A lower bound for composition multiplicities for the Springer resolution

ABSTRACT

Let $\pi :\widetilde{\mathcal{𝒩}}\to \mathcal{𝒩}$ be the Springer resolution of the nilpotent cone for a semisimple connected algebraic group G over ℂ and k be an arbitrary field. What happens to ${\pi }_{\ast }{\underline{k}}_{{}_{\widetilde{\mathcal{𝒩}}}}[dim\widetilde{\mathcal{𝒩}}]$ if the decomposition theorem fails for it? We show that in this case, some additional (with respect to the case char k=0) composition factors of this direct image in the (abelian) category of perverse sheaves may emerge. These factors emerge from the $Z_G\left(x\right)∕Z_G{\left(x\right)}^0$-composition factors of the radicals of certain intersection forms and from that of the top cohomologies of Springer fibres (in the non-semisimple case).

KEYWORDS:

• Cohomology
• nilpotent cone
• perverse sheaves
• Springer resolution

MATHEMATICS SUBJECT CLASSIFICATION:

• 55N30 (Sheaf cohomology)

Notes

¹Use the shifted versions δ(a)⋅B and δ(a)⋅A of B and A, respectively.

²Use the shifted versions c⋅B and c⋅A of B and A, respectively.

³Part (1) corresponds to ${\phi }^{-1}(B{s}_1{s}_2{s}_{1}B∕B)\cap {\tilde{S}}_{x,0}^1$, part (2) to ${\phi }^{-1}(B{s}_2{s}_{1}B∕B)\cap {\tilde{S}}_{x,0}^1$ and part (3) to ${\phi }^{-1}(B∕B)\cap {\tilde{S}}_{x,0}^1$.