where V is a symplectic space such that G = GL(V ) and H = Sp(V ). In previous papers, the authors constructed character sheaves on X , based on the explicit data. On the other hand, there exists a conceptual definition of character sheaves on X based on the idea of Ginzburg in the case of symmetric spaces. Our character sheaves form a subset of Ginzburg type character sheaves. In this paper we show that these two definitions actually coincide, which implies a classification of Ginzburg type character sheaves on X . * supported by ANR JCJC REPRED, ANR-09-JCJC-0102-01.But by comparing (1.3.1) and (3.5.2) in Appendix (or in [SS]), such A is exactly the same A appearing in the decomposition of (ψ m ) * α * 0 E. Then by the discussion in the proof of (3.6.1*) in Appendix, we see that A| Y m−1 gives no contribution to the former factors in (3.6.1*) for (ψ m−1 ) * α * 0 E. Hence it gives no contribution for the former factors of (1.3.3) as asserted. This proves (1.3.3) for m.By considering the case where m = n ′ , we obtain the following result, which is an analogue of Proposition 3.6 in [SS] (see Appendix).is a semisimple complex on Y ′ , equipped with W ′ Eaction, and is decomposed asWe consider θ-stable parabolic subgroups P ⊂ Q of G. We follow the notation in 4.2. In particular,Then V L is a quotient of V ′′ as P θ -module. The following transitivity holds.Proposition 5.2. Let K ∈ D H (X ). Then there is an isomorphism in D P θ (X L ) Res X X L ,V ′′ ,P K ≃ (Res X M X L ,V ′ M ,P M • Res X X M ,V ′ ,Q )K.

An algebraic study of extension algebras

“…We note that Shoji-Sorlin [SS12] independently proved the last part of Theorem E by a completely different method. Also, Theorem E implies [AH08] Conjecture 6.4.…”