# About generalized Springer theory for spin groups

I am interested in the detailed computation of the generalized Springer theory for spin groups (type B or D). The last sentense in Section 14 of Lusztig's Intersection cohomology complex on a reductive group he wrote "It is likely that, if $(C,\mathcal{E})\in\mathcal{N}_{\chi}^{(0)}$ and $g\in C$, then the sizes of the Jordan cells $\beta(g)\in SO(V)$ give the partition $N=1+5+9+13+...$ or $N=3+7+11+15+...$."

In other words, the sizes of Jordan blocks of the unipotent orbits that support those new cuspidal local systems that live in $Spin_N$ but not $SO_N$ were predicted as above. Has this computation been completed somewhere? And if so what are the local systems attached to these orbits? Thanks!

• a more precise reference might help - there are many monumental papers by Lusztig... – Dima Pasechnik Feb 20 '17 at 17:11
• As Dima suggests, a citation would be helpful. Though Lusztig wrote many follow-up papers involving character sheaves, his formulation of the "generalized Springer correspondence" was probably first given in :ams.org/mathscinet-getitem?mr=732546 – Jim Humphreys Feb 20 '17 at 18:17
• Exactly the first paper. Thanks for the suggestion! – Cheng-Chiang Tsai Feb 20 '17 at 18:17
• P.S. Your question is reasonable, but if you edit it further please omit the first word, write "detailed" in place of "detail", change "lives" to "live", and change "system" to "systems". (And it wouldn't hurt to explain Lusztig's notational conventions briefly.) – Jim Humphreys Feb 21 '17 at 0:43
• Thanks! I rephrase the second paragraph a bit, which should now explain (what we need for) Lusztig's notation. – Cheng-Chiang Tsai Feb 21 '17 at 2:30