aa r X i v : . [ m a t h . R T ] J a n A NOTE ON HESSENBERG VARIETIES
KARI VILONEN AND TING XUE
We give a short proof based on Lusztig’s generalized Springer correspondence of someresults of [BrCh, BaCr, P].Let g denote a complex semisimple Lie algebra and let us write G “ Aut p g q for theidentity component of its group of automorphisms. Note that the group G is adjoint. Let uswrite B for the flag variety of g and let us fix a Borel subalgebra b Ă g . Let B Ă G be thecorresponding Borel subgroup. Consider a Hessenberg subspace V Ă g , i.e., a B -invariantsubspace of g . It gives rise to a G -equivariant bundle(1) H : “ G ˆ B V Ñ B . We have a G -equivariant projective morphism(2) π : H Ñ g , p g, x q ÞÑ Ad p g q x the “universal” family of Hessenberg varieties. We can also consider a perpendicular Hes-senberg subspace V K and we similarly get(3) ˇ π : ˇ H : “ G ˆ B V K Ñ g . We now assume that b Ă V so that V K Ă n “ r b , b s . By the decomposition theorem [BBD]we have(4) Rπ ˚ C H “ ‘ IC p g rs , L i qr´s ‘ terms with smaller support ;here the L i are irreducible G -equivariant local systems on the regular semisimple locus g rs and r´s denotes cohomological shifts. We also have(5) R ˇ π ˚ C ˇ H “ ‘ IC p O , E qr´s ;here the O are nilpotent orbits of G and the E are G -equivariant irreducible local systems onthe O . By considering the Fourier transform we conclude that all the IC’s appearing in (4)are Fourier transforms of the IC’s appearing in (5).We have the following Conjecture 1 (Patrick Brosnan [X]) . All the IC p O , E q in (5) appear in the Springer corre-spondence. In particular, all the terms in (4) have full support. This conjecture has been proved by Martha Precup and Eric Sommers (for type G seealso [X]). It is not difficult to prove the following weaker statement which suffices for someapplications. KV was supported in part by the ARC grants DP150103525 and DP180101445 and the Academy ofFinland.TX was supported in part by the ARC grant DP150103525.
Proposition 2.
We have p Rπ ˚ C H q| g reg “ ‘ IC p g rs , L i q| g reg r´s . Proof.
The terms in (4) which do not appear in the Springer correspondence come by in-duction from the (nontrivial) cuspidals in Lusztig’s generalized Springer correspondence [L].Recall that we are in the context of the adjoint group. One can now verify, case-by-caseusing [L], that the supports of such terms do not meet the regular locus g reg . (cid:3) The L i come from representations of the Weyl group W and we will now denote them assuch by ρ i . Let us consider IC p g rs , ρ i q restricted to the regular locus. It is a direct summandin ˜ p ˚ C ˜ g reg where ˜ p : ˜ g reg Ñ g reg is the Grothendieck simultaneous resolution restricted to theregular locus. Let t be a Cartan subspace of g . Then we have the following Cartesian square(6) ˜ g reg ˜ p ÝÝÝÑ g reg §§đ §§đ f t ÝÝÝÑ p t { W and so we have ˜ p ˚ C ˜ g reg “ f ˚ p ˚ C t . Thus, as f is smooth, it suffices to analyze p ˚ C t .Let us now consider a stratum O s in t { W associated to a semisimple element s P t . Notethat the fundamental group of O s is the braid group of s W “ N G p L q{ L where L “ Z G p s q .We also write W s “ Stab W p s q . By observing that N G p L q{ L “ p N G p L q X N G p T qq{ N L p T q wehave an exact sequence(7) 1 Ñ W s “ N L p T q{ T Ñ p N G p L q X N G p T qq{ T Ñ s W “ N G p L q{ L Ñ . The fiber of p over this stratum is W { W s . This implies:(8) IC p t { W, ρ i q| O s “ ρ W s i . The righthand side can be viewed as a representation of s W by (7) and hence as a localsystem on O s . Note that, of course, all the IC’s are just sheaves so in the whole analysistalking about IC’s is not really necessary.The considerations above allow us to conclude that(9) H ˚ p H x q “ H ˚ p H y q W s where x is a regular element with semisimple part s and y is a regular semisimple element.In particular, this covers some of the results of [BrCh, BaCr] as well as the palindromicityof the cohomology of regular Hessenberg varieties [P]. References [BaCr] Ana B˘alibanu and Peter Crooks, Perverse sheaves and the cohomology of regular Hessenberg vari-eties, arXiv:2004.07970.[BBD] A. A. Be˘ılinson, J. Bernstein and P. Deligne, Faisceaux pervers, in
Analysis and topology on singularspaces, I (Luminy, 1981) , 5–171, Ast´erisque, 100, Soc. Math. France, Paris.[BrCh] Patrick Brosnan and Timothy Chow, Unit interval orders and the dot action on the cohomology ofregular semisimple Hessenberg varieties, Adv. Math., 329:955–1001 (2018).
ESSENBERG 3 [L] George Lusztig, Intersection cohomology complexes on a reductive group. Invent. Math. 75 (1984),no.2, 205–272.[P] Martha Precup, The Betti numbers of regular Hessenberg varieties are palindromic, Transf. Groups,23(2):491–499 (2017).[X] Ke Xue, Affine pavings of Hessenberg ideal fibers, arXiv: 2007.08712.
School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia,also Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
Email address : [email protected], [email protected] School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia,also Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
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